Deformation and breakup of drops in simple shear flows

Citation for published version (APA):Bruijn, de, R. A. (1989). Deformation and breakup of drops in simple shear flows. Eindhoven: TechnischeUniversiteit Eindhoven. https://doi.org/10.6100/IR318702

DOI:10.6100/IR318702

Document status and date:Published: 01/01/1989

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DEFORMATION AND BREAKUP OF DROPS IN SIMPLE SHEAR FLOWS

R.A. de Bruijn

DEFORMATION AND BREAKUP OF DROPS IN SIMPLE SHEAR FLOWS

DEFORMATION AND BR.EAKUP OF DROPS '

IN SIMPLE SHEAR FLOWS

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan

de Technische Universiteit Eindhoven, op ge­

zag van de Ree tor Magnificus, prof. ir. M.

Tels, voor een commissie aangewezen door het

College van Dekanen in het openbaar te verde­

digen op vrijdag 3 november 1989 te 14.00 uur

door

ROBERT ANTONIE DE BRUIJN

geboren te Heerhugowaard

Dit proefschrift is goedgekeurd door de promotoren

prof. dr. A.K. Chesters

en

prof. dr. ir. L. van Wijngaarden

CONTENTS

1. INTRODUCTION

2. SCALING IAWS FOR THE FLOY OF EMULSIONS

2.1 INTRODUCTION

page

1

5

5

2.2 EMULSIONS WITH CONSTANT INTERFACIAL TENSION 5

2.3 EMULSIONS WITH SURFACTANT ADSORPTION 12

2.3.1 Fundamental equations

2.3.2 Slowly varying flows

2.3.3 Rapidly varying flows

2.4 EMULSIONS CONTAINING SOLID PARTICLES

2.4.l Position of the particles

2.4.2 Particles in the fluid phases

2.4.3 Particles at the interface

2.5 EMULSIONS OF NON-NEWTONIAN LIQUIDS

2.6 CONCLUSIONS

2.7 REFERENCES

2.8 LIST OF SYMBOLS

3. NEWTONIAN DROP BREAKUP IN QUASI STEADY SIMPLE SHEAR

FLOWS

3.1 INTRODUCTION

3.2 LITERATURE AND THEORY

3.2.1 Introduction

3.2.2 Experimental results

3.2.3 Slender body theories

3.2.4 Numerical techniques

3.3 FLOW IN THE COUETTE DEVICE

3.4 EXPERIMENTAL

3.4.1 Description of apparatus

3.4.2 Experimental procedure

3.4.3 Results

3.5 CONCLUSIONS

12

18

20

21

21

22

22

23

28

29

30

33

33

34

34

35

36

37

38

47

47

48

53

59

4.

5.

CONTENTS

3.6

3.7

REFERENCES

LIST OF SYMBOLS

NON-NEWTONIAN DROP BREAKUP IN QUASI STEADY SIMPLE SHEAR.

FLOW'

4.1 INTRODUCTION

4.2 NON-NEWTONIAN FLUID MECHANICS

4.3 LITERATURE REVIEW

4.3.1 Theoretical results

4.3.2 Experimental results

4.3.3 Conclusions

4.4 BREAKUP OF INELASTIC, SHEAR THINNING DROPS

4.4.1 Model liquids

4.4.2 Drop breakup experiments

4.4.3 Discussion

4.4.4 Conclusions

4.5 BREAKUP OF VISCOELASTIC DROPS

4.5.1 Model liquids

4.5.2 Drop breakup experiments

4.5.3 Discussion

4.5.4 Conclusions

4.6 REFERENCES

4.7 LIST OF SYMBOLS

DEFORMATION AND BREAKUP OF NEWTONIAN DROPLETS IN

TRANSIENT SIMPLE SHEAR FLOWS

5.1 INTRODUCTION

5.2 PROBLEM STATEMENT AND BOUNDARY INTEGRAL METHOD

5.3 NUMERICAL METHOD

5.3.1 Choice of mesh

5.3.2 Evaluation of surf ace variables

5.3.3 Evaluation of surf ace integrals

5.3.4 Redistribution of the mesh

5.3.5 Numerical stability and convergence

page

59

61

63

63

63

67

67

70

81

81

81

85

92

100

102

102

110

113

127

127

129

131

131

132

140

141

142

143

145

147

CONTENTS page

5.4 EXPERIMENTS 152

5.4.1 Descript ion of the Couette device 152

5.4.2 Transient flow in the Couette device 153

5.4.3 Experimental procedure 155

5.5 NUMERICAL CALCULATIONS 157

5.6 EXPERIMENTAL AND NUMERICAL RESULTS 158

5.6.1 Step profile response 158

5.6.2 Triangle profile response 168

5.6.3 Sine profile response 175

5.7 DISCUSSION 175

5.7.1 Step profile experiments 175

5.7.2 Triangle profile experiments 186

5.7.3 Sine profile experiments 188

5.8 CONCLUSIONS 189

5.9 REFERENCES 190

5.10 LIST OF SYMBOLS 193

6. NEWTONIAN DROP BREAKUP IN SIMPLE SHEAR FLOW:

THE TIPSTREAMING PHENOMENON 147

6.1 INTRODUCTION 147

6.2 LITERATURE 198

6.3 EXPERIMENT AL 201

6.3.1 Introduction 201

6.3.2 Viscosity ratio 203

6.3.3 Time dependency 203

6.3.4 Acceleration 207

6.3.5 Surfactants 207

6.4 NUMERI CAL 214

6.5 DISCUSSION 218

6.6 CONCLUSIONS 225

6.7 REFERENCES 225

6.8 LIST OF SYMBOLS 226

CONTENTS page

SUMMARY 229

SAMENVATTING 233

ACKNOWLEDGEMENTS 237

CURRICULUM VITAE 239

APPENDIX A VISGOEIASTIC DROP BREAKUP IN LITERATURE 241

APPENDIX B DROP BREAKUP EXPERIMENTS 255

APPENDIX G EVALUATION OF SINGUIARITIES IN BOUNDARY

INTEGRAL METHOD 269

1. INTRODUCTION

This thesis is part of a long term investigation aimed at modelling

the operation of emulsifying devices in the food industry. In

these devices two imrniscible liquids are mixed to obtain a

distribution of droplets of one of the liquids in the other. Many

everyday products are on a micro-scale dispersions of one fluid in

another. Examples are margarine, an emulsion of water droplets in a

partly crystallised oil, mayonnaise, an emulsion of oil droplets in

a water phase and ice cream, which is a dispersion of gas bubbles

in a partly crystallised water phase. For all of these products the

averaged size and the size distribution of these droplets and

bubbles can affect many product properties. Hence there is

considerable interest in the modelling and optimising of these

emulsification processes. The results described in this thesis are

not restricted to the food industry. They can also be applied to

other emulsification processes e.g. polymer blending and oil

recovery, and even to flow phenomena that involve deformable

particles e.g. red blood cells.

The obvious approach to model the operation of emulsifying devices

is, on the one hand modelling flow and mixing in the devices

concerned and on the other hand drop break-up and coalescence as

local processes. Drop break-up and coalescence can be considered as

local processes when these phenomena are governed by the flow and

material properties in the immediate surroundings of the drops

involved. This is generally the case in emulsification processes.

When both approaches are integrated a full mathematica! model of

the emulsifying operation will result. The advantages of such an

approach are twofold. First the local processes need only be

modelled in a few well defined types of flow (e.g. laminar simple

shear flow, laminar extensional flow and turbulent flows) and the

results are thus applicable to a range of emulsifying devices

operating under the same types of flow. Second the modelling of the

flow and mixing in a particular device can be used for the

modelling of other physical processes as well (e.g. heat transfer

2

and crystallisation). This modular approach will only be successful

when both the flow and mixing in the emulsifying device and the

local drop break-up and coalescence processes can be modelled

accurately. With the take off of computational fluid dynamics

accurate modelling of flow and mixing in process units has recently

become possible. In the knowledge of the drop break-up and

coalescence processes, however, a number of very important aspects

are still not understood.

The work in this thesis aims to descrlbe some aspects of the

viscous break-up of droplets in slmple shear flows. Viscous break·

up is valid when the drop deformation and break-up is induced by

viscous shear stresses and inertial effects can be neglected. This

situation generally applies to the break-up of small droplets in

highly viscous llquids. Three of those aspects in the viscous

break-up of droplets in simple shear flows have been identif ied

which are poorly understood and are very important for the

modelling of emulsification processes. First the break-up of non­

Newtonian drop phases. This is important since many food products

exhibit a markedly non Newtonian behaviour. The available data on

non-Newtonian drop break-up are however scarce and inconsistent.

Second the break-up of droplets in transient shear flows is not yet

well described. This behaviour must however be known if

emulsification processes are to be described in which droplets

experience changes in the flow conditions during time scales in

which the droplet can deform and break-up. Third the origin of the

tipstreaming phenomenon in simple shear flow is not clear, nor is

it known when this mode of break-up will occur. This knowledge is

however important since tipstreaming will result in a greatly

different drop size distribution from the normal mode of drop

break-up. The work described in this thesis is an attempt to

understand and describe these aspects of viscous drop break-up in

simple shear flow.

This thesis starts with an introductory chapter on basic equations

and sealing laws for the flow of emulsions (chapter 2). These

should be known for both theoretical and experimental studies in

this area. The following situations are considered: emulsions with

constant interfacial tension, emulsions with surfactant adsorption,

emulsions containing solid particles and emulsion of non-Newtonian

liquids. For each of these systems the practical limitations of

application of sealing laws to the local processes of drop break-up

and coalescence are considered. Next a Couette device, which was

developed for experimental investigation of drop break-up in simple

shear flow, is described (chapter 3). The Couette device operates

on the principle of two counter rotating cylinders. A stagnant

layer is thus created in the gap between the two cylinders which

permits statie observations of droplets in quasi steady and

transient simple shear flows. This device has been successfully

tested by measurement of Newtonian drop break-up in quasi steady

simple shear flow. In the fourth chapter of this thesis the Couette

device was used to study experimentally the break-up of non­

Newtonian droplets in quasi steady simple shear flows. Two

particular types of non-Newtonian behaviour were studied: shear

thinning liquids with viscosities obeying the power law equation

but with negligible fluid elasticity and viscoelastic liquids with

substantial elasticity combined with a shear rate independent

viscosity. These particular types of non-Newtonian drop phases were

chosen in order to separate the effects of shear rate dependent

viscosities and fluid elasticity. For most liquids these effects

occur simultaneously, which makes it very difficult to separate

these effects. In chapter 5 of this thesis the deformation and

break-up of Newtonian droplets in transient simple shear-~s is

studied. This investigation is mainly of numerical nature but is

supported by experimental work. A computer programme is developed

which calculates the evolution of Newtonian droplets in any

transient shear flow. The programme is based on the boundary

integral method by which the creeping flow equations inside and

outside the droplet are transformed into a form that only involves

quantities at the drop interface. This method is based on the

Fourier solution to the creeping flow equations and uses volume

potential theory. This programme is used to calculate the shape of

3

4

droplets with various viscosity ratios as a function of time for

various shear rate profiles: step profiles, triangular profiles and

sinusoidal profiles. The work was supported by drop deformation

triangular profiles of the shear rate. In the final chapter of this

thesis a special mode of drop break-up, tipstreaming, is

investigated. Tipstreaming is an experimentally observed mode of

drop break-up in which the droplet takes, upon increasing the shear

rate a sigmoidal shape and a stream of very small droplets is

ruptured off the tips of the drop. The investigations in this

chapter are aimed at unravelling the causes of this phenomenon and

use both experimental and numerical techniques.

2. SCALING LAWS FOR THE FLOW OF EMULSIONS

2.1 INTRODUCTION

For both theoretical and experimental study of emulsification

processes sealing laws for multiphase flow should be known. On the

theoretica! front these laws indicate the number and the nature of

the dimensionless groups concerned, while on the experimental front

they indicate the feasability of conducting observations at

practically convenient time and space scales. In this introductory

chapter the sealing laws are derived from the fundamental equations

governing liquid flow for the following multiphase systems:

emulsions with constant interfacial tension

emulsions with surfactant adsorption

- emulsions containing solid particles

- emulsions of non-Newtonian liquids.

For each system the practical limitations of application of sealing

laws to the sealing of emulsion break-up and coalescence are

considered.

2.2 EMULSIONS WITH CONSTANT INTERFACIAL TENSION

In this section the sealing laws will be derived for emulsions with

constant interfacial tension in which colloidal effects may be

neglected. Although the results obtained are not new, see for

example Chesters (1975), the derivation is illustrative of the

approach required, which will be extended to more complex

situations (sections 2.3 and 2.4).

The equations governing isothermal fluid flow in emulsions are the

equations of motion in both phases and the kinematic and stress

boundary conditions at the droplet interface. For isothermal,

incompressible Newtonian fluids the equations of motion are given

by the Navier-Stokes equation and the continuity equation.

5

6

The first is given by:

Dy p­

Dt

811 p - + p (y • V) l.! = - v p + '1 v2 y + Eext

8t [2-1)

in which Eext denotes the external body forces per unit volume. In

what follows the only external body force considered is that due to

gravity, Egrav = pg, but the approach can be extended, if relevant,

to other body forces such as the fictitious ones, (centrifugal,

Coriolis) in systems viewed with respect to rotating axes.

The latter is given by

V.J.!= 0 [2-2)

The kinematic interface condition, neglecting interfacial mass

transfer is:

Yc = l.!d = Yinterface [2-3]

yielding the kinematic condition for the displacement of a material

point in the interface in the absence of interphase mass transfer

D~interf ace

~~~~~- = Yinterface = Yc [2-4]

Dt

From the assumptions of a massless interface and a constant

interfacial tension the stress equilibrium equations reduce to a

balance between interfacial tension forces and viscous traction

exerted by the fluid phasès. The equilibrium of the tangential

stresses is given by

!.t,d = Q [2-5]

where !.t,c and !.t,d are the tangential stresses exerted by/on the

continuous and dispersed phase respectively. The equilibrium of the

normal stresses is given by

[2-6]

where Tn,c and rn,d are the normal stresses exerted by/on the

continuous and dispersed phase respectively. The fluid stresses L can be calculated from the flow pattern and the constitutive

equation. For an incompressible Newtonian liquid the constitutive

equation is given by

in which Óij is the Kronecker delta defined by Óij

ó ij = 0 ( i;o<j ) .

[2-7]

1 {i=j) and

Similarity criteria can be derived by making the equations

dimensionless. All variables can be made dimensionless by a

combination of U, Land p, where U and L denote a characteristic

velocity and length scale of the system.

u u=-

u u L

. t, p

L'il, p = pu2' 1 Ri x

T = ' x L L [2-8)

In absence of external forces other than gravity this leads to the

following dimensionless equations.

7

8

In each fluid:

aü -= + <li • v) 11 at

(g : unit vector in direction of g)

v • 11 0

At the interface:

À Lt,c · Lt,d Q

À rn,c

in which the following dimensionless groups appear.

Re - p U L q·l (Reynolds number)

Fr u2 g·l L·l (Froude number)

We p u2 L ~-1 (Weber number)

À 11d 11·1 (viscosity ratio)

[2-9)

[2-10]

[2-11)

[2-12)

[2-13]

[2-14]

Re is a measure of the relative importance of inertial to viscous

forces, Fr of the relative importance of inertial to gravitational

forces and We of the relative importance of inertial to interfacial

forces. Since equations [2-9) to [2-11] apply to both phases they

yield the dimensionless coefficients

(where no subscripts are used, the dimensionless parameters refer

to the continuous phase). The complete set of dimensionless

parameters describing the flow in and around a single emulsion

droplet is thus

Re, Red, Fr, Frd, À and We

This set can be reduced and modified into a more f requently used

set (since every parameter can be replaced by any combination of

itself with other parameters):

Re, Fr, We, À and~

in which

~ - Pd p-l (density ratio)

The advantage of the dimensionless description follows from the

fact that systems characterized by the same value of the

dimensionless parameters and having the same dimensionless initial

and boundary conditions will behave exactly the same, apart from a

sealing factor. In the case of time dependent flows these boundary

conditions should also include the externally applied time scales.

These are often described by the dimensionless Strouhal number:

St - L/UT. For any practical situation it is almost impossible to

satisfy all five similarity criteria simultaneously. This becomes

clear if we replace Re, Fr and We by the following three

independent dimensionless coefficients:

We Fr

9

10

M, the Morton number, is a very special dimensionless parameter

since, apart from the gravitational constant, it consists of fluid

properties only. Sealing at constant gravity is thus only possible

for liquids having the same value of M. From Table 2.1 it will be

clear that only a few liquids can be scaled completely. However it

is often possible to relax one or more of the similarity criteria

without significant effect on the flow properties. One of these

cases will be discussed briefly.

Many emulsions of practical interest consist of very small

droplets. As a consequence the Reynolds and the inverse of the

Froude number based on the droplets are very small, indicating that

neither the gravitational forces nor the inertial forces affect the

flow in and around a droplet. For such flows the exact values of Fr

and P are irrelevant. Then the only relevant dimensionless

coefficients are those combinations which do not contain p or g,

namely n and À, where

u ~ n = We Re-1 = ~­o

(Capillary number)

Under these conditiQns it is often possible to find model liquids

which allow sealing. Generally this can be done by maintaining À

but increasing viscosities and decreasing the velocities and

keeping o approximately constant. Using these sealing criteria a

large increase in length and time scales can be obtained which

permits visualisation of the phenomena such as break-up which are

generally very difficult to monitor in practical emulsions due to

the very small time and length scales involved. For emulsions with

constant interfacial tension silicone oils are a very flexible

model liquid, since a large range of viscosities are available

commercially (see Table 2.1).

TABLE 2.1 Fluid properties

Fluid M [-J

Water 1. 0 103 1.0 10-3 7. 3 10-2 2.5 10-11

Glycerol 1.3 103 1. 8 6.3 10-2 3 .2 102

Corn. syrup (25% water) 1.3 103 1.2 10-1 2.9 10-2 6.4 10-2

Corn. syrup (10% water) 1. 4 103 2.8 3.5 10-2 1. 0 104

Ethanol 7 ,9 102 1.2 10-3 2,3 10-2 2.1 10-9

Mercury 1.4 104 1.6 10-3 4. 7 10-1 4.4 10-1•

Sunflower Oil 9.2 102 6.0 10-2 3.4 10-2 3.5 l0-3

Castor 011 9. 7 102 7. 7 10-l 3.6 10-2 7. 6 101

Silicone Oil 47 v1*) 8. l 102 1.0 l0-3 1.7 10-2 2.5 10-g

Silicone Oil 47 VlOOO*) 9.7 102 1. 0 2.1 10-2 1.1 103

Silicone Oil 47 v100.ooo*) 9. 7 102 l.O 102 2.1 io-2 1.1 1011

*) ex Rhone Poulenc

11

12

The sealing criteria derived above are only valid when colloidal

effects can be neglected. This is more restrictive than it at first

appears: in any rupture process, Van der Waals forces furnish the

final destabilization. This applies to the rupture of a filament of

liquid connecting a splitting drop and also to the rupture of a

thin film of liquid separating two coalescing drops. In the farmer

case the exact level of the Van der Waals farces will have

negligible influence on the rupture time since the final pinching

phase is so fast. In the case of coalescence, however, the last

stage of film thinning is typically the slowest and the magnitude

of the Van der Waals forces will then be of major influence on the

process. Consequently, while the preceding relations can be applied

with confidence to the sealing of drop break-up processes, this

will not in genera! be the case for coalescence processes.

2.3 EMULSIONS WITH SURFACTANT ADSORPTION

2.3.l Fundamental equations

The interfacial tension of an emulsion droplet may usually not be

regarded as constant when surfactants are present. This is

especially so when the fluids are in motion and the emulsion

droplets are being deformed. Droplet deforrnation leads to

enlargernent of the interfacial area thus tending to change

concentration of the adsorbed surfactant. Surfactant adsorption

will compensate such changes in concentration hut can be a much

slower process than droplet deforrnation, thus leading to non­

equilibrium surfactant concentrations at the interface and to

surface tension gradients.

Such gradients rnodify the stress boundary conditions at the droplet

interface (equations (2-5] and [2-6]). The equations describing

fluid flow in these ernulsions are thus not only the equations of

motion in both phases (equation [2-1] and [2-2]) and the kinematic

and stress equilibrium conditions at the droplet interfaces (the

modified verslons of equation [2-3), [2-5) and [2-6]). When

surfactants are present one should also take into account the

surfactant mass conservation relations and the interfacial

equations of state. If it is asswned that the surfactant is soluble

in the continuous phase only, the mass conservation relations at

the interface and in the continuous phase should be valid. For the

continuous phase the dif fusion equation in a flowing material is

given by

De

[2-15]

Dt

in which c, denotes the surfactant concentration in moles

surfactant per m3 solvent in the bulk and V the surfactant

diffusion coefficient and use has been made of the diffusion law of

Fick:

diffusional mass flux - ID 'ï/c

The surfactant mass conservation relation at the interface is given

by

Dr l DA 8c

- r +ID [2-16]

Dt A Dt 8n

in which

r: surface concentration of surfactants [moles m-21

A: area of interfacial element.

The first right hand term in equation [2-16) denotes the change in

surface concentration due to interfacial dilation, while the second

term denotes the change due to mass transport between bulk and

interface. The bulk and surface mass conservation relations [2-15]

13

14

and [2-16] are connected by the condition that the surface

concentration is in instantaneous equilibrium with the bulk

concentration immediately adjacent to the interface (cs>· For low

surface concentration this relation can often be described by the

Langmuir adsorption isotherm:

r [2-17]

in which rw and c1 are material constants.

This relation is only valid when lateral interactions between the

adsorbed molecules can be neglected.

For rather dilute solutions of simple surfactants (r << r00 ) this

equation reduces to

r" r [2-18]

This equation is known as the two dimensional Henry equation.

The equations of state of the interface relate the surface

concentration of surfactant to its effect on the interfacial

tension. When the adsorption is given by the Langmuir equation [2-

17], the surface equation of state is often given by the Frumkin

equation which is derivable from the Langmuir adsorption equation.

[2-19]

r which for low adsorption ~ << 1 reduces to

roo

r a - RT r~ RT r [2-20]

which is the ideal surface equation of state. In the equations

above, a0 is the interfacial tension in the absence of adsorbed

surfactant. These equations lose their validity if the adsorbed

surfactant molecules start interacting. The validity range of these

equations is thus for proteins and ether macromolecules rather

small. For small surfactant molecules the validity range is usually

larger. For such molecules the error in the Frumkin equation is

often smaller than 15% as long as r;r00 < 0.25. However, the

decrease in interfacial tension is usually still very small in that

range. Typically:

- a < 5 m Nm-1.

The scope for application of sealing criteria is thus rather

limited due to the small range of conditions under which surfactant

behaviour can be modelled easily.

If the interfacial tension is not constant the tangential stress

equilibrium equation [2-5] should be modified to include

interfacial tension gradients:

Lt,c - Lt,d - Vs a [2-21]

These gradients can effectively irnrnobilise the droplet interface

and are responsible for the increased drag coefficient observed for

bubbles in the presence of surface active materials.

15

16

The basic equations governing the flow of emulsions with adsorbed

surfactants have now been given.

In order to derive similarity criteria, these should be made

dimensionless using equation [2-8] and additionally

ë c r n a - r =-

' n= and ä

Co r"' L ao [2-22]

Thus yielding:

aü <i! V) - -v P + Re·l v2 Fr·l -+ . ,!! ,!! + g

at [2-23]

(2-24]

ij • u 0 (2-25]

-Uc ud = Uïnterf ace [2-26)

À Ît,c - it,d = 0 Vs ä (2-27]

[2-28]

Dë [2-29]

Bt

Dr Dt

(2-30]

or r [2-31]

a = 1 + Nsp ln (1-r) or a = l · Nsp r [2-32]

Complete similarity will only be obtained if all dimensionless

parameters are equal and if the same dimensionless boundary and

initial eonditions apply. Besides the dimensionless parameters

eneountered in the previous seetion, viz. Re, Fr, n, p and À some

new parameters appear whieh are defined by

Pe U L

ID

(or equivalently

roo No

L.e00

NA e1

eoo

Ngp -RT r00

ao

(Peelet-number)

v Se - sinee Pe - Re• Se).

ID

(Distribution - number)

(Adsorption - number)

(Surfaee pressure - number)

Pe deseribes the relative importanee of eonveetive surfaetant

transport to diffusive transport.

No deseribes the distribution of surfactant over interface and

bulk.

NA is a parameter indieating the proximity to saturated

adsorption.

17

18

Nsp• finally, describes the lowering of the interfacial tension

(the so-called surface pressure) relative to the interfacial

tension of a clean interface, as a function of surfactant

adsorption.

In what fellows the sirnplification which was introduced in the

previous section will also be assumed to be valid, namely that

Re << 1 and Fr << 1. It was already discussed that for the flow of

emulsions these simplifications are usually allowed. The set of

dimensionless groups thus reduces to:

Full sealing of the flow of emulsions with surfactant adsorption is

very difficult. The scope for sealing is limited by the difficulty

involved in keeping both Q and Pe or equivalently Se constant

during scale-up. As was shown in the previous section upscaling

with Q constant can be done by increasing viscosities. To keep

Se v/ID constant ID should thus be increased by the same factor

as v, but unfortunately higher viscosities tend to decrease ID. It

is thus very difficult to scale the surfactant adsorption process

completely.

In the following some special cases will be considered which allow

all relevant similarity criteria to be fulfilled.

2.3.2 Slowly varying flows

The case of slowly varying flows applies when diffusion processes

dominate over convection processes: Pe << 1. In this situation the

surfactant concentration will at all times be uniform throughout

the bulk, including the region adjacent to the interface. Thus r

and a will be constant at the drop interface, albeit with a

reduced interfacial tension.

'When overall surfactant gradients are maintained the conditions Pe

<< 1 is very restrictive. It implies that

U L 'Y

Pe = --~ << 1

ID ID

in which the characteristic length scale L represents a global

length scale. In this situation the flow velocities need to be so

low that the droplet will hardly be deformed. This case is thus

hardly relevant to emulsification processes.

However when no overall surfactant gradients are maintained,

surfactant molecules only need to diffuse in a small layer around

the droplet to ensure uniform surfactant concentration. In this

situation the characteristic length scale can be given by the

layer thickness L that contains as much surfactant as is needed to

cover the interface with an equilibrium adsorption:

L r/c

The above relation implicitely assumes that the layer thickness is

small compared to the drop size. It will be seen that this

condition is usually satisfied. r and c are related via the

adsorption isotherm. In the dilute case, for example, [2-18]

yields

The value of c1 varies considerably from one surfactant to

another, but is typically such that L is of the order of microns

or less, which is often smaller than the drop size. At large

surfactant concentrations r does not increase linearly with c

anymore and consequently L becomes even smaller. For example,

consider a surfactant at high concentration, such that

19

20

where N denotes the Avogadro number and Am the area occupied by

one adsorbed surfactant molecule. The latter is generally of the

order of 50 ft2, yielding r~ - 3 • 10-6. For a bulk concentration

of 30 mole/m3 the diffusive length scala becomes 0.1 µm. Fora

typical diffusion coefficient of ID - io-10 m2s-l this implies that

the flow can be considered as 'slowly varying' for shear rates up

to io-4 s-1. Clearly there are many practical situations that

satisfy these conditions of slowly varying flows.

2.3.3 Rapidly varying flows

Flows will be termed rapidly varying when convective mass

transport dominates diffusion (Pe >> 1). In this case mass

transport between bulk and interface will hardly occur. Variations

in surface concentration and interfacial tension thus will only be

due to changes in the interfacial area. The exact values of Pe is

thus not important as long as Pe >> 1. If in addition the

surfactant adsorption can be described by a linear adsorption

isotherm, the values of NA and Nn are not important either. The

set of relevant dimensionless groups is therefore reduced to: n, À

and Ngp.

From the example described in the case of slowly varying flows

(section 2.3.2) the applicability of Pe >> 1 is restricted to

concentrated solutions and extremely high shear rates (~ >> lo-4).

For low surfactant concentrations combined with high shear rates

one can however often apply this simplification successfully.

2.4 EMULSIONS CONTAINING SOLID PARTICLES

In many emulsions solid particles are present in the bulk phase or

at the interface as fat crystals in fat spreads and creams.

In this section sealing laws will be derived for these systems to

be able to study the effects of these particles on droplet break-up

and coalescence processes.

2.4.1 ~Qsition of the particles

The equilibrium particle position is determined by minimal total

free energy. However, minimizing the interfacial free energy Fs

also gives correct equations. Fs is defined by

[2-33]

where A denotes the interfacial area, u the interfacial tension and

the subscripts p, c and d refer to particle, continuous and droplet

phase respectively. The particle position proves to be determined

by the adhesion constant C.

c [2-34]

Four ranges of equilibrium positions and corresponding C-values can

be discerned.

c < -1 particles are completely wetted by the continuous

phase.

-1 < C < 0: particles are positioned at the interface, largely

wetted by the continuous phase.

21

22

0 < c < 1

G > 1

particles are positioned at the interface, largely

wetted by the droplet phase.

particles are completely wetted by the droplet phase.

For !Cl < 1, the adhesion constant is equal to the eosine of the

contact angle, as follows from the Young's equation

Ucd • COS 0 [2-35)

The contact angle is the angle between the cd and pd interfaces,

measured in the liquid with the higher density.

2.4.2 Particles in fluid phases

Yhen the particles are completely wetted by one of the fluid phases

the additional sealing requirements can usually be fulfilled.

It will be obvious that scaled-up particles should have the same

shape, density, relativa size and concentration and should be

wetted by the same fluid phase as the original particles.

As we have seen both gravitational and inertial forces are often

negligible for emulsions. However, care should be taken that this

still holds for scaled-up emulsions containing solid particles.

Even if the gravitational and. inertial forces are negligible for

the original emulsion it is thus better to use neutrally buoyant

particles.

2.4.3 Particles at the interface

The contact angle determines, as has been argued, the position of

the particles in the interface. Yhen sealing these emulsions one

should not only fulfill the requirements described in the previous

section, but also keep the contact angle constant. Since

macroscopie particles can sometimes be coated to give the required

surface properties, this can probably be satisfied. However, when

sealing dynamic processes, particles may move relative to the

interface. Thus, not only statie contact angles, but also dynamic

angles may occur.

Indications are that differences between dynamic and statie contact

angles scale with the capillary number (E.B. Dussan V 1979, Hoffman

1975) and so to the first approximation will be covered by the

foregoing sealing requirements.

A certain additional influence of system scale is however to be

expected, which will exclude vigorous sealing (De Gennes, 1985).

Serious sealing difficulties may be expected if the process of

entry of particles into the interface from one of the liquids is to

be scaled, since this again involves the rupture of a thin film of

liquid (see section 2.2).

2.5 EMULSIONS OF NON-NEWTONIAN LIQUIDS

In this chapter we will consider the sealing laws for emulsions in

which one or bath of the phases are non-Newtonian. Sealing of such

emulsions is many times more difficult than for emulsions

consisting of Newtonian liquids. The description of the flow of

non-Newtonian liquids involves a modified equation of motion in

which generally a rather complicated constitutive equation is

inserted. The modified equation of motion is given by;

Dy

P--~ - VP+ V.z. +Eext

Dt

[2-36]

23

24

The extra stress tensor L is related to the flow field by the

constitutive equation. For a Newtonian liquid this constitutive

equation is given in equation [2-1) and can be characterised by a

single fluid property, the dynamic viscosity. For non·Newtonian

liquids one generally needs more than one fluid property to

characterise the constitutive equation. The sealing of such

liquids therefore becomes increasingly difficult. In this chapter

we will only consider a certain type of non-Newtonian behaviour,

which includes on the one hand purely viscous behaviour with a

shear rate dependent viscosity, described by the power law model,

and on the ether hand viscoelastic behaviour. This behaviour can be

modelled by the so·called Criminale-Erickson-Filbey (CEF)

constitutive equation (Schowalter (1978)). This constitutive

equation is a simplification of the retarded motion expansion,

valid for steady flows. The CEF constitutive equation is in general

given by:

q(,Y) N1(Ï') N2()') N1 ( )') 0

L =--Q + 2 ('1-- + --) !2 !l - --!l [2-37)

'Î .y2 -y2 72

In this equation !l is the rate of strain tensor defined by:

[2-38)

.Y is the magnitude of the rate of strain tensor, defined by:

-Y = 2 J<Il:Q) [2-39]

fi is the corotational time derivative of the rate of strain tensor

and is defined by:

[2-40]

In this equation E is the rate of spin tensor defined by:

[2-41]

In equation [2-37] r1, N1 and N2 are the viscometric material

functions. These functions can be obtained from rheometrical

experiments in simple shear flows. In such a flow r is the

tangential stress measured, N1 is the first normal stress

difference and N2 is the second normal stress difference. In this

chapter we will only consider the following forms of the

viscometric material functions:

[2-4la]

[2-4lb]

[2-4lc]

This form of these functions was chosen to include the well known

power law liquid for which ~-0. The power law liquid exhibits a

shear rate dependent apparent viscosity, but does not show any

normal stress differences. The functions also include the so-called

Boger type liquids which exhibit an apparent viscosity which is

almost shear rate independent and show a first normal stress

difference.

To obtain the sealing rules for a liquid obeying the CEF

consitutive equation with the above defined viscometric material

functions equation [2-37] should be inserted in equation [3-36] and

the resulting equation should be made dirnensionless. All variables

can be made dimensionless by a combination of U, Land p (see also

equation [2-7]).

u

L .Y--i'

u

p

P u2

-t u --t L

L -~ u

LV

- L W=-W - u-

[2-59]

25

26

In absence of external forces other than gravity this leads to the

following dimensionless equation:

aü ~ + <ii • v)j! at

- - l 7n-l 1 -m-2 ° -V P + - '1 R + -- ('7,;, (2,R·,R - ,R)

Re* Re** '

[2 51]

In this equation Re* and Re** are modified Reynolds numbers

describing the ratio of the viscous stresses to inertial stresses

and the ratio of elastic stresses to inertial stresses

respectively. These modified Reynolds numbers are defined by:

Re* -K (U/L)n

[2-52] Pd u2

Re** -ic(U/L)m

[2-53] Pd u2

If we only consider the droplet phase to be non-Newtonian, the

other dimensionless equations are given by the continuity equation

and the kinematic and stress equilibrium conditions at the droplet

interface:

v . .l,i o

!!interface

- * - - À** - - 0 2 Rt,c - 2 À Rt,d <R·R - R>t,d

2 ~n,c - 2 À* ~n,d

[2-54)

Q [2-55)

We Re·l (R1-l + R2-l)

[2-56]

In these equations the dimensionless parameters À* and À** denote

the modified viscosity ratios, describing the ratio of the viscous

forces in the drop to the viscous forces in the continuous phase

and the ratio of elastic forces in the drop to the viscous forces

in the continuous phase respectively. These dimensionless

parameters are defined by:

K (U/L)n-1

~

** ~ (U/L)m-1 À =

The following dimensionless parameters have thus appeared in the

above equations:

Re, Re*, Re**, Fr, À*, À**, We, m, n.

This set of parameters can be modified and reduced into a more

generally used set of dimensionless parameters (since every

parameter can be replaced by any combination of itself with other

parameters):

Re, Fr, We, SR, ~. À*, mand n.

As compared with the situation described in section 2.2 for the

sealing of Newtonian liquids, there are 3 extra dimensionless

parameters viz. n, m and SR, while instead of the viscosity ratio À

a modified viscosity ratio À* should be used. This very much

restricts the possibilities for sealing. Many emulsions of

practical interest consist of very small droplets, thus allowing

neglect of the inertial and gravity forces. In this situation the

set of dimensionless parameters reduce to:

Sealing of emulsions containing inelastic shear thinning droplets

is only possible (since SR m = 0) if liquids are used, which have

the same value for the power law index n. If such liquids are found

sealing will still be less straightforward than for emulsions

consisting of Newtonian liquids since the modified viscosity ratio

À* is not constant (as is the viscosity ratio À) but also depends

on the flow conditions.

Although fully inelastic shear thinning liquids do not exist, there

are several shear thinning liquids that are almost inelastic. 27

28

Examples are e.g. solutions of Carboxy Methyl Cellulose (CMC),

certain solutions of the poly acrylic acid Carbopol in water/corn

syrup mixtures pnd dispersions of the silica particles Aerosil in

mineral oils. These model liquids can be made with power law

indices varying in the range 1-0.3.

Sealing of emulsions containing visco-elastic droplets obeying the

CEF constitutive equation is only possible when liquids are used

which have the same values for the indices m and n. In general this

is almost impossible. However a special class of model liquids bas

been developed that exhibit an almost shear rate independent

viscosity (n = 1) combined with elastic effects described by m = 1.5 - 2. These so-called Boger liquids are made of solutions of

poly-acryl-amide in water/corn syrup mixtures (Boger 1977/1978,

Boger and Nguijen 1978 and De Bruijn chapter 4 of this thesis.

2.6 CONCLUSIONS

Sealing laws for emulsion break-up and coalescence processes have

been derived for:

- emulsions with constant interfacial tension

- emulsions with surfactant adsorption

- emulsions containing solid particles

- emulsions of non-Newtonian liquids.

Sealing laws can be applied without great difficulties to

emulsions with constant interfacial tension.

Sealing of emulsions with surfactant adsorption is usually only

possible in "slowly" varying flows in which surfactant diffusion

dominates convection resulting in an equilibrium surface tension

all over the droplet, or in dilute "rapidly" varying flows in which

adsorption and desorption processes may be neglected.

Sealing of emulsions of non-Newtonian liquids is usually only

possible when a homologous series of liquids is available with the

same eonstitutive equation (e.g. shear thinning liquids and Boger

liquids).

Sealing of emulsions eontaining solid particles will beeome

diffieult if the proeess of entry of partieles into the interface

from one of the liquids is to be sealed.

2.7 REFERENCES

1. D.V. Boger, A highly elastie eonstant-viscosity fluid, J.

Non-Newtonian Fluid Meeh. 3, 87-91, (1977/1978).

2. D.V. Boger and H. Nguyen, A model viseoelastie fluid, Polym.

Engng. Sei. 18, 1037-1043, (1978).

3. R.A. de Bruijn, Newtonian drop break-up in quasi steady

simple shear flow, chapter 3 of this thesis.

4. R.A. de Bruijn, Non-Newtonian drop break-up in quasi steady

simple shear flow, chapter 4 of this thesis.

5. A.K. Chesters, The applicability of dynamic similarity

criteria to the isothermal, liquid-gas two phase flows

without mass transfer, Int. J. Multiphase Flow 2, 191-212,

(1975).

6. E.B. Dussan V., On the spreading of liquids on solid

surfaces: statie and dynamic contact lines, Ann. Rev. Fluid

Mech. 11, 371-400, (1979).

7. P.G. de Gennes, Wetting: staties and dynamics, Rev. Modern

Physics 57,827-863, (1985).

8. R. Hoffman, A study of the advaneing interface. 1. Interface

shape in liquid gas systems, J. Colloid Interface Sci. 50,

228-241, (1975).

9. G.G. Ngan, and E.B. Dussan V., On the nature of the dynamic

contact angle: an experimental study, J. Fluid Mech. 118,

27-40, (1982).

10. W.R. Schowalter, Mechanics of non-Newtonian fluids, Pergamom

Press, Oxford, 1978.

29

30

2.8 LIST OF SYMBOLS

interfacial area

adhesion constant

material property in Langmuir

adsoprtion isotherm

c surfactant concentration

~ rate of strain tensor

ID diffision coefficient

E body force per unit volume

Fs interfacial free energy

Fr Froude number

g gravitational acceleration

K consistency index

k Boltzmann constant= 1.3806 io-23

L length scale

M Morton number

m elastic power law index

NA Adsorption number

No Distribution number

Nsp Surface pressure number

N1 first normal stress difference

N2 second normal stress dif ference

n power law index

n normal vector

P pressure

Pe Peclet number

R Gas constant = 8.3143

Ri radius of curvature

Re Reynolds number

SR stress ratio

St Strouhal number

T temperature

t time

U characteristic velocity

!.! velocity

[moles m·3]

[moles m·3]

[s" 1 J

[m2 s·l1

[N m·3]

[Nm]

[ l [m

[Pa.sn]

[JK-1]

[m]

[ J [. l [ ·]

[ - J

[ - l [Pa]

[Pa]

[. l

[ - l [Pa]

[ -1 [JK·l mo1·l1

[m]

[ -1 [ - l [ - 1 [KJ

[s]

[m s· 11 [m s·l1

LIST OF SYMBOLS (Continued)

H rate of spin tensor

We Weber number

x position

fi density ratio

r surface concentration of

surf actants

r 00 material constant in langmuir

adsorption isotherm

~ rate of shear

Óij Kronecker delta

q dynamic viscosity

U contact angle

K elasticity index

À viscosity ratio

p density

a interfacial tension

L stress tensor

TI tangential stress function

0 Capillary number

SUB SCRIPTS

c continuous phase

d disperse phase

ext external

grav gravitational

n normal

0 original

p particle

s surf ace

t tangential

[ l [ - l [rn)

[ - J

[moles rn- 2 J

[rnoles rn" 2 J

[s-1]

[ - l [Pa.s]

[rad]

[Pa.sm]

[ l [kg m-3)

[N rn·l]

[Pa]

[Pa]

[ - l

31

32

SUPERSCRIPTS

dimensionless quantity

*•** modified

3. NEWTONIAN DROP BREAKUP IN QUASI-STEADY SIMPLE SHEAR FLOWS

3.1 INTRODUCTION

The present chapter deals with the modelling of the break-up of

Newtonian droplets in a Newtonian continuous phase in quasi steady

simple shear flows. Break-up criteria for droplets in simple shear

flows are available from the literature (both theoretical and

experimental) provided both the droplet and the continuous phase

are Newtonian, the shear rate is increased very slowly (quasi

steady simple shear) and the interfacial tension is constant over

the entire drop interface. These results are surveyed in section

3.2 and can serve as a starting point for further studies. These

are considered necessary because several important effects which

are known to have a great influence on drop break-up are not yet

sufficiently understood. In many emulsifying devices regions exist

with strongly varying shear rates. An example is the flow in almost

any stirred vessel. When strongly varying shear rates are exerted

on the droplet within time scales which are of the order of the

characteristic droplet break-up time very different break-up

criteria and modes of break-up were observed (Grace, 1982). These

effects are very important for the description of emulsification in

process units. Droplet break-up is also greatly affected when the

droplet or continuous phases are markedly non-Newtonian. This is

often the case for food processing where markedly non-Newtonian

ingredients are used.

Therefore an experimental device was developed to study the break­

up of droplets in simple shear flows experimentally.

The device works according to the Couette principle with two

concentric counter-rotating cylinders and is described in sections

3.3 and 3.4.

The Couette device has been tested by reproducing experimental

results described in the literature. The Couette device has also

been used to study the break-up of non-Newtonian drops in quasi­

steady simple shear flows (chapter 4) and to study the break-up

33

34

of drops in transient simple shear flows (chapter 5). The devices

is also used the study the tipstreaming phenor.1enon observed in

simple shear flows (chapter 6).

3.2 LITERATURE AND THEORY

3.2.1 Introduction

Taylor (1932) laid the foundations for the theoretical description

of the deformation of a small viscous droplet in a viscous

suspending liquid. Following the work of Einstein (1906) on the

viscosity of a fluid containing solid particles Taylor derived a

theory on the viscosity of an emulsion. These theories are based on

the method derived by Lamb (1932) using spherical harmonies to

describe the solution of the creeping flow equations in and around

the drops. Taylor assumed that the radii of the droplets were small

enough for the creeping flow equations to be valid (Reynolds number

based on the droplet smaller than unity) and that the deformation

of the droplets is very small. Taylor irnposed tangential stress

continuity between the suspending and the dispersed fluid phase and

a discontinuity in the normal stresses balanced by the Laplace

pressure. Furthermore Taylor assumed no slip conditions at the

interface for the velocities. These boundary conditions are still

generally applied, although they are usually not valid in the

presence of surfactants, see chapter 2 of this thesis. From the

velocity and stress distribution in and around a spherical droplet

Taylor derived a linear dependency of the deformation of a droplet

on the dimensionless shear rate. This relation is based on the

Laplace pressure distribution needed to compensate the normal

stress differences across the spherical drop interface, assuming

that the (small) deformation does not affect the normal stress

distribution.

This small deformation theory was extended and generalized by Cox

(1969). Cox adjusted the normal stress boundary condition by

balancing the fluid stresses with the interfacial tension forces on

the deformed surface and proposed a problem formulation in terms of

the drop excentricity. Cox examined the general problem of drop

deformation in transient linear shear flows within the limit of

first order ellipsoidal deformation of the drops. A major gain of

Cox's work is the possibility to expand it to higher order in the

drop excentricity. This has been done by Frankel and Acrivos

(1970), Barthes-Biesel (1972) and Barthes-Biesel and Acrivos

(1973). They extended Cox's model to the second order in the drop

excentricity. The resulting relation for the deformation does not

have a solution above a certain critical capillary number.

This can be interpreted as a prediction of the critical capillary

number at which drop break-up occurs. The critical capillary

numbers predicted by Barthes-Biesel and Acrivos (1973) are given in

figure 3.3 and will be compared with our experimental results.

3.2.2 Experimental results

The first experiments regarding the break-up of Newtonian droplets

in quasi steady simple shear flows were performed by Taylor (1934),

using a parallel band apparatus. Most of the later investigators

used a Couette device consisting of two coaxial counterrotating

cylinders. The advantage of the this design is the simpler

experimentation. Experimental studies on drop break-up in simple

shear flow were among others performed by Rumscheidt and Mason

(1961), Torza et al. (1972), Karam and Bellinger (1968) and most

extensively by Grace (1982). Grace's results will be used to

compare with our results (see figure 3.3). Experimental results

show that above a viscosity ratio of about 4 break-up is impossible

in quasi steady simple shear flow. A minimum in the capillary

35

36

number at which break-up occurs of about 0.5 is obtained for

viscosity ratios slightly lower than unity. For low viscosity

ratios break-up becomes increasingly difficult.

In simple shear flows four different modes of drop deformation and

break-up can be observed. These were first classified by Rumscheidt

and Mason (1961). For all four modes the droplet will assume an

ellipsoidal shape at very low shear rates with the principal axis

at an angle of 45 degrees with the flow direction. At very low

viscosity ratios this ellipsoidal deformation generally passes into

an elongated shape with an almost cylindrical centre section. Such

droplets generally break-up in two almost identical and equisized

drops with some small satellite drops in between. At viscosity

ratios around unity the deforming drop will generally form a neck

in the middle, which will progressively thin until two identical

daughter drops and some much smaller satellite drops are formed. At

viscosity ratios exceeding 4 a third mode of drop deformation is

observed. Upon increasing the shear rate the principal axis of the

drop rotates until the droplet is alligned with the direction of

the flow and a steady deformation is maintained. Finally a fourth

mode of drop deformation and break-up can sometimes be observed for

low viscosity ratios. In this mode, called tipstreaming, the

droplet takes, upon further increasing the shear rate, a sigmoidal

shape with sharply pointed ends from which very small fragments are

released. This mode, however, is not believed to be the normal

break-up mechanism in simple shear, but is related to the presence

of interfacial tension gradients at the drop surface (chapter 6 of

this thesis).

3.2.3 Slender bogy theories

Taylor (1964) also initiated the use of slender body theory to

describe the low Reynolds behaviour of droplets with a very low

viscosity ratio, which are subjected to high shear rates. This

usage was based on experimental observations that such droplets can

become very long and slender. This analytica! technique represents

the effect of the droplet on the flow by a distribution of

singularities along the drop axis and calculates the type, the

strength and positions of these singularities, together with the

position of the drop axis, that match the boundary conditions.

Later the slender body theory was refined and extended by

Buckmaster (1972,1973), Acrivos and Lo (1978), Hinch (1980), Hinch

and Acrivos (1980) and by Khakhar and Ottino (1986). First this

theory was applied to the case of an inviscid drop positioned in an

axially symetric pure straining flow. Later the theory was extended

to low viscosity drops and other linear shear flows, hyperbolic

flow and simple shear flow. The slender body theory has also been

applied to higher Reynolds numbers to establish the effects of

inertia forces. These theories predict a strongly decreased

effective shear rate just outside the droplet due to the presence

of the relatively non viscous droplet. Therefore the viscous farces

exerted on the droplet are very much reduced and high critica!

capillary numbers will result. For sirnple shear flows Hinch and

Acrivos (1980) predicted Ocrit - 0.054 À2/3.

3.2.4 Nurnerical technigues

Numerical techniques have been used to describe the deforrnation of

droplets in linear sbear flows. These studies were generally based

on a boundary integral rnethod by wbich the creeptng flow equations

inside and outside the drop are transformed into a form that only

involves quantities at the drop surface. This technique, derived by

Ladhyzhenskaya (1963) and further described by Youngren and Acrivos

(1975), Rallison and Acrivos (1978) and by Rallison (1981), is

based on the Fourier solution of the creeping flow equations, with

the use of volume potentials. Mathematica! problems arise with this

method for very small or very large viscosity ratios, since the

37

38

boundary integral bas neutral eigensolutions for these two

extremes. For axisymmetric problems the calculations are very much

simplified, since the drop surface can then be described by a curve

and less grid points are needed to obtain an accurate solution.

Another simplification occurs for simple shear flow since then one

of the terms in the boundary integral vanishes and the velocities

can be solved without a matrix inversion, substantially reducing

the computational time needed. For this situation Rallison (1981)

predicted a critica! capillary number of 0.42. This numerical

technique has also been applied in chapter 5 of this thesis

resulting in a predicted critica! capillary number between 0.45 and

0.50 fora viscosity ratio of 1.

3.3 FLOW IN THE COUETTE DEVICE

For the drop break-up experiments the simple shear flow is

generated in a Couette device. The flow in this device is not an

exact simple shear flow due to the curvature of the cylinders and

due to their finite length. In the following both effects will be

described.

First, however, the quasi steady flow pattern in the gap between

two infinitely long cylinders will be calculated from the basic

hydrodynamic equations. In this configuration the Navier-Stokes

equations can best be expressed in cylindrical coördinates. Due to

the symmetry of the problem (8/Bt ( ) = 0, Vz 0, vr = 0, 8/Bz ( )

= 0) these equations reduce to only one relatively simple

equation:

2 a v _____.J. 1

+ 2 r

ar ar

v

_1!. - 0 2

r

(3-1]

A solution to this equation is given by

v (r) </>

C r -1 r

or in terms of the angular velocity w - vq,/r:

w (r) c 1

c -2.

2 r

The constants C1 and Cz can be calculated from the boundary

conditions (counterrotating cylinders) given by

[3-2a]

[3-2b]

[3-3a]

(3-3b]

in which the subscripts I and II denote the radius of the inner and

outer cylinder respectively. Note that wr and wrr are defined in

such a way that they have a positive value. This results in:

2 2 w R + w R

I I II II c

1 2 2 [3-4a]

R • R II I

2 2

c -2 2 2 [3-4b]

R - R II I

The loc al shear rate in the gap is given by:

8 v 2

(r) _j_

-y - r 8r r 2

[ 3-5]

r

39

40

Since this shear rate is not constant across the gap the shear rate

at the position of the drop should be calculated. In the

experiments the stagnant layer is adjusted to coincide with the

drop position. The position of this layer r 8 can be calculated from

v~ - 0 resulting in:

r [3-6] s

The local shear rate at the stagnant layer can thus be calculated:

2 2 R w + R w

I I II II 7 (r ) - 2 [3-7]

s 2 2 R - R II I

The variation of the shear rate over the finite radius of the

droplet, R, can be obtained from Eq. (3.5]:

r

7 (r + R) ~ s r

s

s 7 (r )

+ 2R s [3-8]

Since in our apparatus rs is approximately 4.5 cm and the droplet R

is always smal~er than 0.5 mm the variation in shear rate over the

droplet is always smaller than 2%. Larger droplets should

preferably not be used because when the drop diameter is larger

than about 10% of the gap width the flow around the droplet will be

substantially influenced by the vicinity of the cylinders.

The description of the flow, which is given above, is only valid

when the Reynolds number of the flow based on the gap width is

small enough to prevent the occurence of Taylor vortices

(Schlichting, 1968). The relevant Reynolds number for this flow

is:

Re -

p 'Y (R • R ) c II I

'1 c

2

< 10

taking typical values for our experiments

Pc io3 kg ro·3

'Y < 91 s·l

Rn Rr - 1.05 cm

'Ic > 1 Pa.s

[3-9]

The Taylor instability will thus not occur in the Couette device

because

80 [3-10]

The calculations above are valid for the flow in a gap between two

infinitely long cylinders, but the Couette device has only a finite

length. In the following the end effects of top and bottom will be

described. The top of the Couette device usually does not disturb

the flow pattern, since it is open to air. Only when the continuous

phase evaporates and forms a film on top of the liquid, top effects

will occur. Bottom effects always occur, since the bottom rotates

with the outer cylinder. In order to know to which height these end

effects disturb the flow pattern, we have calculated the flow

numerically for a simplified geometry of the Couette device since

the sizes are chosen by RI 4 cm - 5 cm, height - ~ and the

bottom is chosen to be flat. The equations will be solved for a

stationary inner cylinder and a rotating outer cylinder. In this

configuration the Navier-Stokes equations can best be expressed in

cylindrical coördinates. Due to the symmetry of the problem (8/8t

( ) 0, Vz - 0, vr - 0) these equations reduce to one equation:

2 a v 2 ~ l 8v y u

0 [3-11] + + 2 r ar 2 2

ar r 8z

41

42

The boundary conditions for vq, (r, z) are:

v </>

(r, o) - w II

r [3-12a]

v

"' (r, "") = c r [3-12b]

1 r

v (R , z) - 0 [3-12c] </> I

v (R . z) =w R [3-12d]

"' II II II

These equations can be made dimensionless with the length scale

RII-Rr

and the velocity scale wrr Rrr: Thus the equation to be solved is:

2 LJ. 1 ~ -1L

= 0 + + 2 ç a ç 2 2 a ç ç 8 ï

in which

v P.

v w R II II

r ç

R - R II I

z ï

R R II I

with the boundary conditions.

v (Ç, 0) -

1J (Ç • oo)

R - R II I

R II

w R II II

. ç

. ç + R w (R - R ) IIII II I

1 ç

[3-13]

[3-14a]

[3-14b]

0 [3-14c]

v cez, n - i [3-14d]

This equation is numerically solved by a difference scheme. The

discretization was done as follows:

u Vm,n

v v av m-1,n m+l,n

ae 2 h

2 v 2 v + v Q_:Q m-1,n ID,!! m+l,n

2 2 ae h

2 v - 2 v + v Q_:Q m,n-l m,n m,n+l

2 2 ar h

in which h is the step size of€ and\. Substitution of this

discretization in Eq. [29] and reorganizing the terms results in

v m,n+l

2 h_

(4+.,,2) v ~ m,n

-1l v - (l+

m,n-1 2€

-1l v - (1 - ) v

m+l,n 2€ m-1,n

[3-15]

This equation was numerically solved with h - 0.025 in 100

iterations. The results are shown in Fig. 3.1 in which the velocity

is plotted as a function of the height in the cylinder for various

positions in the gap. From these results it can be concluded that

bottom ef f ects have disappeared at a height equal to about one gap

width (error< 1%).

43

44

0.80

0.60

0.40

0.20

··.\" -\\", \ \ "' 1 . "' . \ ......

----~~---

l • ........

\ \ ----\ \ \ \ \ . \ \ \ ' l \

\ \ \ ''\ \ .,_ ' -\ ---------------------\ \

','-.,, , __ --. ... _____ " _________________ _

0.00 '------'-------'-------''------'-------' 0.00 0.45 0.90 1.35 1.80 2.25

--> Height (Dimless}

1/6 gap 2/3 gap

i/3 gap 516 gap

1/2 gap

Fig. 3.1.a Effect of bottom on flow pattern in Couette device for various posit1ons in the gap. Velocity ratio of cylinders: ·1:4

' .... '-\ ', ·. \ .... . \\ \ ....

l\ ". \ 0.80 '1 ' ... "'

\ . ' \ 1 •• '-

---

\ \ "·. '-, 1 \ ·. -.... ____________ _

\ . \ \

0.60 \ \

\ \ \ \\ \ .

0.40

\ \. \ ' " '-...,......._

\ ----'\ ----------------

\\",_

0.20 '-....._ ""....__ ______ ... " _______________ _

0.00 .~--J__,,

0.00 0.45 0.90 1.35 1.80

--> Heigth [Dimless}

1/6 gap 213 gap

1/3 gap 5/6 gap

1/2 gap

Fig. 3.1.b Effect of bottom on flow pattern in Couette device for various posit1ons in the gap. Velocity ratio of cylinders: -1:1

2.25

45

46

' ' ·.\ ', ·. \ '·. \

0.80 .\ · ... \ ,\ \ '' \

\ \ " ' \ "-. \' ' ~ \ '-...._

0.60 \ \ -----

\ ' ' \ \ \

0.40 ......... " .. ". " .. ··-···.

\ \ \ ' \ \ \ ' " '\. \ "'-,

\ '"'-......,._ \ ---\\, ----------------

''--, __

-------------~------------0.20

0.00 ~----~-____J_ ____ -1. _____ ._ ___ ___.

0.00 0.45 0.90 1.35 1.80 2.25

--> Height [Dim.less}

1/6 gap 213 gap

1/3 gap 516 gap

1/2 gap

Fig. 3.l.c Effect of bottom on flow pattern in Couette device for various positions in the gap. Velocity ratio of cylinders: -4:1

3.4 EXPERIMENTAL

3.4.1 Description of apparatus

Por the measurement of deformation and break up of individual

droplets in simple shear flow a Couette type apparatus has been

built, consisting of two concentric counterrotating cylinders.

Thus a stagnant layer is created which permits statie optical

observation of a droplet in a flow field (See Fig. 3.2).

The diameter of the inner and outer cylinder are respectively:

Rr - 39.75 mm and R11 50.25 nun. The height of the inner cylinder

is 50 mm. The inner cylinder is made from aluminium, while the

outer cylinder is made from precision-bore glass which permits

direct visual observation both from the side direction and from

beneath. The cylinders are mounted in ball bearings.

The cylinders are driven by ribbed belts from two permanent magnet

D.G. servomotors with tach (Electro Graft Corporation type E-652-

00-004) each with a reduction of 1:30. The motors are controlled by

a speed control unit (E-652-M) which also has a remote control

facility requiring a voltage range of 0-10 V with a resolution of

1:1000. We used this remote facility to control the motor speed

from a personal computer. The actual rotation speeds were measured

independently and were available in a digital read out. The maximum

attainable shear rate is 91.2 s-1 at a rotation speed of the

cylinders of 100 rev./min.

The drops were observed either by viewing in radial direction

through the cylinder wall or along the z-axis by viewing through a

mirror placed at an angle of 45° beneath the annular slit. In the

case of observation through the cylinder wall we used a small

cuvette with one curved surface and one flat surface to minimize

distortion effects by the curved glass surface of the outer

47

48

cylinder. The advantage of observation in radial direction is that

in the case that the densities of the two phases do not exactly

match, the drop is kept more easily in focus, but its orientation

in the gap between the cylinders is not exactly known. The drops

were illuminated with a cold light source (Schott type KL 500) with

flexible glassfiber light transmitters. We used a Zeiss stereo­

microscope (type SV) with an objective with a focal length of 100

mm and a zoom mechanism. On the microscope a Sony camera (type DXC

1850P) and a Contax RTS-camera were mounted which made recording of

the experiment on video possible. A video timer monitored the time

elapse continuously. The overall maximum magnification of the

optica! system was 500 times with an accuracy on 1%.

3.4.2 Experimental procedure

For a drop break-up experiment the following procedure was used.

The Couette apparatus is filled with the continuous liquid phase. A

drop of the dispersed phase is inserted from a syringe (Pasteur

pipette) usually in the middle of the gap and halfway the heigth of

the inner cylinder. Because it appears to be difficult to insert

just one drop, all unwanted drops are sucked out of the continuous

phase by a syringe. One or both cylinders are slowly rotated until

the droplet is visible on the video screen. The drop radius is

measured from the video screen. Next the video recording is started

and the rotational speeds of both cylinders are slowly increased in

such a way that the droplet remains (almost) stagnant at the center

of the video screen. From the ratio between the inner and outer

cylinder speeds the place of the stationary drop in the gap can be

calculated and the speeds of the cylinders can be further

increased. When drop break-up occurs the rotational speeds from

the cylinders are taken from their digital output and the rotation

Figuur 3.2a

Fig. 3.2.a Couette device: top view

49

50

PARTICLE ü'lder

considerotion

{

\

CAM. ERA LENSE

-- j- ---~~JJSTABl.E l~ROR

Fig. 3, 2, b Couette device: front view

OUTER"' CILINa;:R~

and video recording are stopped. At that time the temperature of

the liquid in the gap is measured. After removing all unwanted

fragments by suction with the syringe a next experiment can be

started with one of the fragments or with a freshly inserted

droplet.

The viscosity as a function of shear rate of the Newtonian liquids

has been measured with a Haake type CV 100 viscometer using a

concentric cylinder geometry (type ZC 15). This apparatus was

thermostatted at 23°C and the shear rate range was from about 1 to

80 s·l.

Two different methods have been applied to measure the interf acial

tension between the used liquids;

a) The Wilhelmy plate method, where a silver plate with

circumference of 0.06 m is drawn from one liquid into the other

and the excess force, corrected for gravity effects, gives

directly the interfacial tension. This method is not suitable

for very low interfacial tensions and can have an accuracy of

0.1%.

b) The sessile drop method, which is based on the drop deformation

under gravity forces. After measuring height and width of the

droplets the interfacial tensions can be calculated with the aid

of the tables given by Bashforth and Adams (1883). An accuracy

is claimed of 0.1%. High interfacial tensions, however, are

difficult to measure.

All interfacial tensions were measured with respect to a silicone

oil with a viscosity of 1 Pas as being representative for all

silicone oils. This because the surface tension of these oils

against air is found to be constant at different viscosity values.

The droplet and continuous phases used are shown in Table 3.1. The

silicone oils (type Rhodorsil 47 ex Rhone-Poulenc Chimie Fine) were

made at different viscosity values by blending a high and a low

viscosity batch so covering a viscosity range from 0.9 to 65 Pas.

51

Table 3.1 Physical data of model liquids

nr. droplet pbase viscosity

[Pa.s)

Castor Oil 8. 7 • 10-l

2 Sunflower Oil 5. 7 • 10-2

3 Ethylene Glycol 1.6 • 10-2

4 Aniline 3. 7 10-3

5 Corn Syrup/Water (90/10) 2.8

6 Corn Syrup/Wat:.er (85/15) 7 .8 10-1

Corn Syrup/Water (80/20) 2.9 • 10-1

Corn Syrup/Water (75/25) 1.2 • io-1

9 Silicone Oil 2.1 lo1

10 Silicone Oil 1.6 • 101

11 Silicone Oil 1.4 • 101

12 Silicone Oil 1.2 • lol

13 Silicone Oil 9.3

14 Silicone Oil 6.4

nr. continuous phase viscosit.y

{Pa.sJ

15 Corn Syrup/Water (92/8) 6.0

16 Silicone Oil. 9.0 • 10-1

17 Silicon-e Oil 5. 7

18 Silicone Oil 6.0

19 Silicone Oil 1.1 • 101

20 Silicone Oil 1.6 lo1

21 Silicone OH 2. 7 • io1

22 Silicone Oil 3.8 101

23 Silicone OH 4.0 • io1

24 Silicone Oil 4.3 io1

25 Silicone Oil 6. 5 • 101

52

The corn syrup solutions were based on a concentrated corn syrup

(type Globe 01170 ex GPC Netherlands B.V.) which was diluted with

water so covering a viscosity range from 0.12-6.0 Pas. Further

ethylene glycol, aniline, castor oil and sunflower oil were used in

standard quality.

3.4.3 Results

Using the procedure as given in paragraph 3.4.2 the critica! shear

rate for drop break-up was measured. The results are given in Table

3.2. The results, averaged per combination of droplet and

continuous phase are visualized in Fig. 3.3. From this figure it

will be clear that there is a very good correlation between the

Capillary number 0 (the dimensionless shear rate at which drop

break-up occurs) and the viscosity ratio.

• • + + ++ •

--+

Fig. 3.3

LEGEND •"' own experi.rtentol results +-Grec", 1982 6. - Barthes-Bl.eseL, 1972 X• Rol.ll.son, 1981 17 • Hl.noh + Acr i, vos, l 980

+

+

IJl­+

+

+ +~

+ +" + + +. _..+;+•

Comparison of drop breakup data

53

54

This correlation can be used as a break-up criterion to predict

maximum droplet sizes in simple shear flows. The correlation

obtained is in agreement with the one measured by Grace (1982).

This comparison shows that the scatter in Grace's results is far

larger than in our results (Fig. 3.3). The theoretica! calculations

by Hinch and Acrivos (1980) and Barthes-Biesel and Acrivos (1973)

appear to predict break-up at somewhat lower shear rates than

determined experimentally. Qualitatively, however, their

correspondence with experimental results is very good. The

numerical simulations by Rallison and those described in chapter 5

of this thesis are in very good agreement with the experimental

results.

From the break-up criteria obtained it will be clear that drop

break-up in simple shear flows is most efficient when the droplet

viscosity is a little lower than the continuous phase viscosity

At viscosity ratios above about 4 break-up in simple shear flow is

not possible at all (e.g. preparation of diluted, oil in water

emulsions is not possible in simple shear flows).

A simple 4 parameter function has been used to fit the experimental

data. The function features the theoretically predicted behaviour

Ocrit - Àa for low values of À, together with a maximum viscosity

ratio, Àmax• above which break-up will not occur:

log Ocrit C2

C1 + a log À + ~~~~~~~~ log À - log Àmax

[3-16]

Curve fitting resulted in a = -0.733, Amax 9.27, C1 -1.560 and

C2 -1.135 with a MSQ of 0.008. (See Fig.3.4). The fit however, is

not very good close to Àmax· A much better fit was obtained by a 5

parameter function

log Ocrit C1 + a log À + C3 (log A)2 + log À

C2

log Àmax [3-17]

with Àmax = 4.08, a = 0.0994, C1

0.124 and MSQ 0.002.

-0.506, C2 = -0.115 and C3 =

The 5 parameter function gives a much better representation close

to Àmax (see Fig.3.4) but to very low À is not justifiable.

For À - 1 this correlation predicts Ocrit 0.48.

Fig. 3.4

•

LEGEND •- own experi..ttenlal res:ulls V= 4 parameter fl,t A • 5 parameter f(, t

Drop breakup correlations

î

55

Table 3.2 Drop break-up experiments

drop cont. " R 7 0crit phase pbase {m Nm- 1 J [mml [s-11 1-1 [-)

20 4 .1 0.143 1.11 5.4 • 10-1 0. 84

0.120 1. 60 0. 75

0.102 1.97 0.79

0.095 1.98 o. 73

0.065 3. 01 0. 77

0.061 3.39 0. 81

23 4.1 0.227 0.60 2.2 • 10-2 1.3

0.148 0.99 1. 4

0.102 1. 31 1.3

0 .075 1. 59 1.2

0.062 2.04 1. 2

0.048 2.59 1.2

0.042 3.24 1.3

0,039 3.23 1.2

0 .021 6. 70 1.4

2 17 2.5 0.475 1.41 1. 0 • 10-2 1.5

0.324 1.63 1.2

0.256 2.14 1.2

0.200 2. 91 1.3

0.152 3. 74 1. 3

0.117 4. 70 1.3

0.092 6.26 1.3

2 20 2.5 0.338 1.20 3, 6 • 10-3 2,6

0 ,234 2.26 3.4

0.130 3.11 2.6

0.090 4 .15 2.4

0. 443 0,85 2.4

0.398 1.20 3.1

0.262 1. 54 2.6

2 23 2. 5 0.315 1.41 7 .1

o. 176 1. 94 5.5

0.113 3.00 5. 4

0.083 4 .17 5.5

17 13' 5 0' 542 19. 10 2.8 • 10-3 4. 3

0.484 18.54 3.8

0' 418 22. 71 4 .0

0,277 34. 78 4 '0

20 13. 5 0.559 13 .06 1.0 • 10-3 8. 7

0. 447 15.94 8. 4

0,308 23 .63 8.6

0. 345 32. 10 9,3

23 13. 5 0.533 15.35 4.0 10-4 24

0. 308 22.91 21

20 5. 0 0.301 33 .83 2. 3 • 10-4 33

18 35 0 452 5. 93 4. 7 10- 1 0.46

0. 352 8.06 0. 49

56

Table 3.2 Continued (a)

drop cont. q R 'r ). noi:it phase phase fm Nm-ll !nrnl [s-1] [-] [-]

5 19 35 0.390 4. 37 2.5 • 10-l 0.55

0.265 5.84 0.50

0.506 3.14 0.52

0.345 4.63 0.52

0.233 6.21 0. 47

21 35 0.410 2.17 1.1 • 10-1 0.68

0.265 3.24 0.65

5 24 35 0.330 1.89 6, 5 • 10-2 0. 77

16 33 0,464 36.57 8. 7 • 10-l 0,46

0.367 48.11 0.48

6 18 33 0.506 5.17 1.3 10-l 0 .57

0,486 7.36 0.52

0.303 9.27 0.51

0.235 11. 95 0.51

0.185 14. 78 0.50

19 33 0.412 4.97 6.9 • 10-2 0. 70

0.310 6.95 0. 74

0.220 9.18 0. 70

22 33 0.366 2.80 2.0 • 10-2 1.2

0.285 3.87 1.3

0.209 5.02 1.2

25 33 0.355 2.09 i.2 • io-2 1.5

0.280 2. 75 1.5

0.175 3.91 1.3

0.140 4.98 1.4

18 31 0.431 8.67 4.8 • 10-2 o. 72

0.339 11.06 o. 73

0.262 13.92 0.71

0.205 17 .26 0.68

19 31 0.383 4.32 2. 6 • 10-2 1. 3

0.261 12.31 1.2

22 31 0.383 4. 75 7 ,6 • 10-3 2.2

0,395 6,47 2.3

0.215 7. 75 2.1

25 31 0.429 3. 21 4.5 • 10-3 2.9

0,322 3 .94 2.6

0.190 6.54 2.6

16 29 0.512 32.93 l.3 • 10-l 0.52

18 29 0.461 13 .12 2.0 10-2 1.3

0.357 16.11 1.2

0.281 18.-5 1.1

0.217 23.35 1.1

0.173 31. 70 1.1

8 19 29 0.366 12.08 t. l • 10-2 1.7

0.295 15.90 1.8

0,227 21.15 1.9

57

Table 3.2 Continued (b)

drop cont. R " 0 crit phase phase [m Nm-1 ] [l1lll) r.-1 1 [-) [-J

22 29 o. 402 8.41 3.1. 10-3 4.4

0.322 9.09 4.2

0 .251 11.90 3.9 g 15 38 0.540 57 .96 3. 3 5.1

0.640 45. 93 4.8

0. 555 63.83 5.6

10 15 38 0.434 14.91 2.6 1.1

0.352 19. 77 1.2

0.285 25.38 1. 2

0.217 32.85 1.2

11 15 38 0.470 11.63 2.3 0.85

0.381 15.12 0.89

0.298 19.09 0.87

0.238 24.36 0 .88

12 15 38 0 .446 9.83 2.0 0,69

0.343 12.15 0.66

0.274 16. 00 0.69

0.214 20.01 0.68

0.187 21.21 0.62

13 15 38 0.393 8.64 1.5 0.55

0 .323 10.87 0. 57

0. 243 13.69 0.54

0.193 17 .80 0.55

0.154 22.65 0.56

14 15 38 0. 447 6.16 l.o 0.45

0.331 7 .98 0.43

0.270 9. 88 0.43

0.208 12.48 0.42

0.175 16.43 0. 46

58

3.5 CONCLUSIONS

Reliable break-up criteria are available to predict droplet break­

up in quasi steady simple shear flow, provided both liquid phases

are Newtonian and surfactants are absent. The break-up criteria are

fully described by the drop capillary number 0 and the viscosity

ratio À.

Simple shear flow is found to be most efficient for the break-up of

droplets with a viscosity ratio in the range 0 1 < À < 1. For

droplets with a viscosity of more than 4 times the viscosity of the

continuous phase, break-up is not possible at all in simple shear

flows.

The experimental results obtained with the developed Couette

apparatus show a very small scatter and are in agreement with

results obtained elsewhere. It is thus concluded that a well

functioning apparatus has been developed that will be used to

investigate the effects of non-Newtonian drop phases on drop break­

up (de Bruijn, chapter 4 of this thesis), to investigate the effect

of transient simple shear flow on drop deformation and break-up

(chapter 5 of this thesis) and to study the tipstreaming phenomenon

in simple shear flow (chapter 6 of this thesis).

3.6 REFERENCES

1. A. Acrivos and T.S. Lo, Deformation and break-up of a single

slender drop in an extensional flow, J. Fluid Mech. 86, 641-

672, (1978).

2. D. Barthes-Biesel, Deformation and burst of liquid droplets

and non-Newtonian effects in dilute emulsions, Thesis Stanford

University, Michigan, USA, (1972).

3. D. Barthes-Biesel and A. Acrivos, Deformation and burst of a

liquid droplet freely suspended in a linear shear field, J.

Fluid Mech., 61, 1-21, (1973).

59

60

4. F. Bashforth and J.C. Adams, An attempt to test the theories

of eapillary aetion, University Press, Cambridge, (1883).

5. R.A. de Bruijn, Sealing laws for the flow of emulsions,

ehapter 2 of this thesis.

6. R.A. de Bruijn, Non-Newtonian drop break-up in quasi steady

simple shear flows, ehapter 4 of this thesis.

7. R.A. de Bruijn, Deformation and break-up of Newtonian droplets

in transient simple shear flows, ehapter 5 of this thesis.

8. R.A. de Bruijn, Newtonian drop break-up in simple shear flows

the tipstreaming phenomenon, ehapter 6 of this thesis.

9. R.G. Cox, The deformation of a drop in a general timedependent

fluid flow, J. Fluid Meeh., 37, 601-623, (1969).

10. A. Einstein, Ann. Physik, 19, 289, (1906).

ll. N.A. Frankel and A. Aerivos, The eonstitutive equation for a

dilute emulsion, J. Fluid Mech., 44, 65-78, (1970).

12. H.P. Graee, Dispersion phenomena in high viscosity immiseible

fluid systems and applieation of statie mixers as dispersion

deviees in sueh systems, Chem. Eng. Commun. 14, 225-277,

(1982).

13. E.J. Hinch, The evolution of slender inviscid drops in an

axisymmetrie straining flow, J. Fluid Mech., 101, 545-553,

(1980).

14. E.J. Hinch and A. Aerivos, Long slender drops in a simple

shear flow, J. Fluid Mech. 98, 305-328, (1980).

15. H.J. Karam and J.C. Bellinger, Deformation and break-up of

liquid droplets in a simple shear field, Ind. Eng. Chem.

Fundam. 7, 576-581, (1968).

16. D.V. Khakkar and J.M. Ottino, Deformation and break-up of

slender drops in linear flows, J. Fluid Mech., 166, 265-285,

(1986).

17. O.A. Ladyzhenskaya, The mathematieal theory of viscous

incompressible flow, Gordon and Breach, New York, (1963).

18. H. Lamb, Hydrodynamics, 6th ed. Dover press, New York,

(1945).

19. J.M. Rallison, Note on the time dependent deformation of a

viscous drop which is almost spherical, J. Fluid, Mech., 98,

625-633, (1980).

20. J.M. Rallison, A numerical study of the deformation and burst

of a viscous drop in general shear flow, J. Fluid Mech., 109,

465-482, (1981).

21. J.M. Rallison, The deformation of small viscous drops and

bubbles in shear flows, Ann. Rev. Fluid Mech. 16, 45-66,

(1984).

22. J.M. Rallison and A. Acrivos, A numerical study of the

deformation and burst of a viscous drop in an extensional

flow, J. Fluid Mech., 89, 191-200, (1978).

23. F.D. Rumscheidt and S.G. Mason, Particle motions in sheared

suspensions, XII deformation and busrt of fluid drops in shear

and hyperbolic flow, J. Coll. Int. Sci. 16, 238-261, (1961).

24. H. Schlichting, Boundary layer theory, McGraw-Hill, New York,

1968.

25. G.I. Taylor, The viscosity of a fluid containing small drops

of another fluid, Proc. Roy Soc. A 138, 41-48, (1932).

26. G.I. Taylor, The formation of emulsions in definable fields of

flow, Proc. Roy Soc. A 146, 501-523, (1934).

27. G.I. Taylor, Conical free surfaces and fluid interfaces, Proc.

Int. Congr. Appl. Mech., llth, p 790-796, Munich (1964).

28. S. Torza, R.G. Cox and S.G. Mason, Particle motions in sheared

suspension, XXVII transient and steady deformation and burst of

liquid drops, J. Coll. Int. Sci. 38, 395-411, (1972).

29. G.K. Youngren and A. Acrivos, On the shape of a gas bubble in a

viscous extensional flow, J. Fluid Mech., 76, 433-442, (1975).

3.7 LIST OF SYMBOLS

C1 C2 C3: constants

h step size of Ç and Î

m Ç-position

[s-1, m2.s-l]

[ - ]

[ - J

61

62

LIST OF SYMBOLS (continued)

n

p

R

t

v

°' i'

ï

'ld· À

v

ç p

17

v

(l

w

'lc

r, ;/>, z

Fr

Re

ï-position

pressure

drop radius

radius of inner and outer cylinder

position of stagnant layer

time

velocity

constant

shear rate

normalized axial position

viscosity of drop and continuous

viscosity ratio

kinematic viscosity

normalized radial position

density

interfacial tension

dimensionless velocity

Capillary number

angular velocity

cylindrical coordinates

Froude number

Reynolds number

phase

[ - l [Pa]

[m]

[m]

[m]

[s] [m s-lj

[ l [s-1]

[ - ]

[Pa.s]

[ - ]

[m2.s-l]

[ - l [kg.m-3]

[N.m-1]

[ l [ - l [s-1]

4. NON-NEWTONIAN DROP BREAK-UP IN QUASI STEADY SIMPLE SHEAR FLOW

4.1 INTRODUCTION

The present chapter deals with the modelling of the break-up of

non-Newtonian droplets in a Newtonian continuous phase in quasi

steady simple shear flow. The break-up behaviour of Newtonian

droplets in a Newtonian continuous phase in quasi steady simple

shear has been described in the previous chapter. The break-up of

non-Newtonian droplets is especially relevant to food processing

where markedly non-Newtonian ingredients are used. In this chapter

theoretical and experimental results available in the external

literature are surveyed. These results, however, do not give a

complete understanding of non-Newtonian drop break-up. Therefore

the break-up of two types of non-Newtonian droplets was studied

experimentally in the Couette device described in chapter 3 of this

thesis: droplets which combine a strong shear-thinning behaviour

with very low normal stress differences (low elasticity) and

droplets which combine high elasticity with an almost constant

(i.e. shear-rate independent) viscosity. These two types of droplet

rheology were chosen to be able to separate the effects of shear

thinning behaviour and of droplet elasticity which usually occur in

a combined way.

4.2 NON-NEWTONIAN FLUID MECHANICS

The basic equations describing the flow of non-Newtonian droplets

in a Newtonian continuous phase are the equations of motion and the

mass conservation relations in both droplet and continuous phase

and the boundary conditions which apply to the droplet interface

complemented by the constitutive equation of the fluid. The

constitutive equation of a liquid relates the stresses in the

liquid to the deformation of the liquid. The constitutive equation

of a Newtonian liquid is relatively simple and the equation of

motion of a Newtonian liquid thus reduces to the well known Navier-

63

64

Stokes equations. Since there are many types of non-Newtonian

behaviour a large number of constitutive equations have been

formulated to describe these liquids. The basic equations

governing the flow are discussed in more detail in chapter 2 of

this thesis.

In this chapter we shall present the rheometrical data with a

simplified description of the constitutive equations that is only

valid for steady simple shear flows. The stresses in a liquid can

be described by the stress tensor !.

t" 7 12

'"J !. 7 21 7 22 7 23 7 31 7 32 7 33

[4-1]

The deviatoric stress tensor representing the deviation of the

stress from an isotropic state (described by a single scalar, the

pressure) as a consequence of flow is defined in such a way that

1"11 + 1"22 + T33 = 0. Since only the deviatoric stresses lead to

drop deformation attention will be focussed on these, omitting the

prefix deviatoric for convenience. Further it can be shown that the

stress tensor is symmetrie in the absence of extraneous moments, so

that Tij

that r31

Tji· For a simple shear flow it can further be shown

1"32 = 0. The stress tensor is thus fully determined by

only three independent material functions of the

shear rate -)<.

r12 7 21 7 11 - 7 22

7 22 - r33

1"1 ()')

Ni ()'2)

N2 ('Î2)

[4-2-a]

[4-2-b]

[4-2-c]

r1 ('Î) is the shear stress function, N1 ('Î 2 ) is usually called the

first normal stress difference and N2 C.Y 2 ) the second normal stress

difference; 1"1 (j) is an uneven function and Ni (1' 2 ) and N2 (j2 )

are even functions of i.

For a Newtonian liquid the material functions reduce to

[4-3-a]

[4-3-b]

in which the dynamic viscosity ~ is independent of the shear rate.

Inelastic, shear-thinning liquids can in a certain shear-rate

regime often be described by the power law equation:

[4-4-a]

[4-4-b]

in which K is the consistency index and n the power law index.

In most viscoelastic liquids the second normal stress difference is

much smaller than the first normal stress dif ference and can

therefore be neglected. This is for instance done in the upper

convected Maxwell model which can be described by the following

material functions

Tl - ~ i N1 - 2 8 ~ i2

N2 - o

(4-5-a]

[4-5-b]

in which ~ is called the dynamic viscosity and 8 the relaxation

time of the upper convected Maxwell fluid. However it is very

difficult to find model liquids which fully correspond to this

constitutive equation. In practice the ratio of r1/t is not exactly

constant and the first normal stress difference does not vary

exactly proportional to ~2. Thus more general material functions

are needed. We will therefore describe our fluids with the

following material functions.

65

66

o.

[4-6-a]

[4-6-b]

[4-6-c]

in which ~ is the elasticity index and m the elastic power law

index. This turns out to be a rather good description of the

viscometric data of the non-Newtonian model liquids used in our

studies. These material functions obey the Criminale-Erickson­

Filbey constitutive equation, which is a simplification of the

retarded motion expansion, valid for steady state flows

(Schowalter, 1978). This constitutive equation is thus not suited

to describe transient phenomena, but it was adopted for the ease

with which the viscometric data of the model liquids used in

present and other studies could be described.

The use of dirnensionless groups in studying the dynamics of non­

Newtonian liquids cannot be as straightforward as it is for

Newtonian liquids due to uncertainties in the constitutive

equations of practical liquids. However three dimensionless numbers

are aften used: the Weissenberg number Nwe• the stress ratio SR and

the Deborah number De. The Weissenberg number is a measure of the

relative importance of elastic and viscous effects:

N We

y

L [4-7]

in which 0 is a natural time, characteristic of the liquid, and V

and L are a characteristic velocity and length, respectively.

If the constitutive equation of the liquid considered is well known

e.g. the Maxwell upper convected liquid, the definition of the

natural time is straightforward and the relation with fluid

elasticity is clear. When normal stress differences and shear

stresses are only known in a limited shear rate regime it is aften

better to use the dimensionless stress ratio SR

s R T

[4-8]

12

When the stress ratio is plotted against shear stress one can

compare the relative importance of fluid elasticity of different

viscoelastic fluids. The Deborah number is the ratio of a natural

fluid time to a characteristic observation or process time. The

process time should describe the flow experienced by material

points, thus it is a time scale related to the movement in the

direction of the flow, whereas the process time scale in the

Weissenberg number, L/V, is generally related to the velocity

gradient perpendicular to the flow direction. When De << 1 the

material behaves in a fluid-like manner and when De >> 1 the

material behaves solid-like.

4.3 LITERATURE REVIEW

4.3.1 Theoretical results

Very few theoretical approaches to the break-up of non-Newtonian

droplets in simple shear flow have been pursued. This is probably

partly due to the great complexity of the problem because it

involves three dimensionality, free surfaces, non-stationary flow

and complex constitutive equations.

Van Oene (1972) used a modified interfacial tension to account for

the fluid elasticity. The scientific basis of this theory is very

weak. The model is based on an expression for the recoverable free

67

energy Fg for constrained recovery in shear flow which was

developed by Janeschitz-Kriegl (1969) from an expression for the

stress tensor in polymerie liquids. For an upper convected Maxwell

liquid in simple shear flow one obtains:

[4-9]

Van Oene's rather qualitative argument was that when a droplet is

formed in a continuous phase the recoverable free energy changes

by

[4-10]

in which R denotes the radius and the subscripts d and c denote the

drop and continuous phase respectively. This change in recoverable

free energy was taken into acccount as an additional contribution

to the interfacial free energy a, which is the free energy of

formation of the interface per unit area

R a* - a + - (N - N )

6 l,d l,c [4-11]

This model predicts that elasticity of the droplet phase stabilizes

the droplet. This conclusion at least qualitatively corresponds

with experimental evidence (see section 4.3.2). This model also

predicts that elasticity of the continuous phase destabilizes the

droplet. However for a very large continuous phase first normal

stress difference, the model predicts a negative modified

interfacial tension. This, however, will not occur since when a*

reduces smaller droplets will be broken up, so negative values for

a* will not be obtained.

More theoretica! results are available for the description of the

break-up process of droplets or jets which already possess an

extended cylindrical shape.

Chin and Han (1980) considered the effects of fluid elasticity

(Maxwell model) on the break-up in non-uniform shear flow. They

considered a long, neutrally buoyant, liquid cylinder in a

continuous phase subjected to a cylindrical Poiseuille flow. The

long droplet is located on the axis of the tube. They assumed

axisymmetric disturbances which are periodic in the flow direction

and which grow or decay exponentially in time. They performed a

linear hydrodynamic stability analysis to determine the stability

or instability of the disturbance. They found that the stability

region of such a cylindrical droplet increases when the capillary

number based on the droplet increases. This can be interpreted as

either the stabilizing effect of increased shear rate in the

Poiseuille flow or as the destabilizing effect of an increased

interfacial tension. The stability region increases when the

continuous phase elasticity increases. Chin and Han (1980) also

analysed the stability of elongated droplets when they are

disturbed by waves of all wave-lengths under the condition of

inertialess flow. They concluded that the growth rate of the

disturbance is higher for increased droplet elasticity and for

decreased capillary numbers. As the droplet phase viscosity becomes

greater than the continuous phase viscosity the growth rate of

disturbance becomes slower. If it is assumed that the fastest

growing wavelength is responsible for drop break-up it can be

concluded that the smaller the droplet phase viscosity and the

larger the droplet phase elasticity, the larger the broken droplets

will be.

Bousfield et al (1986) have described the surface tension driven

break-up of viscoelastic jets. They found that disturbances on

viscoelastic jets grow more rapidly at short times than on

Newtonian jets, but that the growth is retarded at longer times due

to the development of large extensional stresses. The formation of

satellite drops was found to be retarded by the fluid elasticity.

69

70

4.3.2 Experimental results

Gauthier et al (1971) were the first to report introductory

experimental results on the deformation and break-up of non­

Newtonian drops in a Newtonian continuous phase in simple shear

flows (see Appendix A). For shear thinning droplets they did not

found any deviation from the Newtonian behaviour. For viscoelastic

droplets their experiments indicate an increase of the critical

capillary number which they attributed to the droplet elasticity.

Since they performed just a few experiments on non-Newtonian

droplets which they have documented very poorly their results are

not very conclusive.

Tavgac (1972) has performed experiments with viscoelastic drop

phases (mostly Poly Acryl Amide (P.A.A.) in water/glycerine) in

Newtonian silicone oils (See Appendix A). He used a Couette device

with a gap of 7.65 mm allowing shear rates up to 90 Tavgac

described the rheological behaviour of his model liquids by a

modified 5 parameter Bird-Carreau model with the following material

functions.

~

r1 (i) - ~ 2 2

[4-12a]

p=l 1 + À l,p

i

2 2 ~ À . 1

N <7) - E p 2.p

1 2 2 [4-12b]

p~l 1 + À i l,p

Nz (~) - 0 [4-12c]

in which

ri ri p 0 "' >.

1,k

_2_ >. >. "'1 l,p 1 (p + 1)

_2_ À >.

(p + 1)"'2 2,p 2

The critical capillary number at break-up will thus depend on 5

dimensionless numbers:

n crit [

I'/ 0 n -,

crit ric a ' 1 "' ' 2

>. 1 >1, (o' Do), ,:J [4-13]

For the solutions of P.A.A. in water/glycerine used by Tavgac the

ratios a2/a1 and >.1/-'2 are approximately constant. For those

liquids the functional dependence thus reduces to

0 crit

[4-14]

Tavgac presented his experimental data as plots of Ocrit vs De for

the various drop phases used. To compare his results with ours we

had to estimate the power law constants K, n, K and m from the 5

Bird-Carreau parameters. This can be done numerically since in the

relevant shear rate regime (0.1 s·l < t < 100 s·l) the shear

stresses and normal stress differences of the model liquids used by

îavgac vary with tn and tm. The model liquids exhibit strong shear

thinning behaviour (0.19 < n < 0.79) combined with considerable

fluid elasticity.

71

72

There are several inaccuracies in Tavgac's work, the most striking

experimental error being the large drop sizes. Although the gap

width of the apparatus is only 7.65 mm, he used droplet diameters

varying between 0.56 < D < 4.22 mm. Most experiments were done with

droplets occupying far more than 10% of the gap. This will lead to

wall effects. Especially the conclusions on the observations with

drop phase T7 are erratic due to these wall effects. For this drop

phase the critical capillary numbers were thought to be lower than

the Newtonian limit which was assumed to be Ocrit 0.8 for De < 1

with a minimum for De= 0.25. According to our results (chapter 3

of this thesis) the Newtonian limit is however much lower (Gerit

0.5). The data for De< 0.2, which show a decrease in critical drop

size with increasing elasticity are erratic due to wall effects.

Hence the critical capillary numbers smoothly increase with

increasing droplet elasticity. Further, Tavgac does not correct

for the shear rate variations in the gap. The average shear rate is

determined instead of the shear rate at the stationary layer, which

may result in an additional error of up to 14%.

Tavgac correlates his experimental results by

[4-15)

with C1 and 0 0 ,crit dependent on the fluid system. From this

correlation he predicts the existence of a critical drop size,

below which break-up can not occur, given by

C À a 1 1

R [4-16] crit 17

c

C1 was found to decrease with increasing viscosity ratio. Therefore

we also plotted rvs 7crit for each fluid system (See Fig. 4.1).

Fluid systems T3, T4 and T8 appear to indicate critical drop radii.

For the other fluid systems rcrit is expected to be much lower than

the drop sizes studied.

10

" "

î . (j)

~ 1 -e 8-'b .

.

Fig. 4.La

+

shear ra te [ 1 /sJ

Viscoelastic drop breakup results obtained by Tavgac (1972). Critical capillary number vs. shear rate+ Drop phase: 0.1% Poly Acryl Amide

+ +

+

...

Elmendorp (1986) bas performed some experiments witb viscoelastic

drop pbases (PAA in water) in a Newtonian silicone oil (See

Appendix A). The viscoelastic model liquids sbowed botb a sbear

thinning behaviour and nonnal stress differences. He used a Couette

device for his experiments, with a gap width of 6 mm. The data were

presented in a plot of l/rcrit vs i. The experiments were done with

drop sizes between 0.8 < D < 3.4 mm. Wall effects due to too large

droplets will tbus be present. For these reasons bis results are

very inconclusive.

73

74

Fig. 4.1.b

shear rate ( 1/s)

Viscoelastic drop breakup results obtained by Tavgac (1972). Critical capillary number vs. shear rate. Drop phase: 0.4% Poly Acryl Amide

Prabodh and Stroeve have performed experiments with a viscoelastic

drop phase (PAA in Corn Syrup) in Newtonian silicone oils (See

Appendix A). They used a cone and plate device allowing shear rates

up to 400 s-1, and determined the shear rate at which a droplet of

known initial size broke up. The viscoelastic model liquid they

used however probably degraded when subjected to shear rates above

60 s-1. They observed that some drops would not break-up during

shear and would become greatly extended. These droplets sometimes

did break-up when the shear rate was suddenly decreased to zero.

They have presented their experimental data in a plot of Ocrit vs 7

for various continuous phase viscosities and in a plot of Ocrit vs

10~.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~--.

" •

1 ; . .

Fig. 4.Lc

+ " + + ....

++* + ++

+ ++- +

shear rate [ 1 /s]

Viscoelastic drop breakup results obtained by Tavgac {1972). Critical capillary number vs. shear rate. Drop phase: 0.75% Poly Acryl Amide

À for various critical shear rates. From these two plots the

experimental results could be traced. From their data Prabodh and

Stroeve concluded that below a viscosity ratio of 0.5 the droplet

elasticity has a stabilizing effect but above this viscosity ratio

the viscoelastic drops are less stable than Newtonian drops. We do

not agree with that conclusion since our Newtonian drop break-up

data slightly differ from their data. At all viscosity ratio's the

capillary numbers for the viscoelastic drops are above the ones

obtained by ourselves for Newtonian droplets (chapter 3 of these

thesis).

75

76

.

Fig. 4.1.d

+ +

+

+* ++ ++++

shear rate [ 1 /s]

Viscoelastic drop breakup results obt:ained by Tavgac (1972). Critical capillary number vs. shear rate. Drop phase: 1.0% Poly Acryl Amide

A detailed analysis of their drop break-up data (see Appendix A)

throws up many questions. They state that for systems with a

viscosity ratio À < 1 the critical capillary number varies almost

linearly with the shear rate (Ocrit - Gonst. ~). This statement was

found to be based on observations of the break-up of equally sized

droplets: e.g. droplets with a radius of 21 µm dispersed in the

same continuous phase were found to break-up at shear rates varying

from 40 to 265 s-1. In our opinion this is either due to an

enormous scatter in their results or, more likely, due to the fact

that these data refer to observations of drop break-up after

reducing the shear rate to zero. At any rate these statements are

10 •

1 Il)

~ 1 -

e 8-..,

" -"

0.1 0.1

Fig. 4.1.e

++ +

+ ++

shear rate [ 1 /s)

10

Viscoelastic drop breakup results obtained by Tavgac (1972). Critical capillary number vs. shear rate. Drop phase: 2.0% Poly Acryl Amide

misleading. For exactly the same reason their conclusion that the

capillary number increases with increasing first normal stress

difference for the drop phase, evaluated at the imposed critical

shear rate in the continuous phase, is misleading.

The data obtained by Prabodh and Stroeve do not give a broad view

on the effect of elasticity of the droplet phase on droplet break­

up since the stress ratio SR only varies between 1.3 and 2.0. Their

results show a stabilizing effect of the droplet elasticity.

100

77

78

1-

î fl)

~ ' -f . 8- . -tJ

Fig. 4.Lf

+

" +

+

shear rate [1/s]

Viscoelastic drop breakup results obtained by Tavgac (1972). Critical capillary number vs. shear rate. Drop phase: 2. 6% l'oly Acryl Amide

Mirmohammad-Sadeghi (1983) has performed experiments with a

viscoelastic drop phase (P.A.A. in Corn Syrup) in Newtonian

silicone oils (See Appendix A). He used the cone and plate device

which was also used by Prabodh and Stroeve. The experimental

procedure used by Mirmohammad-Sadeghi was as follows. A very dilute

but coarse emulsion was brought into the device, a slow shear rate

ramp was applied (ramp up times of 0.75 s, plateau stages of

5-120 s and ramp down times of 0-60 s). Afterwards the maximum drop

size in the emulsion was determined microscopically. The ramp up

and ramp down times were chosen such that they did not affect the

resulting drop sizes. Mirmohammad-Sadeghi observed that

1 ....

Fig. 4.1.g

+

l-+ ++

shear ra te ( 1/s]

Viscoelastic drop breakup results obtained by Tavgac (1972). Critical capillary number vs. shear rate. Drop phase: 1.0% H.E.C.

viscoelastic droplets are stretched out to greater length prior to

break-up as compared with Newtonian droplets.

The experimental data were presented in a plot of 0 vs 7 for

various drop phases with increasing polymer concentration. From

this plot the experimental results could be traced. The data were

also plotted as Ocrit vs Ocrit • SR. The stress ratios varied with

polymer concentration from 0.05 to 1. All viscoelastic drop break­

up experiments were performed at similar viscosity ratios (0.55 <À

< 0.77). From the data it was concluded that the critical capillary

number was not constant for each droplet phase, but varied with the

79

80

10

'" " "

~ . fl)

iè 1 -e 8--b .

Fig. 4.1.h

++ ++ +

•+ ++ +

shear rate [ 1 /s)

Viscoelastic drop breakup results obtained by Tavgac (1972). Critical capillary nwnber vs. shear rate . Drop phase: 1.5% Kelzan

shear rate. It was also concluded that at low shear rates droplet

elasticity destabilizes, but the opposite is true at high shear

rates.

Several question marks should be put here. The critical drop sizes

determined are sometimes as large as 300 µm (diameter) where as the

maximum width of the gap between cone and plate is about 450 µm.

These data are consequently obtained in inhomogeneous shear flows

since the droplet diameter should ideally be smaller then 1/10 of

the gap width to avoid wall effects. This means that many data are

thus unreliable. Only the data obtained at very low shear rates (<

10 s-1) show critical capillary numbers which are significantly

below the corresponding Newtonian values. Especially these data

correspond to far too large droplets. Consequently the conclusion

of the destabilizing effect of droplet elasticity is not correct.

4.3.3 Conclusions

Very few theoretical approaches to the break-up of viscoelastic

drops in simple shear flows are available. More extensive studies

are available for the break-up of long cylindrically shaped

droplets in axisymetric non uniform shear flows.

Several experimental studies on the break-up of viscoelastic drops

have been reported. These studies, however, are inconclusive due to

limited observations, experimental errors, and coupling of the

effects of shear rate dependent viscosity and droplet elasticity.

Contrary to the conclusions of some authors, the available data

indicate that droplet elasticity has a stabilizing effect on break­

up though it may accelerate the initial growth of disturbances.

4.4 BREAK-UP OF INELASTIC, SHEAR-THINNING DROPS

4.4.1 Model liguids

Our investigations are aimed at characterizing the effects of non­

Newtonian droplet rheology on the break-up of droplets in simple

shear flows. There are many types of non-Newtonian behaviour but we

have selected two typical behaviours: liquids that are shear

thinning (a strong dependence on the shear-rate of the viscosity)

but exhibit very slight normal stress differences (negligible

elasticity) and liquids that have a nearly constant viscosity

(shear rate independent) but exhibit pronounced elasticity at

increasing shear rates (large normal stress differences). These

liquids were chosen to discern between the effects of two

frequently encountered non-Newtonian phenomena: shear rate

dependent viscosities and normal stress differences. In the

81

82

previous section on available literature it was shown that these

effects are usually coupled. In this chapter the break-up of small

inelastic shear thinning droplets obeying Eq. [4-4] will be

discussed. As derived by dimensional analysis of the basic

equations in chapter 2 of this thesis the break-up behaviour of

such droplets in a Newtonian continuous phase can in most relevant

cases be described by just three dimensionless groups:

11 'Î R

the capillary number: (l -

_c __

n-1 Ki_

- an apparent viscosity ratio: A a

- the power law index: n

,., c

The break-up of viscoelastic droplets will be discussed in section

4.5.

To study the inelastic shear-thinning behaviour, model liquids for

the drop phase had to be developed exhibiting this behaviour in the

shear rate range 1-100 s-1, being the range applied in the Couette

device. Following Acharya et al. (1976) solutions of polyacrylic

acid were used, which are commercially available as Carbopol (ex

B.F. Goodrich). At neutral pH value these Carbopol solutions built

up a gel structure but at low or high pH value no gel structure is

formed and they then exhibit strong shear-thinning behaviour

combined with low elasticity. Carbopol 941 was used because this

type gives very high viscosity values. Four different types of

these solutions were used:

1. Solutions of Garbopol 941 in water at various concentrations

(1.0 wt%, 0.5 wt% and 0.25 wt%). These solutions had a low pH.

2. Solutions of Carbopol 941 in an aqueous solution of sodium

hydroxide (1.5 wt%) at various concentrations. (0.75 wt%, 0.64

wt%, 0.5 wt% and 0.35 wt%) These solutions had a high pH.

3. Solutions of Garbopol 941 in a mixture of glycerol (80 wt%) and

water (20 wt%) at various concentrations (1.0 wt%, 0.5 wt% and

0.25 wt%).

4. Solutions of Carbopol 941 in a mixture of corn syrup (type Globe

01170 ex CPC Netherlands BV) (± 50 wt%) and water (± 50 wt%) at

various concentrations (1.0 wt% and 0.75 wt%).

All solutions were made in batches of 200 g by stirring the

solutions for one hour at moderate agitation. To prevent

bacteriological deterioration pentachlorophenol was added.

In all experiments the continuous phase was a Newtonian liquid.

Silicone oils (type Rhodorsil 47 ex Rhone-Poulenc Chimie Fine) were

used. Different viscosities were obtained by blending a high and

low viscosity batch.

The rheological properties of the liquids were measured with a

Haake type CVlOO viscometer using a concentric cylinder geometry

(type ZC 15) and an Instron rotary rheometer model 3250 using a

cone and plate geometry (diameter 60 mm, cone angle 2.4°). The

apparatuses were thermostatted at 23°C. The shear rate range was

from about 1-80 s·l. With the Haake, one measures apparent

viscosities as a function of shear rate, while the Instron measures

simultaneously tangential and normal farces (i.e. both viscosity

and normal stress differences). The elastic properties of the

droplet phases were characterized by normal stress differences in

steady shear experiments since these (large) deformations are more

similar to the deformations occuring in the droplet during droplet

deformation and break-up than the (small) deformations occuring in

oscillatory shear experiments. The rheological measurements were

fitted to the material functions given in Eq. [4-6]. The

rheological properties of the model liquids are tabulated in Table

4-1. For some drop phases normal stress differences could be

83

84

measured and these liquids are thus not strictly inelastic. However

for all experiments the ratio of the normal stress difference to

the tangential stresses was always less than unity. In section 4.5

it will be shown that the effect of droplet elasticity on drop

break-up is then negligible. Where no values are given for the

normal stress difference measurements they were below the accuracy

limit of the instrument. The apparent viscosities of the shear

thinning drop phases are shown in Fig. 4.2. The rheological

properties of the drop phases did not change signif icantly in the

time over which the experiments were performed (less than 5%)

Two different methods have been applied to measure the interfacial

tension between the liquids used;

a) The Wilhelmy plate method, where a silver plate with

circumference of 0.06 m is drawn from one liquid into the other

and the excess force, corrected for gravity effects, gives

directly the interfacial tension. This method is not suitable

for very low interfacial tensions and can have an accuracy of

0.1%.

b) The sessile drop method, which is based on the drop deformation

under gravity forces. After measuring height and width of the

droplets the interfacial tensions can be calculated with the aid

of the tables given by Bashforth and Adams (1883). An accuracy

is claimed of 0.1%. High interfacial tensions, however, are

difficult to measure.

All interfacial tensions were measured with respect to a silicone

oil with a viscosity of 1 Pas as being representative for all

silicone oils used. This because the surface tension of these oils

against air is found to be independent of the viscosity.

Interfacial tension measurements are very difficult for shear­

thinning liquids since they show very high viscosities at the very

low shear rates occuring during these measurements. Reproducible

measurements, however, were obtained when the Carbopol solutions

were first subjected to very high shear rates (- 104 s·l). This

shear treatment strongly reduced the apparent viscosities, thus

allowing interfacial tension measurements, and, hopefully, did not

greatly affect the interfacial tension. The measurements are

tabulated in Table 4.1.

4.4.2 Drop break-up experiments

The measurements of deformation and break-up of individual droplets

were performed in a Couette device with two counter-rotating

cylinders. The flow in this unit approximates simple shear flow.

The apparatus and the adopted procedure have been fully described

in chapter 3 of this thesis. The experimental results are tabulated

in Appendix B. Only these results are given where fracture of

droplets was observed. Tipstreaming (see chapter 6 of this thesis)

was also observed for the solutions of Carbopol in glycerol/water.

Tipstreaming was always found to occur at capillary numbers in the

range 0.4 < 0 < 0.8, but these data are not given in Appendix B.

For the various types of shear-thinning drop phases the drop break­

up data are plotted in a dimensionless form in Fig. 4.3. In these

figures the critical capillary n1.llllber at drop break-up, 0, is

plotted versus the viscosity ratio Àa· As a reference a fit through

the Newtonian drop break-up data (chapter 3 of this thesis) is also

plotted (curve). This function is given by

log Ocrit

in which C1

Cz

C3

C4

c 3

C + C log À + 1 2 log À + C

1.560

0.733

1.135

0.967

[4-17]

4

85

TAB LE 4.1 Model liquids used for shear-thinning drop break-up

Dro:elet ;:ehase~

Nr. Description K n x m

[Pa.sn) [-] [Pa.sm) [-) [mNn- 1 J

ST.D.1 C941.Hz0 1.0% 10 0.25 0. 75 26.5

ST.D.2 C941.llz0 0.5% 4.8 0.29 0.5 0. 75 28

ST.D.3 C941.Hz0 0.25% 1. 95 0.34 29

ST.D.4 C941.HzO.NaOH 0. 75% 4. 0 0.42 1. 5 0.55 29

ST.0.5 C941.HzO.NaOH 0.64% 1.9 0. 44 0.55 29.5

ST.D.6 C941.HzO.NaOH 0. 5% 0 .64 0.45 0.5 0.55 30

ST.D.7 C941.llzO.NaOR 0.35% 0.35 0. 48 30.5

ST .D.8 C94l.Glyc.Hz0 1.0% 3. 7 0. 44 2.4 0.62 21

ST.D.9 C941.Glyc.ll2o o.5% 0.95 0.61 0. 4 0. 7 23

ST .D.10 C941.Glyc,H20 0.25% 0.33 0 .65 24

ST.D.11 C941.CS 50.l!zO 0. 75% 20 0.42 0.3 0.9 38

ST.D.12 C941.CS 49.HzO 1.0% 29 0.47 1. 7 0.8 35

ST .D.13 C941.CS 48.HzO 0. 75% 30 0. 48 0.3 1.0 35

Continuous phases

Nr. Desc.ription K n x m [Pa.sn) [-) [Pa.sm] [-J

ST.C. l Silicone Oil 0. 9

ST.C.2 Silicone Oil 2.6

ST.C.3 Silicone Oil 5.67

ST.C.4 Silicone Oil 6.3

ST.C.5 Silicone Oil 16.0

ST.C.8 Silicone Oil 40 .0

86

. ëi5 0 ü (j)

> Î

Shear thinning liquid Carbopol 94 î .H20

-+- 1.0% - -A- 0.5% -e- 0.25%

îEî

! 1 " i

1

îEO ~ ~

~ 1 r 1

î r

t

shear rate ( î /sec)

Fig. 4.2.a Rheological properties of shear thinning model liquids. l.0%, 0.5% and 0.25% Carbopol 941 in water

87

(ij cü û. ___..

::>-, +-' (j)

0 ü (j)

s

88

îEî

Î

Î 1

Shear thinning liquid Carbopol 94 î .NaOH

-+-- 0.64% --è.- 0.5% -e-- 0.35%

~ t

~ ~~ 1

~ &~ 1-

~ r G-..

' ',,& ~ r

'&... ........... ' "A-'s.. -

! ' -1:.. " f- '-s._ &.

~ ' ~il. t 's..

'G--s._

1E-J 0

1 i 1 j Il 1 1 j 1 !

Î îO îOO

shear rate ( î /sec)

Fig. 4.2.b Rheological properties of shear thinning model liquids. 0.64%, 0.5% and 0.35% Carbopol 941 in water/NaOH

>, +-' (j)

0 ü (j)

>

1Eî

1EO

1E-î

1E-2

Shear thinning liquid Carbopol 94 î .H20/glycerol

-!- î.0% --6.- 0.5% -e- 0.25%

1 10 100

shear rate ( î /sec)

Fig. 4.2.c Rheological properties of shear thinning model liquids. 1.0%, 0.5% and 0.25% Carbopol 941 in water/glycerol

89

-(j)

cv Q_ ..._...

:>-, +-' (j)

0 u (j)

>

90

100

10

1

0. Î

Shear thinning liquid Carbopol 94 î .H20/cornsyrup ---+- 0.75% - -&- 1.0% -e- - 0.75%

50%cs 49%cs 48%cs

~

' ' '" ~'..__ ' ' ''!è.

'o

1 10 100 1000

shear ra te ( 1 /sec)

Fig. 4.2.d Rheological properties of shear thinning model liquids. 0.75%, l.0% and 0.75% Carbopol 941 in water/corn syrup (50%/49%/48% corn syrup respectively)

àï 10

~ c » .._ Jg lï !U 0

1

I t:. J

/

viscosity ratio

+ 1.0% A 0.5% 0 0.25%

Fig. 4.3.a Drop breakup results for various shear thinning drop phases. 1.0%, 0.5% and 0.25% Carbopol 941 in water

To calculate an apparent viscosity ratio Àa, the apparent droplet

viscosity was taken to be that at the critica! shear rate icrit

(applied to the continuous phase at the moment of drop break-up)

n-1 K •

À [4-18] a '7

c

91

92

viscosity ratio

+ 0.75% A 0.64% 0 0.50% + 0.35%

Fig. 4.3.b Drop breakup results for various shear thinning drop phases. 0.75%, 0.64%, 0.5% and 0.35% Carbopol 941 in waterfNaOH

4.4.3 Discussion

The solutions of Carbopol in water and in the aqueous solution of

hydroxide usually showed normal drop break-up, while the solutions

of Carbopol in water/glycerol mixtures showed almost always

tipstreaming. The reason for this has not been studied, but it may

be related to the presence of surface active materials (see chapter

6 of this thesis), which is indicated by the rather low interfacial

tension. The solutions of Carbopol in water/corn syrup mixtures

were intended to extend the results to higher viscosity ratios.

--+-- 0.75% 50%cs

viscosity ratio

~ 1.0% 49%cs

~~ + + +

0 0.75% 48%cs

Fig. 4.3.c Drop breakup results for various shear thinning drop phases. 0.75%, 1.0% and 0.75% Carbopol 941 in water/corn syrup (50%/49%/48% corn syrup respectively)

The experimental results show the following:

1. There is a reasonably good correlation between the measured

critica! capillary number 0 and the apparent viscosity ratio

calculated with Eq. [4-18].

2. The scatter in the shear-thinning break-up data is larger than

the scatter in the Newtonian break-up data (chapter 3 of this

thesis).

3. At low viscosity ratio's the critica! capillary number varies

approximately by O - Àa-2/3, being the theoretically predicted

dependence for Newtonian droplets. 93

94

4. The value of the experimentally determined critical capillary

numbers are systematically higher f or the shear thinning

droplets than for the Newtonian droplets.

5. The deformation of the shear-thinning droplets prior to break-up

was generally somewhat larger than for the Newtonian droplets.

The mode of break-up, however, was very similar.

6. The critica! capillary number for apparent viscosity ratio's

above 0.5 increases rapidly.

The first observation is somewhat remarkable since from dimensional

analysis (chapter 2 of this thesis), for ideally inelastic shear­

thinning liquids one predicts that the critical capillary number is

a function of two dimensionless groups: the apparent viscosity

ratio Àa and the power law index n. For these model liquids the

power law index varied between 0.25 ~ n ~ 0.48. On closer

examination, however, there appears to be a slight tendency for the

lower power law index drop phases to have a somewhat higher

capillary number.

The second observation is partly misleading because in the

graphical representation of the Newtonian drop break-up data the

critical capillary numbers obtained for a given combination of a

droplet and continuous phase were averaged. For shear thinning

droplets this is not allowed since the apparent viscosity ratio is

not constant for one combination of liquids.

The fourth observation cannot only be interpreted by a higher shear

rate at which drop break-up occurs, but can also be interpreted by

an apparent viscosity ratio which is too high (i.e. the data points

have been shifted to the right). This interpretation implies the

presence of effective shear rates within the droplet, that are

higher than those applied to the continuous phase. To obtain an

empirical correlation for these break-up results we have taken the

ratio of the internal shear rate ~int and the applied shear rate 'l

to be constant:

'lint - C ~ (4-19)

The modified viscosity ratio is thus given by

n-1 n-1 K .

"int K -Y

n-1 n-1 À c c À [4-20] ml '1 '1 a

c d

The constant C has been determined by statistica! analysis of the

break-up data of the solutions of Carbopol in water and in the

aqueous solution of sodium hydroxide. A least squares method for

non-linear functions based on a Marquardt algorithm was used. The

best fit was obtained for

c - 5.1 [4-21]

The thus calculated modified viscosities are tabulated in Appendix

B and the results are graphically presented in Fig. 4.4 for all

drop phases. Below a modified viscosity ratio of 0.1 the shear

thinning drop break-up results can be well described by the

Newtonian drop break-up criteria, provided the modified viscosity

ratio is used. Above this modified viscosity there appears to be a

discrepancy between the shear thinning drop break-up data and the

Newtonian drop break-up criteria. This discrepancy is even more

pronounced for the results with the solutions of Carbopol in the

water/corn syrup mixtures.

For spherical Newtonian droplets the flow inside the droplet can be

obtained analytically with the method described by Taylor {1932).

This analysis was done by Bartok and Mason. For a droplet placed

in a simple shear flow with the velocity components

u -y y

v-o

w - 0

[4-22a]

[4-22b]

[4-22c]

95

96

they derived the following internal velocity field.

2 2 2 2 (x + y + z ) x

u = "I 5 - 4-+ (2>. - 1) 4 (À + 1) 2 2

R R

[4-23a]

2 2 2 2 x (x + y + z )

4 L_ + v = "I 5 (2.x + 5) 4 (À + 1) 2 2

R R

[4-23b]

z xy w = "I 4-

4 (À + 1) 2 [4-23a]

R

To f ind the internal shear rates one should look at the rate of

strain tensor ~. which is defined as ~ = ~ (1 + in which 1 is

the velocity gradient tensor defined as 1 = 8y/84.

2 2 2 2 xy Sx + 8 y + 5 z 3zy

-3 2 2 2

R R R

2 2 2 "/ 8 x + 8y + 5 z 2 xy 3zx

~ = -3 --4 (). + 1) 2 2 2

R R R

3 zy 3 zx 4 xy

2 2 2 R R R

(4.24]

Generally the flow in the droplet is thus not a simple shear flow.

Only for some special positions in the droplet the flow reduces to

simple shear. The maximum rate of shear in a simple shear flow is

• 5 • "( =--y int 2

[4·25]

which is obtained for: z = o, y = o, x = ± R and À = o. Generally

the strength of a flow field is given by the norm of the rate of

strain tensor: IRI. The norm will be defined by

IRI = )2 R !l.1

[4-26]

For simple shear flow the norm thus defined is equal to the shear

rate. The strength of the internal flow field varies in the

droplet. As a measure of this strength the maximum norm in the

droplet was chosen. The maximum in the norm can be found by

standard analytica! techniques to find the extremes of a function

under a constraint. After tedious calculation it can be shown that

the global maximum of the norm of the velocity gradient tensor of

the internal flow field is, for all viscosity ratios, obtained for

x - ± l:t !2îl. y ± l:t /2îl and z - 0 [4-27]

The maximum norm is given by

1121 max

- .:y (Hl)

J7 [4-28]

Elmendorp (1986) has also given an analysis of the rate of shear in

ellipsoidal drops. In the limit of spherical droplets, however, his

analysis predicts zero rate of shear since his analysis assumes

rigid rotation for spherical droplets. Especially for low viscosity

ratios this assumption is wrong.

97

98

The maximum norm of the velocity gradient tensor will be used to

calculate a theoretically modified viscosity ratio Àm2 by taking

the droplet viscosity at the shear rate given by 111max:

À m2

By

is

K ( 111 )

combining

obtained:

n

max

I'/ c

Eq.

- ---1 - n

À + m2

n-1

[4-29)

[4-28] and [4-29] an implicit expression for Àm2

1

[:cl 1 ----

1 - n 1 n À "Î J7 ~ 0 [4-30]

m2

The value of Àm2 was numerically calculated for each break-up

experiment with a standard Newton-Raphson zero point technique.

The results are tabulated in Appendix B and are graphically

presented in Fig. 4.4. From these results it will be clear that

when Ocrit is plotted versus Àm2 there is much better

correspondence to the Newtonian break-up criteria than when the

apparent viscosity ratio Àa is used. For viscosity ratios close to

unity the description in terms of Àm2 is also in better agreement

than the description in terms of Àml· This is due to the prediction

of Eq. [4-13] that for high viscosity ratio 1 s (À> J7 - 1 ~ 1.6)

the maximum norm of the velocity gradient tensor becomes smaller

than the applied shear rate.

The theoretically obtained shear rates in spherical Newtonian

droplets are lower than the empirically determined internal shear

rate based on the best fitting modified viscosity ratio. This is

probably due to the shear thinning characteristics of the droplet

phases. For shear thinning liquids it is well established (e.g.

Crochet et al. 1984) that the velocity changes are restricted to a

smaller distance than for Newtonian liquids, resulting in higher

(ij 10 _Q

E 2

0

viscosity ratio

Fig. 4.4.a Drop breakup results for shear thinning drop phases plotted for Apparent viscosit:y ratio À•

maximum shear rates. It is thus very likely that the shear rate in

the droplet near the interface, which is probably the most relevant

shear rate, is higher for a shear thinning droplet than for a

Newtonian droplet.

Although the analysis of the flow within the droplet may offer an

explanation for the shear rate at which break-up occurs it has not

explained why break-up occurs at larger critical drop deforma­

tions. This effect might be due to the elongational viscosity of

the drop phase. Many non-Newtonian liquids are known to have a

ratio of the elongational to simple shear viscosity that is (much)

larger than 3 as it is for Newtonian liquids. During the final

stages of pinch-off and break-up, the elongational component of the

99

100

viscosity ratio

Fig. 4.4.b Drop breakup results for shear thinning drop phases plotted for Modified viscosity ratio Àm1

flow in the droplet will become very important. When the

elongational viscosity is much higher, larger drop deformations

prior to break-up are thus expected.

4.4.4 Gonclusions

Solutions of Garbopol 941 in water and in aqueous solutions of

sodium hydroxide are appropriate model liquids exhibiting strong

shear thinning behaviour in the shear rate range 1-100 combined

with very low normal stress differences.

For viscosity ratios below 0.1 the critica! capillary number at

which drop break-up occurs for shear-thinning droplets can be

correlated by the Newtonian break-up criterion (Eq. [4-17))

viscosity ratio

Fig. 4.4.c Drop breakup results for shear thinning drop phases plotted for Modified viscosity ratio À""'

provided a modified viscosity ratio Àml (Eq. [4-20]) is used to

account for the internal shear rate in the droplet, which is higher

than the applied shear rate.

For viscosity ratios of the order of unity the critical capillary

number at which break-up occurs rises with increasing viscosity

ratio. This rise starts for shear thinning droplets at lower

viscosity ratios than for Newtonian droplets. This is probably due

to the fact that for viscosity ratios greater than unity the

internal shear rate becomes considerably lower than the applied

shear rate.

101

102

4.5 BREAK-UP OF VISCOELASTIC DROPS

4.5.l Model liguids

As discussed in the review of the literature on non-Newtonian drop

break-up, there are only a few studies available on the break-up of

viscoelastic drops in simple shear flows. None of them is very

conclusive. The current studies were aimed at discerning the

effects on droplet break-up in simple shear flows of two frequently

encountered types of non-Newtonian behaviour of the droplet phase:

shear rate dependent viscosities and normal stress differences. The

first study was discussed in section 4.4. In this section a study

of viscoelastic droplet break-up will be discussed in which model

liquids for the droplet phases were used that exhibit hardly any

shear rate dependency for the viscosity combined with considerable

normal stress differences (obeying Eq. [4-6]). As derived by

dimensional analysis of the basic equations in chapter 2 of this

thesis, the break-up behaviour of such droplets in a Newtonian

continuous phase can in most relevant cases be described by five

dimensionless groups:

'7 'Î R

- the capillary number: 0 _c __

a

- an apparent viscosity ratio: À

- the power law index: n

- the elastic power law index: m

- the stress ratio: S R

a

n-1 K...i.._

'7 c

Our present studies are aimed at further reducing this number by

taking n = 1 and m is approximately equal for all droplet phases.

For such model liquids the choice of the shear rate at which the

apparent viscosity ratio should be determined is much less

ambiguous than it was for the inelastic shear thinning droplets.

Hence only three dimensionless groups will be left n, Àa and SR.

For the study of viscoelastic drop break-up, model liquids for the

drop phases had to be developed that show high normal stress

differences combined with shear rate independent viscosities in the

shear rate range 1-100 s·l, being the range applied in the Couette

device, Such liquids are called Boger-liquids since Boger (1977,

1978) was the first to report such liquids. Boger suggested the use

of small amounts of the polyacrylamide Separan (ex Dow Chemical

Company) in corn syrup. Choplin et al. (1983) prefered the use of

solutions of the polyacrylamide Pusher (ex Dow Chemica! Company) in

mixtures of glycerine and water. They reported several

disadvantages of the separan/corn syrup mixtures: flocculation of

the polymer, crystallization of the syrup and irreproducable

results from batch to batch. In the studies of shear thinning drop

break-up, however, glycerine solutions resulted in tipstreaming,

which is outside the scope of the present studies. We circumvented

all these problems by using small amounts of Pusher in water/corn

syrup mixtures. The viscosity of the water/corn syrup mixture was

adjusted by adding various amounts of water to a corn syrup (type

Globe 01170 ex CPC Netherlands B.V.). The elasticity of the model

liquids was varied by adding various amounts of polyacrylamide

(type Pusher 700 ex Dow Chemical Company). To prevent bacteriologi­

cal deterioration sodium azide (NaN3) was added at 0.02 wt%. The

solutions were made homogeneous, after adding the polyacrylamide to

the corn syrup/water mixture, by alternately stirring very gently

and stopping. This procedure was continued for one week.

103

104

In all experiments the continuous phase was a Newtonian liquid.

silicone oils (type Rhodorsil 47 ex Rhone Poulenc Chimie Fine) were

used to have little problems with traces of surface active

materials. Different viscosities were obtained by blending a high

and a low viscosity batch.

The rheological characterization was performed in the same way as

described in section 4.4.l for the shear thinning drop break-up

experiments. To measure the high normal stress differences a

careful and systematic rheometric method was applied. Since it is

known that real overshoot behaviour (Lockyer and Walters, 1976) can

be encountered and that the waiting time between two measurements

with one sample affects this overshoot behaviour (Stratton and

Butcher, 1973) the following approach was adopted. After a sample

was introduced between the cone and plate the sample was

periodically subjected to 60 s rest and 60 s steady shear with

shear rates increasing per step. After 60 s of steady shear the

overshoot was generally levelled off (See also Michele, 1978} and

hence the readings were taken after 60 s of steady simple shear

flow. The rheological properties of the model liquids were found

to be very stable in time, provided they were made according to the

previously described procedure. They are tabulated in Table 4.2,

and graphically represented in Fig. 4.5. The smaller the water

content of the corn syrup/water mixture and the lower the

polyacrylamide level, the more the model liquid has a shear rate

independent viscosity. When 0.5% polyacrylamide is added the

viscosity is clearly shear rate dependent. The elastic power law

index m varies between 1.2 and 1.7 but does not systematically vary

with the corn syrup/water mixture viscosity or the polyacrylamide

level.

The interfacial properties were measured with the Wilhelmy plate

method as described in section 4.4.1.

TAB LE 4.2 Model liquids used for viscoelastic drop break-up

Droplet phases

Nr. Desoription K n " m " [Pa.sn] [-) [Pa.sm] [-] [mNm-1 l

VE.D.l 75. 25/ 0.12 29

VE.D.2 /0.05% 0.175 0.01 1.47 29

VE.D.3 /0 .1% 0,29 0.95 0.06 l.57 29

VE.D.4 /0.2% 0.85 0 .84 0.6 1.4 28

VE.D.5 /0,5% 4 .15 0.65 8 1.2 27

VE.D.6 BO. 20/ 0.29 l 31

VE.D.7 /0. 05% 0.53 0.88 0.19 1.48 31

VE.D.8 /0 .1% 0. 78 1 0.51 1.55 31

VE.0.9 /0.2% 1.67 0,84 2.1 l.47 30

VE.D.10 /0. 5% 5.5 0.64 16 1.19 29

VE.D.11 85.15/ o. 78 l 33

VE.D.12 /0. 05% 1.15 1.05 0.16 1.37 33

VE.D.13 /0.1% 1.2 1.05 0.48 1. 40 33

VE.D.14 /0.2% 2 0.99 1.2 1. 57 32

VE.D.15 /0. 5% ll. 7 0.67 55 1.21 31

VE.D.16 90.10/ 2.8 1 35

VE.D.17 /0.1% 4.1 0.98 l. l 1. 58 35

VE.D.18 /0 .2% 8.15 0.90 12 1.40 34

VE.D.19 /0.4% 5.6 0.98 3.9 1. 75 33

Continuous phases

Nr. Description K n " m

rea .• n1 [-) [Pa.sm) [-)

VE.C.1 Silicone 011 0.9

VE.C.2 Silicone Oil 6.0

VE.C.3 Silicone Oi l 11.36

VE.C.4 Silicone Oil 26.60

VE.C.5 Silicone OH 38.11

VE.C.6 Silicone Oil 43.2

VE.C.7 Silicone Oil 64.6

105

106

100

10

viscoelastic 1iquid PAA in 75/25 CS/H20

shear rate ( î /sec}

···+·· visc. 0.05%

·A" visc. 0.1%

O·· visc. 0.2%

+ ·· visc 0.5%

--.-.. norm. 0.05%

-it- norm. 0.1%

--+--norm. 0.2%

-6.- norm. 0.5%

Fig. 4.5.a Rheological properties of viscoelastic model liquids. 0.5%, 0.2%, 0.1% and 0.05% in corn syrup (75%)/water mixture

-ro fb ({) ({) (])

.P ({)

ro E '-0 c

viscoelastic liquid PAA in 80/20 CS/H20

1000 ....------

100

10

."+" visc. 0.05%

""b. · visc. 0.1%

··O··· visc. 0.2%

".+·· visc 0.5%

_,._norm. 0.05%

-e- norm. 0.1%

norm. 0.2%

-±.- norm. 0.5%

100 1000

shear rate ( î /sec)

Fig. 4.5.b Rheological properties of viscoelastic model liquids. 0.5%, 0.2%, 0.1% and 0.05% in corn syrup (80%)/water mixture

107

108

~ ~ (f) (f) <l.J !o... +-' (f)

ro E !o...

0 c -

viscoelastic lîquid PAA in 85/ î 5 CS/H20

1

0.1 '--1-'-'-u.u.M.~..J....J..lfl!tl 1 1!1111!1

' " "...1

···+·· visc. 0.05%

···/::.· · visc. 0.1%

O·-- visc. 0.2%

··+·· visc 0.5%

_,.,._norm. 0.05%

-.-norm. 0.1%

-+-norm. 0.2%

--A- norm. 0.5%

0.1 10 100 1000

shear rate ( î /sec)

Fig. 4.5.c Rheological properties of viscoelastic model liquids. 0.5%, 0.2%, 0.1% and 0.05% in corn syrup (85%)/water mixture

ro § 0 c -

viscoelastic liquid PAA in 90/10 CS/H20

shear rate ( î /sec)

".+" visc. 0.1%

""t;,.". visc. 0.2%

".O··· visc. 0.5%

".+··· visc 0.4%

-....-.. norm. 0.1%

_._norm. 0.2%

.........,__norm. 0.5%

-~- norm 0.4%

Fig. 4.5.d Rheological properties of viscoelastic model liquids. 0.5%, 0.2%, 0.1% and 0.05% in corn syrup (90%)/water mixture

109

110

4.5.2 Drop break-up experiments

The measurements of deformation and break-up of individual droplets

were performed in a Couette device with two counter-rotating

cylinders. The apparatus and the adopted procedure have been fully

described in chapter 3 of this thesis. In first instance drop

break-up experiments were performed with a wide range of combina­

tions of 19 droplet phases and 7 continuous phases.

The experimental results are tabulated in Appendix B, where the

apparent viscosity ratio Àa and the stress ratio SR are calculated

at the critical shear rate at which droplet break-up occured.

To compare these data with the behaviour of Newtonian droplets, the

critical capillary number at which break-up occured has been

plotted versus the apparent viscosity ratio in Fig. 4.6 together

with the 5 parameter fit through the Newtonian drop break-up data

(chapter 3 of this thesis).

log 0 crit

in which

C1 -0.506

C2 -0.0994

C3 0.124

C4 -0.115

C5 -0.611

c 2

C + C log À + C (log À) + 1 2 3 log À + C

5

[4-31]

For some droplet phases, especially those with a high corn syrup

level and a high polyacrylamide level it was observed that break-up

in stationary simple shear flow was impossible below a certain drop

radius. Therefore additional systematic experiments were performed

(j; 10 .0 E :J c

» '--

~ u (IJ u

viscosity ratio

Fig. 4.6.a Drop breakup results for viscoelastic model liquids.

Fig. 4.6.b

0.5%, 0.2%, 0.1% and 0.05% in corn syrup (75%)/water mixture used as drop phase

+ + •

viscosity ratio

Drop breakup results for viscoelastic model liquids. 0.5%, 0.2%, 0.1% and 0.05% in corn syrup (80%)/water mixture used as drop phase

1E1

111

,_ 10 Q)

.Q

§ c

>-,_ ~ ëi ro ü

112

viscosity ratio

Fig. 4.6.c Drop breakup results for viscoelastic model liquids. 0.5%, 0.2%, 0.1% and 0.05% in corn syrup (85%)/water mixture used as drop phase

+ +

viscosity ratio

Fig. 4.6.d Drop breakup results for viscoelastic model liquids. 0.5%, 0.2%, 0.1% and 0.05% in corn syrup (90%)/water mixture used as drop phase

to determine this critical drop radius for various combinations of

droplet and continuous phase. These results are also tabulated in

Appendix B. Some of the results have been visualized in Fig. 4.7.

4.5.3 Discussion

The experimental results on viscoelastic drop break-up show the

following features:

1. The critical capillary number at which break-up occurs does

clearly not only depend on the apparent viscosity ratio, as for

Newtonian droplets, but also on droplet elasticity (See Fig.

4.6).

2. The critical capillary numbers for the viscoelastic droplets are

higher (sometimes much higher) than for Newtonian droplets:

droplet elasticity clearly stabilizes the droplet against break­

up. No systematic observations were made of critical capillary

numbers below the Newtonian drop break-up measurements (See Fig.

4.6). This is contrary to (erroneous) conclusions drawn by

Tavgac (1972), Prabodh and Stroeve and Mirmohammed-Sadeghi.

3. The deformations of the viscoelastic droplets prior to break-up

are often much larger than of Newtonian droplets, especially for

the rather elastic droplets. Stable droplets or rather

threadlike objects with a length to width ratio well above 100

have been observed in steady simple shear flows.

4. Droplets with a large length to width ratio, which were stable

in steady simple shear flow, often were unstable when the flow

was stopped abruptly. Droplet break-up according to a Taylor

instability like mechanism occured, resulting in a large number

of equal sized fragments. Such observations however, have not

been included in the tabulated experimental results as drop

break-up measurements, since only droplet break-up in quasi

steady increasing shear flows was being studied.

113

114

Ê s -~ 0 "'O 1E-1 (tl i...

~

shear rate (1/sec)

• break~ observed

o no break~ observect

Fig. 4.7.a Determination of critical drop radius for viscoelastic drop phase VE.D.28 in continuous phase VE.C.35

1E-1

•

0

shear rate ( 1 /sec)

• break-'-'> observed

o no break-1.4' observed

Fig. 4. 7 .b Determination of critical drop radius for viscoelastic drop phase VE.D.28 in continuous phase VE.C.37

115

116

5. For a combination of a certain droplet and continuous phase a

series of measurements generally resulted in one point, or a

vertical series of points, in the plot of Ocrit vs Àa· This is

due to the almost shear rate independent droplet viscosity.

The critical capillary number for such a combination increases

for decreasing droplet sizes and hence increasing rate of shear.

Since m > 1 for all droplet phases, the critical capillary rises

also with increasing stress ratio.

6. The increase in the logarithm of the critical capillary number

seems to be exponential with the logarithm of the stress ratio

and it seerns to becorne larger with increasing viscosity ratio.

This observation was the basis of the function that was used to

correlàte the viscoelastic drop break-up data: Ocrit = f (Àa,

SR). For Newtonian drop break-up, SR o, this function was

taken to be equal to the 5 parameter fit through our Newtonian

drop break-up data (Eq. [4-31]. To account for the effect of non

zero stress ratio the sirnplest possible exponential function was

chosen that yields, for constant stress ratio, a critical

capillary nurnber that rises with increasing apparent viscosity

ratio and that yields for all non zero stress ratios critical

capillary nurnbers higher than for SR o

log Ocrit =log 00 ,crit + C6 exp ((C7 +Cg log À) 2•log SR]

(4-32]

The constants C6, C7 and Cs have been determined by statistical

analysis of the break-up data given in Appendix B. Only those

data in which droplet break-up was actually observed in

increasing quasi steady simple shear flow, were used. A least

square method for non linear functions based on a Marquardt

algorithrn was used. The best fit was obtained for

c6 0.121

C7 1.474

Cs 0.355

Visco-elastic drop break-up in Newtonian continuous phase

fitted functlan OM

1.16

0.66

0.17

-8.37 ll._J~J__JL_J_~j___J~J...._J~J_..J[___J_~1-~-io~.soïlö".....1.~.J..........1:....-...i............,.o.3!I

101110 (p ) 101110 (SrJ

-2.60 -L&a P -1.00

la 10 (a• • rit

Fig. 4.8 llrop breakup for viscoelastic drops as a function of viscosity ratio and stress ratio

'-(J)

.D E

118

:J c

• break-up observed

.. ··-· .. ..... . •••••••

stress ratio

Fig. 4.9.a Drop breakup results for viscoelastic drops as a function of the stress ratio for 0.003 < .l. < 0.01

This function is shown in Fig. 4.8. In Fig. 4.9 the break-up

data are shown together with the empirical correlation for

various ranges of the apparent viscosity ratio. The empirical

correlation is plotted for both the lower and the upper limit of

the apparent viscosity ratio range. Both the data at which

break-up was observed and those at which break-up was not

observed up to the maximum shear rate used are plotted in these

figures. From these figures it follows that the empirica!

correlation is a reasonable representation of the break-up

data.

• break-up observed

stress ratio

Fig. 4.9.b Drop breakup results for viscoelastic drops as a function of the stress ratio for 0.01 < À < 0.02

7. For the droplet phases with the highest polyacrylamide

concentration and for the droplet phases with the highest corn

syrup concentration critica! drop radii have been observed.

10

For these droplet phases the experimental data have been plotted

in Fig. 4.7 as droplet radius vs critica! shear rate at which

break-up occured. The solid circles denote actually observed

break-up conditions. The open circles denote that break-up did

not occur up to the given rate of shear. These figures show that

for the larger drop radii and the lower shear rates the critical

119

120

• break-up observed

0 no break-up observed

• • " .... -~~· . ••• ~rP-.-• -.... . -· •.. ·-------.-.--

Fig. 4.9.c

stress ratio

Drop breakup results for viscoelastic drops as a function of the stress ratio for 0. 02 < À < 0. 05

capillary number is constant (slope = -1) or rises slowly with

decreasing droplet size (or equivalently the capillary number

rises with increasing stress ratio). Ata certain drop size,

however, there appears to be an abrupt change in the behaviour

of the droplets. Even when rnuch higher shear rates are applied

to such droplets, break-up will not occur.

© _Q

E ::l c

• break-up observed

n no break-up observed

stress ratio

Fig. 4.9.d Drop breakup results for viscoelastic drops as a function of the stress ratio for 0.05 <À< 0.1

•

The drop radius at which this change occured were determined

from Fig. 4.7. The critica! drop sizes are tabulated in Table

4.5. The critica! drop sizes tend to increase with the apparent

viscosity ratio, but do not correlate at all with the stress

ratio (See Fig. 4.10). Such critica! drop sizes have been

predicted, on empirica! arguments by Tavgac (1972) (See section

4.3.2) but have to the author's knowledge not yet been observed

or explained theoretically. 121

L QJ

.D E :::J c

122

• break-·up 0 no break-up observed observed

10

/

/ ~ 0

/00 8Sl2> 0 CQ) 0

0 0 0 0 00

0

•o

stress ratio

Fig. 4.9.e Drop breakup results for viscoelastic drops as a function of the stress ratio for 0 .1 < >. < 0. 2

A very qualitative argument predicting critical drop radii is

given below. For Newtonian drops break-up will occur when the

viscous shear stress exerted by the continuous phase is larger

than the interf acial tension f orces or

fl c

7 > !!. or O R

11 'Î R c --->ü

a crit [4-33]

•

10

L (IJ Ll E ::J c

• b:eak-up observed

o no brea'-\-·up observed

/

/! 0

0 0

/. / 0 D

. / 0 CJ 0 0

!/ 0 ~/ 0 0

/. .·: . • z!-_ •

80 • • • . . ____ ...:;. . . . . . .. __.... .--.- .~ ... ·=.=:::=-~ ..... ~ ..... "- _ ....

0.1 0.1

Fig. 4.9.f

• •

stress ratio

Drop breakup results for viscoelastic drops as a function of the stress ratio for 0.2 < À < 0.5

.~

0

For viscosity ratios between 0.1 <À< 1, which is the range in

which most of the data on critical drop sizes were obtained, we

know that Ocrit = 0.5 0 0 ,crit· For viscoelastic drop break-up

the viscous shear stresses exerted by the continuous phase

should also overcome the elastic forces in the droplet, which

are given by

123

w .D E :::J c

124

• break-up observed

•

- l t . . 1 • l J ·~ 1

o no break-up observed

0

8 •

stress ratio

... 0 •

0 0

Fig. 4.9.g Drop breakup results for viscoelastic drops as a function of the stress ratio for 0.5 < À< 2

Let us assume that the viscous shear stresses exerted by the

continuous phase should simply be greater than the sum of the

interfacial forces and the elastic forces inside the droplet:

a m

" .y > n -+ c . /Ç t c o,crit R

[4-34a]

or equivalently

n 1 >

n 1 - Gonst. SR . À o,crit

[4-34b]

However, in some situations the right hand term in Eq. 4-34

increases more rapidly with i than the left hand term. This will

result in the inexistence of a solution. Break-up will thus

only be possible when

R>

0 (J

o,crit

m 'I i-C.iç,Y

c

The critical drop size

such that 'Ic .y - Gonst.

l

" m:T c

.y c

/Ç m

The critical drop size

R crit

·te /Ç":] " c

[4-35]

will be obtained when the shear rate is

iç .ym has a maximum:

[4-36]

is thus obtained for

0 (J

o, crit (4-37]

ntr [c": m] ~ - Const. K.

Although these arguments qualitatively predict critical drop

sizes, the predictions are quantitatively not-correct.

This must be due to the over simplification of Eq. [4-34].

Prabodh and Stroeve have done viscoelastic drop break-up

experiments using high rates of shear and small droplet sizes

but have not reported any phenomenon indicating the presence of

critical drop sizes. This is probably due to the rather low

elastic power law index m of the drop phase used by Prabodh and

Stroeve: m - 1.06 (See Appendix A). Critical drop sizes can

probably only be observed when the elastic forces in the droplet

increase more rapidly with the shear rate than the applied shear

stresses (m > 1).

125

1EO

E' ..s ({) :J 'ö ro ,_

0.. 1E-1

0 ,_ -0

(ij u

:;::;

·= ü

Fig. 4.10

126

• • + +

" 'Î"! 0 • + VED.5 • . ,.. t:. VED.10

+ •• + 0 VE.D.15

+ VE.D.17

+ • VED.18

• VE.D.19

1EO

viscosity ratio

Critical drop radii observed for viscoelastic drop breakup in quasi steady simple shear flow as a function of the viscosity ratio

4.5.4 Conclusions

Droplet elasticity results in higher critical capillary numbers, at

which break-up occurs, than for Newtonian droplets.

This increase rises with droplet elasticity and is stronger for

viscosity ratio's of order unity than for lower viscosity ratio's.

Droplet elasticity results in much larger droplet deformations

prior to break-up. These long cylindrical drops can be stable in

steady shear flow but break up after cessation of the flow.

Critical drop sizes have been observed below which break-up is

impossible irrespective of the magnitude of the shear rate.

This is probably a result of the fact that for the liquids used the

elastic forces in the droplet increase more rapidly with the shear

rate than the shear stress exerted by the continuous phase. The

critical radius increases with the viscosity ratio.

4.6 REFERENCES

1. A. Acharya, R.A. Mashelkar and J. Ulbrecht, Flow of inelastic

and viscoelastic fluids past a sphere. 1. Drag coefficient in

creeping and boundary layer flows, Rheol. Acta 15, 454-470,

(1976)

2. W. Bartok and S.G. Mason, Particle motions in sheared

suspensions. 7. Internal circulation in fluid droplets

(theoretical), J. Colloid Sci. 13, 293-307, (1958)

3. F. Bashforth and J.C. Adams, An attempt to test the theories of

capillary action, University Press, Cambridge, (1883)

4. D.V. Boger, A highly elastic constant-viscosity fluid, J. Non­

Newtonian Fluid Mech. 3, 87-91, (1977/1978)

5. D.V. Boger and H. Nguyen, A model viscoelastic fluid, Polym.

Engng. Sci. 18, 1037 1043, (1978)

127

128

6. D.W. Bousfield, R. Keunings, G. Marucci and M.M. Denn, Non

linear analysis of the surface tension driven break-up of

viscoelastic filaments, J. Non Newt. Fluid Mech. 21, 79-97,

(1986)

7. R.A. de Bruijn, Sealing laws for the flow of emulsions,

chapter 2 of this thesis.

8. R.A. de Bruijn, Newtonian drop break-up in quasi steady simple

shear flow, chapter 3 of this thesis.

9. R.A. de Bruijn, Newtonian drop break-up in simple shear flow

the tipstreaming phenomenon, chapter 6 of this thesis.

10. H.B. Chin and C.D. Han, Studies on droplet deformation and

break-up. 2. Droplet deformation in non uniform shear flow, J.

Rheol. 24, 1-37, (1980)

ll. L. Choplin, P.J. Carreau and A. Ait Kadi, Highly elastic

constant viscosity fluids, Polym. Eng. Sci. 23, 459-464,

(1983)

12. M.J. Crochet, A.R. Davies, and K. Walters, Numerical

simulation of non-Newtonian flow, Elsevier, Amsterdam, (1984)

13. J.J. Elmendorp, A study on polymer blending microrheology,

Ph.D. thesis, Delft University of Technology, (1986)

14. F. Gauthier, H.L. Goldsmith and S.G. Mason, Particle motions

in non-Newtonian media, 1. Couette flow, Rheol. Acta 10, 344-

364, (1971)

15. H. Janeschitz-Kriegel, Flow birefringence of elastico-viscous

polymer systems, Adv. Polym. Sci. 6, 170-318, (1969)

16. M.A. Lockyer and K. Walters, Stress overshoot: real and

apparent, Rheol. Acta 15, 179-188, (1976)

17. J. Michele, Zur Rheometrie viskoelastischer Fluïde mit der

Kegel-Platte Anordnung, Rheol. Acta 17, 42-58, (1978)

18. V. Mirmohammed-Sadeghi, An experimental study of the break-up

of model viscoelastic drops in simple shear flow, M. Sc.

thesis, University of California, (1983)

19. H. van Oene, Modes of dispersion of viscoelastic fluids in

flow, J. Colloid Interface Sci. 40, 448-467, (1972)

20. P. Prabodh and P. Stroeve, Break-up of model viscoelastic

drops in uniform shear flow, pers. conununication by P.

Stroeve, Dep. Chem. Eng., University of California

21. W.R. Schowalter, Mechanics of non-Newtonian fluids, Pergamom

Press, Oxford, 1978

22. R.A. Stratton and A.F. Butcher, Stress relaxation upon

cessation of steady flow and the overshoot effect of polymer

solutions, J. Polym. Sci. Polym. Phys. Ed. 11, 1747-1758,

(1973)

23. T. Tavgac, Drop deformation and break-up in shear fields, Ph.

D. thesis, University of Houston, (1972)

24. G.I. Taylor, The viscosity of a fluid containing small drops

of another fluid, Proc. Roy. Soc. A 138, 41-48, (1932)

4.7 LIST OF SYMBOLS

c. D

~ De

Fg

K

k

L

,k

m

N1

N2

Nwe

n

p

R

C1, .. Cg

u,v,w

constant

drop diameter

rate of strain tensor

Deborah number

recoverable free energy

consistency index

summation index

length scale

velocity gradient tensor

elastic power law index

first normal stress difference

second normal stress difference

Weissenberg number

power law index

summation index

drop radius

critical drop radius

stress ratio

velocity components

[ - l [ml

[ s - l]

[ - l

[J]

[Pa.sn]

[ - ]

[m]

[s-1]

[ - J [Pa]

[Pa]

[ - J

[ - J

[ - J

[m]

[m]

[ - l [m s- 11

129

130

LIST OF SYMBOLS (continued)

v x,y,z

"(

hnt

J'/

J'/c, J'/d

J'/o, J'/p

8

À

Àl, À2,

Àl,p· À2,p

Àa

Àml• Àm2

a

a*

velocity scale [m s-1]

Cartesian coordinates [m]

parameters of Bird-Carreau model (-]

shear rate (s-1]

internal shear rate [s-l]

dynamic viscosity [Pa.s]

dynamic viscosity of continuous/drop phase [Pa.s]

parameters of Bird-Carreau model [Pa.s]

relaxation time [s]

elasticity index

viscosity ratio

parameters on Bird-Carreau model

apparent viscosity ratio

modified viscosity ratio

interfacial tension

modified interfacial tension

shear stress

stress tensor

capillary number

critical capillary number

critical capillary number in Newtonian

limit

[Pa.sm]

[ - l

[s]

[ - l [ - l [N m-1]

[N m-1]

[Pa]

[Pa]

[ - 1

[ - l

[ - l

"

5. DEFORMATION AND BREAKUP OF NEWTONIAN DROPLETS IN TRANSIENT

SIMPLE SHEAR FLOWS

5.1 INTRODUCTION

Most investigations regarding drop deformation and breakup refer

to quasi steady state conditions. These are reviewed in chapter 3.

Very few investigations, however, are available on drop

deformation and breakup in transient simple shear flows. Grace

{1982) reported measurements on drop deformations prior to

break-up for drops subjected to shear rates well above their

critical shear rate and Torza et al. {1972) reported some

measurements on deformation and orientation as a function of time

for a few viscosity ratios. More detailed measurements are

available for other types of linear shear flows (Stone et al.,

1986).

The present investigation is aimed at studying both experimentally

and numerically the behaviour of Newtonian drops suspended in a

Newtonian liquid and subjected to transient simple shear flows at

low Reynolds number. This is relevant to emulsification processes

that involve flows in which a droplet experiences rapidly varying

flow conditions. The investigation involves the development of a

computer program to evaluate the shape of a droplet in a general

transient shear flow. This program is based on the boundary

integral method by which the creeping flow equations inside and

outside the drop are transformed into a form that only involves

quantities at the drop surface. This technique, derived by

Ladyzhenskaya (1963) and further described by Youngren and Acrivos

{1975), Rallison and Acrivos (1978) and by Rallison {1980, 1981),

is based on the Fourier solution of the creeping flow equations,

with the use of volume potentials. Mathematical problems arise

with this method for very small or very large viscosity ratios,

since the boundary integral has neutral eigensolutions for these

131

132

two extremes. For axisymmetric problems the calculations are very

much simplified, since the drop surface can then be described by a

curve and less grid points are needed to obtain an accurate

solution. Another simplification occurs for simple shear flow

since then one of the terms in the boundary integral vanishes and

the velocities can be solved without a matrix inversion,

substantially reducing the computational time needed.

The programme was applied to droplets with viscosity ratios

ranging between 0.5 and 5 and was used to calculate the shape of

the droplet as a function of time for various shear rate profiles,

step profiles, triangular profiles and sinusoidal profiles. The

present investigation also involves experimental work on drop

deformation and break-up in simple shear flows. Experiments have

been performed with droplets with viscosity ratios ranging from

0.01 to S. These droplets were subjected to step and triangular

shear rate profiles.

5.2 PROBLEM STATEMENT AND BOUNDARY INTEGRAL METHOD

The problem under consideration is that of droplets of an

incompressible Newtonian liquid with a viscosity nd' suspended in

another immiscible incompressible liquid with a viscosity nc' The

drop has an interfacial tension G and is subjected to a linear

shear flow at infinity with a velocity ~inf = !·!· Provided the

droplet is small enough, the flow in and around the droplet will

be dominated by the viscosity and inertia and gravity effects can

be neglected. This is allowed when the Reynolds numbers based on

the drop phase and the suspending phase are both small. Fluid flow

can then be described as a quasi steady state problem by the

Stokes or creeping flow equations. The velocities and pressures

are thus governed at each time by the following equations.

div u = 0 everywhere [5.la]

v 't = 0 x not E S [5.lbj

.! P I + l\ v u + v u T € vout [5. lc] !

't = - p I + lld 'il u + 'il u T E vin [5.ld] !

with ~ the velocity, .! the stress tensor with the isotropic part p

the statie pressure and the non isotropic part the flux of

momentum due to a velocity gradient. The boundary conditions for

the deformable droplet are given by

u" = u -u -c

1! 1 -+ inf

! ES

with n defined as the outward normal vector and where V ~

denotes the surface curvature.

[5.2a]

[5.2b]

[5.2c]

To find a general solution for these equations we follow the

approach described by Ladyzhenskaya (1969) and first solve the

Stokes equations for the flow due to a point source of force in

the k-direction applied in a certain point i:

11 v2 ~k(~,l) - (V q(~,l))k

div uk = 0

[5.3a]

[5.3b]

where x is an arbitrary point vector and q a scalar force term, S

the Dirac-S function and ~k a unit vector along the k-th

coordinate axis. All differentiations are carried out with respect

to ! and the point l• the point where the applied force is

concentrated, plays the role of a parameter. Fourier transforms

can be used to find the solutions of these equations. With Q(~)

and Q(~) the Fourier transform of ~(!) and q(~) respectively, this

results in the following set of equations.

133

134

1 - 11 «2 U. k - i « Qk = SJ.k

J j (2n)3/2

k «j uj o

with «Z (~·~>· The subscript j represents the

velocity. Unique solutions of Ujk and Qk can be

equations quite straightforwardly:

k 1

uj <~> n

Q\!l:> i~

(2n)3/2 2 ()l

[5.4a]

[5.4b]

component of the

derived from these

[5.Sa]

[5.Sb]

The inverse Fourier transform of equation [5.5] will give the

fundamental unique solution to the original flow problem.

k u. (x,v) J - "-

In these

in point

the k-th

[5.6a]

[5.6b]

equations u.k(x,v) represents the j-th velocity component J - "-

x, due to a point source of force in point l• acting in

~irection. qk(!,l_) represents the associated pressure

contribution in point x. Ye note for future use that the solutions

~k(!,l_) and qk also sa~isfy equation [5.3] when all the

differentiations are carried out with respect tol instead of~·

The above derived solutions ~k(~ 1 l'._) and qk are Greens functions

and can be used to define the following volume potentials

[5.7a}

[5.7b]

These volume potentials obey the inhomogeneous Stokes equations

div U = 0

[5.8a}

[5.8b]

in which !(~) represents the distribution function of the external

sources of force. In our case !(~) is determined by the

interfacial tension forces which can be described by a layer of

Stokeslets on the drop interface with a strength an V.n.

Next a solution for g will be sought which obeys these

inhomogeneous Stokes equations. Therefore Gauss's theorem is

applied to vector fields of the type u. ~ .. (U) to obtain Greens l lJ -

formula to the Stokes problem, using the following identity

[5.9]

in which the fact is used that both ~ and g are solenoidal (i.e.

divergence is zero). Formula [5.9] can be integrated over a domain

V, which gives

[5.10]

135

136

Now Gauss's theorem is applied to the vector field uk. •· .(U) - U. * k * 1 lJ - k 1 •·· (u) in which •·· (U) is given by the stress tensor•· .(u) lJ - lJ - lJ -

after interchanging ~k and g, interchanging qk and -P and

differentiating with respect to z. This will result in:

k

I [ [ 11 v2 u1 - ~ ] u\ - ui [ n v/ u\ + ~ ] ] dV v a xi a y1

= Is [ 'ij<!!> u\ nj - 'ij *c.!h Ui nj ] dS (5.11)

If we identify U with the volume potential which obeys equation

[5.8a] and ~k w~th the fundamental singular solution (5.6a],

equation [5.11) can be rewritten to:

Uk(!)= Jv ukl.(~,z> f.(z) dV + I ... *(uk) U.(x) n.(z) dS l s 1J - l - J

- Is uk1<!•l> 'ij(!!) nj(X) dS

P(!) Jv qk(!•l> fk(r) dV - Is qk(!•l> 'ij<!!> nj(Z) ds

J a lcx,x>

- 2 ll - Uk(~) n.(}:'.) dS s Cl x. J

J

From the solution [5.6] for ~k it follows that

* k •·. (u (x,v)) lJ - - "-

_:__ (xi-yi)(xj-yj)(xk-yk)

4 n l!-rl5

Using this relation together with the fundamental singular

solution [5.6], equation [5.12a] can be rewritten

[5.12a]

(5.12b]

[5.13]

Uk(~) = Jv uki(~ 1 l) fi(l) dV - IsKijk(E) Ui(l) nj(X) dS

- _1_ J J.k(~) i:.j(X) n.(l) dS 8 n n s J 1 1

with:

[5.14]

This equation can be used to derive the boundary integral equation

for the velocity of the points at the droplet interface by

applying this equation to the exterior of the droplet and to the

interior of the droplet. The exterior equation is formulated with . * inf . the disturbance veloc1ty Q = Q - Q = Q E.x. The interior

equation is formulated with the total velocity Q. In that case the

volume integrals vanish since external forces are applied neither

at infinity nor anywhere else outside the droplet interface. The

surface integrals at infinity vanish as well. Thus only surface

integrals just outside and just inside the interface remain.

Uk(~) E.x - Is(e) Kijk(E) U1<x> nj(X) dS

-1- I J 'k(r) i: .. (y) n1 <x> dS x e: S(e) [5.15a]

8 n n S(e) J - lJ -c

Uk(~) = - J K .. k(E) U1.(X) nJ.(X) dS S( i) lJ

- _1_ J Jjk(!) i: .. (X) ni(X) dS 8 n l"ld S(i) iJ

x e: S(i) [S.15b]

Since velocity continuity is prescribed by the boundary conditions

[5.2b] the velocity approaching the drop interface from the

137

138

exterior or from the interior must be identical. The J and K

integrals are however discontinuous at the drop interface. The

discontinuity in the J-integral is given by the normal stress

boundary condition [5.2c}. The discontinuity in the K-integral is

given by

JS(e) Kijk(E) Ui(X) nj(X) as Is Kijk(E) U1<x> nj<x> dS

- 112 Uk(~)

Is(i) Kijk(E) U1(l) nj(l) dS Is Kijk(E) Ui(X) nj(l) dS

[5.16a)

[5.16b]

Inserting these boundary conditions in equation [5.15) results in

the boundary integral formulation for the velocity of a point x on

the drop interface:

2 2(1-À)

Uk(!9 E.x - I K. 'k(r) ui <;O nj <x> ds l+À l+À s lJ -

11 I J.k(r) n.(y) v. n dS (l+À) 4 Jt nc s J - J -

[5.17]

Vhen À = 1 the equation is very much simplified because the

K-integral term, which itself involves interface velocity terms,

will then vanish. The resulting equation is an explicit expression

for the surface velocities Q(~), which can be solved after

discretising the drop interface with N points. This simplification

has been used by Rallison (1978,1981). In its full generality

solving the velocities at the points on the drop interface

involves the solution of a linear set of 3N equations. Equation

[5.17] has been made dimensionless to facilitate the numerical

solution. The positions have been made dimensionless with the drop

radius R, the velocities with u/nc and the deformation gradient

tensor~ by the characteristic velocity gradient y. The resulting

equation, in which all quantities represent dimensionless

quantities, is:

2 Q

Uk(~} = - E.x l+À.

2(1-À) I -- K. 'k(r} U.(z:} nj(l) dS

l+À. s lJ - l

1

I J 'k(r) n.(v) 'il • n dS J - J L -s 4 lt ( l+À.} [5.18]

Equation [5.18] is only valid for the points at the drop interface

but contains all information necessary to evaluate the full three

dimensional shape of a droplet as a function of time. The time

dependency does not stem from the inertia terms in the

Navier-Stokes equations, since they were neglected in the low

Reynolds assumption. However they stem from the changing boundary

conditions at the drop interface following a change in the applied

undisturbed flow field. The boundary integral formulation can be

extended to yield the velocity in any point inside or outside the

droplet expressed in quantities to be evaluated at the drop

interface. The only additional problem is to find expressions for

the full J-integral terms (for the interfacial velocities only the

jump in the J-integral terms was needed). If the interfacial

velocities have been calculated with equation [5.18), it is

however possible to evaluate the unknown terms éij(y).ni(l)= Tj(l)

for all the points at the drop interface, by solving a set of 3N

linear equations in Tj(l}· These equations are obtainable from

equation [5.15a]. Once these terms have been evaluated equation

[5.15] will give the following expressions for the velocity in a

point outside or inside the droplet.

139

140

2

3

12

+ 12

1 for ~ € vout

[5.19a]

I K .. k(r) U.(l_) n.(y) dS s lJ - 1 J -

1 Is Jjk<E> Tj<l> ds

Jt À '1c Cf I J.k(E) n.(l) V.!!. dS for ~ € Vout

Jt À nc s J J [5.19b]

where l. denotes a point on the drop interface, the integrations

are performed on the drop interface S and the terms Tj(l.) are

defined just outside the drop interface.

5.3 THE NUMERICAL METHOD

A numerical scheme has been developed to solve equation [5.18] for

the velocities at the drop interface. This scheme allows

calculation of the velocities at the drop interface for droplets

subjected to any type of linear flow field. For this scheme the

drop surface S(t) has been discretised by N collocation points x1•

Two meshes have been used in the present investigation, consisting

164 and 266 points distributed over the drop surface. The

evaluation of the surface variables, the normal and curvature of

the surface, is carried out by fitting the drop interface in the

locality of each collocation point by a second order curved

surface of which the surface variables can easily determined. A

method is developed to redistribute the points of the mesh at each

time step, in order to prevent clustering of the mesh points

during deformation of the mesh.

5.3.1 Mesh definition

Two similar meshes have been defined, one consisting of 164 points

(9 rows of 18 points, plus 2 points on the poles), and one with

266 points (11 rows of 24 points, plus 2). The initial

distribution of the collocation points over the spherical droplet

is shown in figure 5.1. The latter mesh has been created in order

to evaluate S(t) more accurately when the surface area of S(t)

increases and the total number of points per surface area

decreases accordingly. Two meshes have been used to test the mesh

dependency and the stability of the numerical method. The number

of triangular regions enclosed by the points is 324 and 528,

respectively, giving a surface area per triangular region of less

than 0.5 % of the total surface area. The points of the mesh are

spaced such that all triangles made up by those points have about

! 0

~ A 1 18

36

Fig. 5.1 Representation in spherical coordinates of the mesh with

266 collocation points that was used in numerical

calculations

141

142

the same surface area which is necessary for numerical stability.

The choice of less points on the first row resulted in a growing

periodical disturbance in the first time steps which proved to be

catastrophic for larger time steps. Both meshes consist of more

collocation points than the mesh used by Rallison (1981).

5.3.2 Evaluation of surface variables

To evaluate the surface variables, the drop interface is fitted in

the locality of each collocation point to a second order curved

surface which is point symmetrie with respect to the origin. This

curved surface can be represented by:

i i 2 i i 2 1 1 2 i i i i i i i i i a 1x l +a 2x z +a 3x 3 +a 4x 1x 2+a 5x 1x 3+a 6x 2x 3-1 Û

g(xi)

[5.20)

The coef ficients of the curved surface are determined for each

collocation point by fitting the curved surface through the point

under consideration and five neighbour points. The normal in each

collocation point can be found by:

nj a g(xi)

i a x j [5.21]

The local curvature is given by:

[5.22]

with R1 and R2 the two principal radii of curvature. These two

radii of curvature can be calculated if the intersections of two

planes, both perpendicular to the tangent plane in x1, and the

. i i second order curved surface, given by g(x1 )=0 with a 1 •• a 6 , are

known. To calculate the intersections a new orthonormal base is

defined. One of the new base vectors is the already known normal i n (Eq. [5.21]) and the other two base vectors can be found from

the tangent plane in xi. Both xi and g(x1) can be rewritten with

respect to this new base and are given i i by y and h(y )

respectively. The principal curvatures (the reciprocal values of

the radii of curvature) can then be calculated by:

- 2h i i h i h i y ly 3 y 1 y 3

+ h . . h2 . l l l

y 3Y 3 y 1 [5.23)

in which the subscript y1 . at the function h(y1) denotes that J

partial differentiation is carried out with respect to the

collocation point y1j. The subscript at j of k1j denotes the

direction in which the curvature can be found. The other

curvatures can be found by interchanging the subscripts 1, 2 and 3

cyclically.

This method for the evaluation of the surface variables will

always result in exact solutions for the normal and curvature for

ellipsoidal spheroids even in the case of negative curvatures.

This method was therefore preferred above the iterative method

used by Rallison (1981), which resulted in errors up to 2%.

5.3.3 Evaluation of the surface integral

The contribution to the J-integral of each collocation point xi is

calculated as follows. First the contribution of the triangles,

which do not have a vertex at x1, is calculated using the

trapezoidal rule. The contribution of these triangles can be

rewritten as contributions of the vertex points xj of the

143

triangle, to the point xi, since the centre of gravity of each

triangle is given by:

xtriangle =113 ( E xvertex ) [5.24]

Those triangles which do have a vertex at xi have to be dealt with

separately due to the 1/r singularity in J for r=O. The integral

however is finite when r approaches zero. An analytical expression

for the contribution of these triangles is given in Appendix Cl.

Once the numerical value for the contribution of such a singular

triangle is known, it can simply be added to the contributions of

all other points.

If the viscosity ratio is 1, the K-integral can be omitted, and

the velocity v1 in each point can be calculated directly by adding

the contribution of the external flow field to the value of the

J-integral. If the viscosity ratio is not equal to 1, the

K-integral has to be evaluated for each component of the velocity

vi. Since the velocity itself is present within the integral, a

system of linear equations with the integral contributions of the

velocity components of all collocation points equaling the sum of

the J-integral contributions and external flow field contributions

has to be solved. This involves matrix solving techniques. To

compose the elements of this matrix, an analytica! expression

similar to the one for the J-integral has to be found for the

K-integral. This expression is given in Appendix C2. Point

symmetry with respect to the origin has been used to reduce the

matrix by a factor 4 and to gain a factor 16 in computational

speed. The complete system of 399 linear equations with 399

unknowns is solved using an iterative Gauss-Seidel matrix solver

technique, with an accuracy of lE-6. The iterative Gauss-Seidel

technique was chosen because of its efficiency for problems of

which the solution changes little per time step and the dominance

of the diagonal elements of the matrix.

5.3.4 Redistribution of the mesh

The collocation points of the mesh behave during the deformation

of the droplet as material points on the surface. Hence they move

away from their original positions. The new positions can be

calculated as follows:

~new = ~old + y(~) at [5.25]

This movement results however, in clustering of points at the ends

of the deformed surface and a depletion at both sides (see

figure (5.2a)). This distribution of points will finally lead to

numerical instabilities at highly deformed surfaces (deformations

greater than 0.5 for a mesh with 164 points, see figure (5.2b)).

2-

-2L -2

Fig. 5.2.a

1 -1 0

1 2

Droplet contours in simple shear flov vith mesh correction

145

146

4.00-

2.40-

0.80-

-0.80-

-2.40

-4.00L -4

Fig. 5.2.b

1 -3

r - -

1 -1 0

- ...,

__ J

1 2

1 3

Droplet contours in simple shear flow without mesh

correct ion

1 4

A better method is to let the collocation points move in the

normal direction:

[S.26]

Vith this method however, the collocation points will also become

unevenly distributed over the drop surface when the droplet is

highly deformed. To prevent clustering of the collocation points

on the surface, the points of the mesh are redistributed at each

time-step. First, all points, which initially had the same

z-coordinate, are given the same relative z-coordinate with

respect to the z-coordinate of the top point using the equation of

the curved surface in the locality of the point. Second, the

points in the z plane of the droplet, are redistributed in such a

manner that they have about the same distance from one another,

making use of the coefficients aij of Eq. [5.20]. Finally all

collocation points on contours having the same z-coordinate are

redistributed similar to the points in the z-plane. Once the mesh

is redistributed, the surface variables are reevaluated.

5.3.5 Numerical stability and convergence

The numerical method described in the previous sections has been

tested on numerical stability and convergence both in the presence

and in the absence of external shear flow fields. In the absence

of an external flow field it was checked that the droplet remained

spherical. It was observed that this depended on the mesh

distribution. Optima! results were obtained with the mesh given in

figure 5.1. Yhen a mesh distribution as in figure 5.3 was used (as

was done by Rallison, 1981) wave-like disturbances were observed,

especially on the two rows of points close to the top points.

These wave-like disturbances showed themselves in inward movement

of some of the points on these rows, combined with an outward

movement of other points. This effect was related to the

asymmetry in such a grid. Some points on the third row are namely

surrounded by 5 triangles with comparable surface area, while

others are surrounded by 5 relatively small triangles and 1

relatively large triangle. The observed wave periodicity coincided

with the mesh periodicity. These wave-like disturbances were

especially noticeable at relatively large time steps. When the

droplet was subjected toa shear flow the wave-like instability

was suppressed by the applied flow. For the present mesh type the

sphericity was very well maintained. The equilibrium deformations

in x-y and x-z direction were for all viscosity ratios very small:

Dxy < 0.0004 and Dxz < 0.003. The somewhat larger deformation Dxz

147

148

is due to an inward displacement of the top points. Vhen an

equilibrium shape is obtained, all points tend to move inward

simultaneously. The relative decrease of the drop radius in a unit

dimensionless time was however quite small: 0.6%. A correction has

been built in to preserve the drop volume by inflating the droplet

radially after every time step.

0 , --- -- --- - --- ' , ' , ' , '

0 i M ~ w n H u '

8

Fig. 5.3 Representation of the mesh used by Rallison (1981)

Effects of time step and mesh point density on numerical

convergence have been investigated in relaxation experiments. For

these experiments the initial shape of the droplet was obtained by

multiplication of the z or the x coordinates by a certain factor

and the relaxation of the resulting ellipsoid was calculated as a

function of the time step size. Axis ratios ranging from 1:1:10 to

1:1:2 have been used. Examples are shown in figures 5.4 and 5.5.

From these results it became clear that a time step size of 0.1 is

small enough for À=l, for x~z a somewhat larger time step can even

be used, hut for X:0.5 a smaller time step is necessary: ót=0.05.

3.20-

1.60

-1.60-

-3.20 L -3.20

Fig. 5.4

Fig. 5.5

1 -1.60 0

1 1.60

1 3.20

Droplet contours at various times during the relaxation in

the absence of externally applied flow of an initially

ellipsoidal drop with an axis ratio of 3:1:1

Droplet contour during the relaxation in the absence of

externally applied flow of an initially ellipsoidal drop

with an axis ratio of 10:1:1

149

150

llhen smaller time steps were used the differences in Dxy and Dxz

were always smaller than 0.001. These required time step sizes are

in line with the relaxation times of the droplets. As a rule of

the thumb time steps of 0.01/À should be used in future

calculations. Yith respect to the mesh density the following can

be concluded from figure 5.6. Since some differences in

deformation (up to 0.03) between the 164 grid point mesh and the

266 grid point mesh were observed, reliable calculations need at

least a 266 grid point mesh. The effect of mesh distribution has

been investigated for X=l by comparison of the relaxation of

ellipsoids with their main axis in the z or the x direction. Only

small differences were observed. For the 164 grid point mesh the

difference in deformation was always smaller than 0.02 and for the

266 grid point mesh they were even smaller than 0.007. These

results indicate a good independence of the numerical solution of

the mesh distribution, especially for the 266 mesh point grid and

it is concluded that for these deformations a numerical scheme has

been developed with which mesh and time step independent results

can be obtained.

Numerical stability and convergence have also been studied for X=l

in step response experiments for the 164 and the 266 grid point

mesh. From the results in figure 5.7 the following can be

concluded. At small deformations (D<0.3) there is only a small

difference between the results obtained with the 164 and the 266

grid point mesh. These data confirm the results obtained in the

relaxation experiments. At larger deformations there is an

increasing discrepancy showing the deficiency of 164 grid point

mesh for larger deformations. This is especially clear from the

fact that above 2=0.40 no stable final deformations could be

calculated with the 164 grid point mesh. This is showed in figure

5.7 by the steady increase of the deformation with time. For the

266 grid point mesh stable deformations could be calculated up to

2=0.4. At large deformations (D>0.4) there should be some doubt

... ó

C"' 0 . ·- 0

ö E 1...

J2 N (J) • 00

Fig. 5.6.a

In

"' ci

Cl ':

"' "' ci

.Q ~ Qó E 1... U1

J2 (J) '"

0

8 ci

"' c ci Cl c;

0.0

Fig. 5.6.b

D = >.-1, z-oxLs><3, 266 polnls 0 ),=1) x-axLsx3, 266 pocnts " >.= 1, z-ax:Î...s*3, 164 pol,nt.s + >. = 1' x-axcs*3, 164 pocnt.s

Time

Relaxation of droplets vith an ellipsoidal initial shape.

Bffects of number and distribution of collocation points

for droplets vith an initial axis ratio of 1:1:3.

5.0 10.0

Time

o = i.=2, z-axl,s><2, 266 pocnts o = >-1 z-axcs><2, 266 pocnts 6 - ~-o:s~ z-axLs~2J 266 poLnts

lS.O

Relaxation of droplets vith an ellipsoidal initial shape.

Effects of viscosity ratio for droplets vith an initial

axis ratio of 1:1:2.

151

152

about the 266 grid point mesh as well and even higher mesh

densities are required to perform reliable calculations.

"! 0

": 0

"' à

:z "' 30 '""' a: ... ::c . er:. 0

a t....M w. c:i"'

N

0

à 0

c

o.o

Fig. 5. 7

5.0 10.0 15.0 TIME

'Y - 0-0.1, • 0-0.2, Il 0-0.3, l'l - 0-0. • 0-0. 0 - 0-0.1, " - 0-0.2, + - O=O. 3, x - 0-0.1, " - 0-0.45,

166 MESH POINTS 166 MESH POINTS 166 MESH POINTS 166 MESH POINTS 166 MESH POINTS 266 MESH POINTS 266 MESH POINTS 266 MESH POINTS 266 MESH POINTS 266 MESH POINTS

Mesh dependency of the results for the deformation

response of droplets to a step like shear rate profile for

X=l.

5.4 EXPERIMENTS

5.4.1 Description of the Couette device

Deformation and breakup experiments were carried out in a Couette

device. This device bas been described in section 3.4.1 of this

thesis. The Couette device consists of two counter rotating

cylinder~. In the gap between the cylinders a linear simple shear

field is generated (see figures (3.2)). The two cylinders are belt

driven. The motor controls can be operated both manually and

automatically. To drive the AC-motors automatically, a personal

computer has been used. The bottom and wall of the outer cylinder

are made of glass in order to visualise the flow phenomena.

5.4.2 Transient flow in Couette device

To evaluate the capabilities of the Couette device, to be used to

measure drop deformation and breakup in transient shear flows, the

response of the fluid flow in the gap between the two rotating

cylinders following a change in rotational speed of the cylinders,

has been calculated numerically. These calculations were performed

to ensure that inertia had no effect on the phenomena observed

between the cylinders. Bottom effects were neglected in this

analysis. The Navier-Stokes equations for this particularly simple

geometry reduce to:

[5.27]

in which U+ represents the velocity in the tangential direction

and r the radial position in the gap. The response to a step like

change in the velocity has been calculated, using the following

initial and boundary conditions:

u+ (r ,t) U+ (Rl,t) U.p (R2, t)

for t < 0 for t ~ 0 for t ~ 0

[5.28a] [5.28b] [5.28c]

These equations were solved numerically by an explicit centra!

differential scheme. This scheme has been used to calculate the

development of the velocity with time and the time required to

obtain 99% of the steady state velocity. The calculations carried

out for various speeds of the cylinders. The development of the

velocity in time is given in figure 5.8 fora velocity ratio of

-1:4. All the calculations show that the velocity is everywhere

within 99% of the steady state velocity after:

[5.29)

153

1.00 Veloai

0.80

0.60

0.40

0.20

Fig. 5.8

154

t - 0.1

t • 0.2

t • 0.5

/~i' / '/ •:

/ 1/:/1 / ///11

// /;'::// / , ' 1

/ / / ..- /1 /'; ! ' ." / 1

/ / I . 11 / / , . I

/ / I l 1

/ / / / 1

" / I / 1 / / • 1

/" / I I 1

"" / / I ' .n2'" "" / • I

1

~ " I I ' "" / I 1 " , . I i

" / / I ' / / / / /

// "/ ./ / ,' / / /./ /1

/ /' • / I

// " ./ ··" / /

/ // ,,./···· --~- .... /'

4.19 4.40 4.61 4.82 5.03

--> RadtJS {Cl'l'I/

Development of the velocity profile in the Couette device

after a step like change of the velocities of the

cylinders. Velocity ratio betveen inner and outer cylinder

is -1:4

For experiments using a silicone oil of 10 Pas as a continuous

phase this implies that the bulk fluid responds within 5 msec. In

the triangle profile experiments in the Couette device actual ramp

times were always more than 100 times the fluid response time. So

inertial effects due to fluid motion do not interfere with the

drop deformation and breakup reported in this investigation.

5.4.3 Experimental procedure

During an experiment a small fluid droplet ( R < 0.5 mm ) is

brought into the gap between the cylinders in the Couette device,

which is filled with another fluid with about the same density as

the droplet. This droplet is held at one place between the two

rotating cylinders in the stagnant zone (i.e. a zone where

U+(r)=O) by adjusting the rotational speeds of the cylinders

manually.

To keep the droplet positioned in the stagnant zone during the

experiments, a preliminary experiment is carried out to determine

the position of the droplet in the gap. This is done by

determining the ratio of the velocities of the 2 cylinders, at

which the stagnant layer coincides with the drop position. This

ratio is used to keep that droplet positioned in this stagnant

layer in subsequent experiments. The deformation of the droplet is

recorded on video together with a display of the rotational speeds

and the time and is analysed afterwards. In order to measure the

deformation using the video image of the particle as accurately as

possible, a high contrast illumination of the particle is chosen.

No distortion of the video image has been observed in any

direction. The error in the deformation measurements varied from

0.1% to 5 % depending on the quality of the video image and the

magnification of the particle, with the errors of most

measurements in the lower range.

155

156

In the experiments two types of fluids have been used.

Polymethylsiloxanate (Rhodorsil silicone oil 47, Rhone-Poulenc)

has been chosen for its Newtonian behaviour, high viscosity and

low dependency of its viscosity on temperature. Corn syrup (Globe

01170 of Cerestar Benelux BV.) has been chosen for its high

viscosity and Newtonian behaviour as well, but its viscosity

temperature dependency is rather strong. A high viscosity (10 Pas)

was necessary for both fluids since otherwise at lower viscosities

the time scales would be too small for the video registration. The

fluid systems used can be found in table (5.1).

TABLE 5.1: Fluid properties

Continuous phase Discrete phase interfacial tension

descript ion viscosity description viscosity [mNm-1 ) [Pas] [Pas]

CS/water 97 so 105 36 CS/water 10.8 so 11.0 37 CS/water 4.8 so 24.0 30 so 24.0 glyc./water 4.8 23

Silicone oil has been chosen as the high viscosity drop phase,

because corn syrup drops tend to become rigid. This may be due to

crystallisation of the droplet resulting from dissolution of the

water phase of the corn syrup droplet in the oil phase. The

viscosity temperature relation was measured with a Haake

rotational viscometer (SV12) and the interfacial tension with the

Yilhelmy plate method. Occasionally the drop phase has been

coloured with Sudan red for photographic reasons. The same

additive ( the fraction which has not been dissolved) could be

used as a tracer material in the drop phase to visualise the flow

pattern in the drop. Three viscosity ratios have been used for the

experiments: 0.01, 1 and 5. For the transient deformation

measurements (experiments 2,3 and 4) the value of the maximum

capillary number has been varied up to 5 times the critica!

capillary numbers concerned, while the value of the time to reach

that maximum capillary number has been varied from 0 to 10

dimensionless time units.

The following experiments have been carried out:

1. The deformation was measured as a function of the shear

rate, which was slowly increased, giving the droplet enough time

to reach its equilibrium deformation at all shear rates (quasi

steady state deformation experiments}.

2. The deformation was measured as a function of time

following a sudden, step like increase of the shear rate (step

response deformation experiments).

3. The deformation was measured as a function of the shear

rate and time, during a triangular shear rate ramp. (triangle

response deformation experiments).

4. The breakup of droplets in transient experiments was

observed and the number of fragments was counted. (breakup

experiments)

5.5 NUMERICAL CALCULATIONS

The program, written in Pascal, was run on a VAX 8530 machine. The

program required 20 seconds of CPU time per time step for the

simpler case of a viscosity ratio equal to 1 and 80 seconds per

time step for other viscosity ratios. Both figures refer to the

denser mesh consisting of 266 collocation points. To test the

program with respect to accuracy and reliability the following

experiments have been carried out, using both meshes under

consideration:

1. Relaxation of an ellipsoidal droplet in the absence of an

applied flow, with time steps 0.5, 0.1, 0.05, 0.01 and 0.005, for

157

158

oblate and prolate ellipsoids with the major axis in x and

z-direction (this results in 2 different meshes due to the

configuration of the mesh)

2. Deformation of the droplet in response to a step like

change in the shear rate for various time steps.

When above tests proved to be satisfactory other simulations were

carried out:

3. Step response experiments for various viscosity ratios for

capillary numbers up to the critical capillary number, with and

without mesh correction.

4. Triangle response experiments for the same set of

viscosity ratios.

S. Sine response experiments for various viscosity ratios.

5.6 EXPERIMENTAL AND NUMERICAL RESULTS

5.6.1 Step profile response

In this section the results of drop deformation experiments in

step profile simple shear flows will be described. The shape of

the droplets will be described by the deformation D=(L-B)/(L+B)

and the orientation of the droplet, given by the angle between the

longest axis of the droplet and the direction of the flow.

For a viscosity ratio of 1 the deformation and orientation of

drops are shown in figure 5.9 as a function of the dimensionless

time for capillary numbers ranging from 0.1 to 0.5. The results

show very good agreement between the experimentally observed

deformations and the numerically calculated ones especially when

the capillary number is smaller than 0.4. At higher capillary

numbers some deviation occurs and the numerical model predicts

unstable droplets where experimental measurements indicate stable

deformations up to 0.68, corresponding with a critical capillary

number of 0.48. Note that the experimental data for 2':0.486 relate

to an unstable situation; break-up occurred after t=36. At high

capillary numbers and shorter times the numerically calculated

deformation is somewhat smaller than the experimentally observed

deformation, but at longer times the numerically calculated

deformations become larger due to the instability of the drop. The

numerically calculated orientation angles are somewhat smaller

than the experimentally observed final deformations.

The general shape of the droplet is even at high capillary numbers

in very in good agreement with the experimentally observed droplet

shape as is shown in figure 5.10 where the grid points with a zero

or positive z-coordinate are projected on the x-y plane for

capillary numbers of 0.45 and 0.60 at various times. Both

experiments and calculations show that droplets attain an

ellipsoidal shape at small deformations followed by a sigmoidal

shape at deformations above 0.6.

For viscosity ratios of 0.5 and 2.0 the numerical results on drop

deformation and orientation as a function of time for various

capillary numbers ranging from 0.1 to 0.4 are given in figures

5.11 and 5.12 respectively. For a viscosity ratios of 5 and 0.01

the numerical and experimental results on drop deformation and

orientation as a function of time are presented in figure 5.13 and

5.14. The effect of the viscosity ratio on the step response is

shown in figure 5.15 fora capillary number of 0.40.

The final drop deformations have also been measured as a function

of the capillary number for the viscosity ratios 0.01, 1 and 5.

The results are given in figure 5.16. For each viscosity ratio the

results refer to a number of different drop sizes.

159

"' 0

" ei

'" 0

z"' s ci E--< cr: ... :c . cc: 0

0

(.._ "' w. 00

"' "' ei 0

0

o.o s.o 10.0

TIME

"'= 0=0.094, EXPERINENTAL • O·O. 194, EX PER IMENTAL ri = 0=0.284, EXPERIMENTAL ~ = 0=0.389, EXPERIMENTAL • - 0=0. 431, EX PER IMENTAL • 0=0. 486, EXPER IMENTAL

15.0

0

~] 2· Ç)

.,; "

z I~ QO ~"' E--<o cr: • E--< "' z"" S:lo ;sîi

0

tR 0

ó "' q

"' ... 0.0

0 = 0-0.1, "'= Cl=0.2, + - n-o .3, x o-o. 4, <:i = 0=0.45,

s.o TIME

NUMERI CAL NUMERI CAL NUMERI CAL NUMERI CAL NUMERI CAL

Fig. 5. 9 Numerical and experimental results for the response of

droplets to a step like shear rate profile for À=l.

•

10.0 15.0

Fig. 5.10

..... 0) .....

Ca•0.45 -> Ca•0.60 ->

t•0.50-1.76-4.27-19.60. t•0.75-3.52-5.78-11.31.

Contours of droplet (J.,:1) at various capillary numbers. Numerical calculations for '2=0.45 and 2-0.60

,,, c:i

0.0 2.0 ~.o s.o TIME

o 0-0 .1, NUMERI CAL " - 0-0.2, NUMERICAL + 0-0.4, NUMERICAL

8'.0 10.0

<; 0

" 0

tri "'

Zo 0.

~~ a: e-:z wc; -"' 0:::"' Ci

t:? 0

"' 0

* 0.0 2.0 4.0 6.0 TIME

Fig. 5.11 Numerical results for the response of droplets to a step

like shear rate profile for >--0.5.

B.O 10.0

~ 0

"' ei

.... zo 0 -f-.. a:"" >= • 0-:0 0 "-WN oei

0

0 0.0 5.0

T!ME

o - 0-0 .1, NUMERI CAL "' - 0-0.2, NUMERICAL + 0-0. 3, NUNERICAL

0-0.4, NUMERICAL

10.0

c: "' " c: R

0

z~ 0

f-.. 0 a: . f-..O z«> w -0-:o 0.

"' "' 0

ei lf)

c: ~

15.0

TIME

Fig. 5.12 Numerical results for the response of droplets to a step

like shear rate profile for >.=2.0.

0

ö "' 0 Il// •• _.,,__,

~---~~~~-,-~~~~-..~~~~--,

TIME

o - 0-0.18, EXPERIMENTAL v - 0-0.40, EXPERIMENTAL " - 0-0 . 80 , EXPER I MENTAL

o.o

o - n-o .1, "' - n-0.2, + - 0-0.4, x 0-0.8,

!O.O

TINE

NUMERI CAL NUMERI CAL NUMERI CAL NUMERI CAL

Fig. 5.13 Numerical and experimental results for the response of

droplets to a step like shear rate profile for À=5.

20.0 30.0

"' ei

N

ei

ei 0

ei_,._~,-~,.---,~~~~~~~~~~~~~ 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7 .0 8.0 9.0 10.0

Time

o = 0=0.38, experimental a = 0=0.76, experimental + - 0::::0.93, experimental " = 0=1.39, experimental x - 0=1.67, experimental

Fig. 5.14 Experimental results for the response of droplets to a

step like shear rate profile for A=0.01.

'1!

"': 0

w ci

z U>

0 ci !--< a:"' >:: • O'.::o 0 {~"' w. o"'

~

ci

"'! ~ *.-..~~~~~~~~~~~~~~~~

0.0 s.o TIME

o - À-0.5, NUMERICAL 8 - À-0.8, NUMERICAL + - }..-1.0, NUMERICAL x - À-2.0, NUMERICAL o }..-5.0, NUMERICAL

o.o 5.0

TINE

Fig. 5.15 Effect of viscosity ratio on the response of droplets to a

step like shear rate profile for ~0.4.

10.0 15.0

(D

;l "' 0:

z"' ~ci ,___, a:: ... >::: • e>::o 0 L_"' w. 00

0

o.o O.l 0.2 0.3 0.4 o.s 0.0 0.2 0.4 0.6 0.8 !.O 1.2 l.4 !.6 !.8 2.0

CAPILLARY NUMBER CAPILLARY NUMBER

o - l\=1.0, experimental " À=1.0, numerical

Fig. 5.16 Final deformation of droplets as a function of the

capillary number for various viscosity ratios.

168

5.6.2 Triangular profile response

In this section the response of a droplet to a triangular shear

rate profile is analysed. In these experiments the shear rate

increases linearly in a time ~inc until it reaches a maximum

value, ymax' followed by a linear decrease of the shear rate in a

time ~dec· The results of two numerical simulations are shown in

the figures 5.17 for a viscosity ratio 1 and for two different

values of i , y •t and 2y it respectively. In these figures max cri er the time is normalised with the value of ~inc for the different

curves in one plot. From the first set of curves (figure 5.17a) it

is clear that for values of ~inc smaller than 5 the droplet will

relax to its spherical shape without breakup. Even for values of

2ycrit the droplet will not break up, provided ~inc is short

enough. For longer times the droplet becomes unstable if ~inc

reaches a value of 5.

The same experiments have been carried out in the Couette device.

These results are plotted in figures 5.18 and 5.19. In these

figures the curves of a very simple first order model of the

droplet response for small deformations (see section 5.6.1) are

plotted as well.

Additional experiments have been carried out in the Couette device

for three values of the viscosity-ratio, respectively 5, 1 and

0.01. For these experiments attention was paid to the number of

fragments obtained after breakup of the droplet. The results for

the viscosity ratios 1 and 0.01 are plotted in figure 5.20. For

the viscosity ratio 5, no results are plotted due to the fact that

in only one case breakup had been observed, which could not be

reproduced.

0.50 \ '• )\ = 1.0 0 = Ocrit '• " •• 1

0.40

t;nc 5.0

0.30 i

1 0.20

0.10

0.5

0.00 0 2 4 6 8 10

->~tm.ft/fh:J

1.00 )\ = 1.0 0 = 2.0*0crit

0.80

0.60 i ! <'l

0.40

0.20

t1nc=0.5

0 2 4 6 8 10

->~tlmlt(r/tlnl':J

Fig. 5.17 Effect of the ramp time Tine on the response of a droplet (À=l) to a triangular shear rate profile (numerical calculations).

169

+

+

Fig. 5.18

170

"" +

1?..00

Response of a droplet (X=l) to a triangular shear rate

profile for 'inc=2 for various values of the capillary

number (experimental observations). The solid line

describes the first order model (section 6.7.2) and the +

describe the experimental deformation.

1.00 kYfflallOI?

,,.

+ 0.96

o.32 +

t Fig. 5.18

f: 0 = 4.00*0crit

240

+

·~o

Response of a droplet ()1..1) to a triangular shear rate

profile for ~inc=2 for various values of the capillary number (experimental observations). The solid line

describes the first order model {section 6.7.2) and the+

describe the experimental deformation

171

"""

t OM

Fig. 5.19

172

O.&O OIPfrnmWI

a:Tinc 1. 1 1 b:Tinc = 1.38

T dec 2.21 t Tdec = 2.49 0.6•

•2 ,. •2 ,. - ---

c: Tine= 1.so Tdec=2.91 t

d: Tine 1.94

T dec= 3.32

Response of a droplet (À=l) to a triangular shear rate

profile for ymax = ycrit for various values shear rate

ramp time ~inc (experimental observations). The solid line describes the first order model (section 6.7.2) and the+

describes the equilibrium deformation.

,.

+

OA&

Fig. 5.19

e: Tine = 2.77

Tciec=3.88

t :T1nc 6.92

T 00c= 7.75 ().64

,. 20 ---Response of a droplet (X.:1) to a triangular shear rate

profile for +max ycrit for various values shear rate

ramp time ~inc (experimental observations). The solid line

describes the first order model (section 6.7.2) and the+

describes the equilibrium deformation.

173

174

Fig. 5.20.a Number of fragments observed after breakup of a drop for

various viscosity ratios as a function of the maximum shear rate and the ramp time. À • 0.01

." S20

" . ••• " " 10

" " ... "

Fig. 5.20.b Number of fragments observed after breakup of a drop for

various viscosi ty ratios as a function of the maximum

shear rate and the ramp time. À • 1.0

••

"

5.6.3 Sine profile response

As an example of the behaviour of droplets in oscillatory flows,

the response of a droplet to a sine like shear rate profile was

studied. The profile used is given by:

2(t)

2(t)

A [l+sin(2nt/T-n/2)]

0

t>O

t<O [5.30]

with T the period of the sine function and A the amplitude. These

calculations were carried out for two viscosity ratios, 1 and 5

and for different values of the period Tand amplitude A. The

results are shown in the figure 5.21, where the deformation

response is given for various values of the capillary number and

various periods T. The time of the periods bas been made

dimensionless with the characteristic drop deformation time )1R/ 11.

5.7 DISCUSSION

5.7.1 Step profile experiments

The experimental results in figures 5.9 and 5.16 on step response

experiments and final deformation experiments at a viscosity ratio

of 1 show stable deformations up to a capillary number of 0.48,

corresponding with a maximum stable deformation of 0.68. These

experimental measurements can be compared with some data available

in the literature. The final stable deformation prior to break-up

measured by Rumscheidt and Mason (1961) was D=0.71 and the

critica! capillary number was found to be 0.45. Grace (1982)

reported a final stable deformation of D=0.72 and a critica!

capillary number of around 0.5. Taylor (1934) reported for a

viscosity ratio of 0.9 a maximum stable deformation of 0.79 and a

critical capillary number of 0.55.

175

176

0.50 À=1

0.40

T= 1.0

~ 0.30

j 0.20

0.10

0.00 0 4 6 12 16 20

---~

0,50

À=5

A = 0.5*Ücrit 0.40

0.30 ~

1 0.20

0.10

0.00 ""'---'----'----'----'---__J

0

Fig. 5.21

4 8 12 16 20

Response of drops of various viscosity ratios to a

sinusoidal shear rate profile for various values of the

amplitude and the period of the sine.

All these data are in close agreement with the present

measurements. At smaller deformations (up to about 0.25) the

experimental results show good agreement with the theories of

Taylor (1934) and Cox (1969), the experimental results by

Rumscheidt and Hason (1961) and the numerical results by Rallison

(1981). These results are compared with the present data in

figures 5.22 and 5.23. At small deformations the predictions by

Cox' formula coincide almost exactly with our experiments.

Comparison with the numerical calculations reveals that there is

very good agreement between the calculations and the experiments

for deformations up to about 0.5. For these deformations not only

the final deformations hut the entire transient response can be

calculated accurately. At higher deformations the numerical scheme

predicts unstable droplets whereas the experiments still show

stable droplets up to D=0.68. It is believed that this instability

is a numerical effect since at these deformations the mesh point

density is too small to describe the shape of the droplet

accurately. This is particularly notable in the x-y plane where

the deformation is largest. At these deformations the

circumference at the x-y plane is much enlarged. This instability

would probably be reduced when a mesh with more grid points,

especially in the circumferential direction in the x-y plane is

used.

These numerical calculations can also be compared with the

calculations by Rallison (1981) who used the same boundary

integral method to develop a similar numerical scheme for drops

having the same viscosity as the surrounding fluid. Rallison used

a grid with 117 mesh points. Rallison has used this scheme to find

the final deformation and orientations of a droplet in a simple

shear flow as a function of the capillary number.

0.11

Fig. 5.22

0.45

0.15

Fig. 5.23

178

Ralllspn

--

0.36

Comparison of numerical drop deformation results with the

data of Taylor (1934), Cox (1969), Barthes-Biesel (1972)

and Rallison (1981)

Comparison of final deformation measurements and

calculations with data from Rumscheidt and Mason (1961)

His calculations predict that stable drop deformations can be

obtained up to a critical capillary number of 0.42. The maximum

stable corresponding deformation is about 0.53, which is not as

close to the experimentally observed values, as is claimed by

Rallison (see figure 5.22, note that the dimensionless time scale

used by Rallison is l/4n smaller than the scale used in this

investigation and the capillary number used by Rallison is 4n

times the definition in this investigation). Some observations by

Rallison are not confirmed by the present investigation. Rallison

claims that smaller time steps were needed to maintain stable

drops when the deformation of the droplet increased. This was said

to be due to the fact that the mesh points come closer to one

another when the droplet becomes heavily distorted. As was already

shown in section 5.3.5 these effects were not observed for the

present scheme. Rallison (1981) also reports drop instability for

capillary numbers below the critica! value, provided the flow is

applied with sufficient suddenness. In the present investigation

nor, as far as we are aware, in any other investigation have these

effects ever been observed for a viscosity ratio of 1. The

experimental results on the response to a step profile for À=l,

some of which are given in figure 5.9, do not show any sign of

"overshoot" behaviour. For this viscosity ratio the deformation in

a step response experiment is a monotonously increasing function

of time. The numerical calculations for step profile responses do

not show any sign of this overshoot behaviour for the capillary

numbers at which stable final deformations were observed. It is

thus believed that this effect observed by Rallison (1981) is a

numerical artefact. The observations on the different modes of

break-up for capillary numbers just above and well above the

critica! value were confirmed in the present investigation.

For low capillary numbers it can be shown (see figure 5.24) that

Shanks transforms (Bender and Orzag, 1978) can be used very

successfully to estimate the final deformation.

179

180

O.OO Shaffls Transtarm

0.48

Q.36

o.24

0.12

Fig. 5.24

Ca•0.35 Shanks Transform

. . . . . . . . . . . . . . ' -.

Calcuiated deformation

Ca•0.20

Calculated deformation

4 8 12 16 20

-> Time /Dltrlle$S)

First order Shanks transforms performed on numerical

calculations at various shear rates for À = 1.0

This is done by extrapolation from the deformation vs time curve

shortly after the start of deformation. Shanks transforms can be

used to determine the sum of a slowly converging series (-z)k.

Yhen three successive terms A of such a series are known the

following relation will produce another series which often

converges much faster:

An+l An-1 - An2

An+l An-1 2An [5.31]

Convergence can sometimes further be speeded up by using higher

order Shanks transforms which are obtained by applying the

transform to the resulting series. Thus third order Shanks

transforms have been used to estimate the final deformations for

those capillary numbers where the numerical scheme becomes

unstable above a deformation of 0.5. The results are given in

table 5.2. The predictions can be compared with the experimentally

observed values for the maximum stable drop deformation of 0.68,

which is obtained for Sè:0.48. Theses predictions have been used

in the comparison of the data in figure 5.16.

TABLE 5.2 Predicted final deformations for À=l

Q D

0.40 0.54 ± 0.05

0.45 0.50 0.64 ± 0.05 0.72 ± 0.12

0.60 1.0 ± 0.3

Fora viscosity ratio of 1 in the step profile experiments the

deformation response indicates a first order behaviour of the

type:

-t1t0 D(t) = Dfinal ( 1- e [5.32]

182

The best fitted values for Dfinal and t 0 are given in table 5.3

and show that the time constant is not very dependent on the

capillary number. A comparison between the experimental and

numerical results and the first order predictions is shown in

figure 5.25.

TABLE 5.3 First order representation of step profile experiments

1 1 1 1 1 1 1 1 1 1 2 2 2 2 5 5 5 Q.8 0.5 0.5 0.5 0.01 0.01 0.01 0.01 0.01

2

0.1 0.2 0.3 0.4 0.45 0.097 0.194 0.284 0.389 0.431 0.1 0.2 0.3 0.4 0.1 0.2 0.4 0.4 0.1 0.2 0.4 0.38 0.74 0.93 1.39 1.67

Dfinal

0.11 0.21 0.32 0.51 0.59 0.10 0.23 0.36 0.52 0.62 0.11 0.23 0.35 0.43 0.11 0.23 0.53 0.47 0.11 0.19 0.46 0.29 0.56 0.67 0.81 0.88

2.2 2.1 2.2 3.2 3.9 1.3 2.4 2.9 2.4 2.8 3.3 3.5 3.8 3.4 6.4 7.2 9.6 2.5 1.6 1.2 2.0 1.2 1.0 1.2 1.0 1.0

type

num. num. num. num. num. exp. exp. exp. exp. exp. num. num. num. num. num. num. num. num. num. num. num. exp. exp. exp. exp. exp.

First order behaviour is also predicted by Cox' theory for the

transient behaviour of droplet in simple shear subjected to low

capillary numbers. Cox derived:

0.60 Oeformation /Dimfess]

----i 0.46~

1

0.36

0.24

Fig. 5.25

~L 4

+··

-> Tme {Dimless]

--- Cox' data

Numerical results

Experlmental results

- ··~ First-order model

Comparison of drop deformation calculations vith simple

first order behaviour

183

184

D [ -40t/19À2 [ 20 ]

Dfinal 1 + e + 2 19 À cos(t-T) - ~ sin(t-T)

[ [ 20 ]

2 2 ]-'h

• ; + (19 À) e -20t/(19À2)]

[5.33]

with 5 ( 19 À + 16)

4 À + 1) [ (20/2)2 + (19À) 2 l

T • tan -1 [-20 ] 19 À 2

which reduces to a simple first order behaviour for small

capillary numbers (2<0.15). For higher values of Q the first order

model is still a very good representation of the results. This

first order representation will also be used for the description

of the response of droplets to other shear rate profiles such as

the triangle profile.

The orientation results are compared with the calculations by Cox

(1969) and Rallison (1981) in figure 5.26. The comparison shows

close correspondence between the experimental observations and the

numerical results from Rallison and the present investigation. The

theoretical results by Cox deviate however clearly, even though

the deformation results are in very good agreement (see figure

5.22).

Fora viscosity ratio of 0.5 the numerical calculations show

deformations that are only marginally smaller than for À=l. The

orientation of the drops is somewhat closer to the streamlines

than for À.al. From table 5.2 it follows that the response times

are clearly faster than for À=l. For À-2 opposite effects are

observed. Note that the effect of increasing the viscosity ratio

on the drop response time is larger than the effect of decreasing

it. This is due to the fact that the drop response time is mainly

determined by the most viscous of the two fluid phases, whereas

the timescale is always made dimensionless with nc.

0

0

"' 0

:li "'! Cl

"' 0

!!; Co 0 •

]~ c: 0 <l> • ·r: 1Z Oo

ó .,, 0

"' "' 0

c:i "!

* 0.0 O.l 0.2 0.3 0.4 0.5

Capillary number

o - own numerical results "' - own experimental results + - theoreticol results by cox, 1969 " - numerical resulls by rollison, 1981

Fig. 5.26 Comparison of droplet orientation results vith the

resultsof Cox (1969), Rumsche1dt and Mason (1969) and

Rallison (1981).

The experimental results for a viscosity ratio of 0.01 show a

critical capillary number of 1.9 and a maximum stable drop

deformation of 0.91. These measurements agree well with

experimental observations by Grace (1982). The response times are

given in table 5.2 and are only a little bit faster than for the

higher viscosity ratios 0.5 and 1 as was expected.

The experimental and numerical results for À=5 (see fig. 5.13) are

in close agreement for small deformations only. A striking result

185

186

is the oscillation in the deformation observed for 9=0.8. For this

situation overshoot is clearly noticeable. These overshoot effects

cannot be due to fluid inertia. As was shown in section 5.4.2 the

shear rates following a step profile change in shear rate, can

initially be higher than the final shear rate. These inertial

effects, however, have died away after 0.5(R1-R2 )R2t~ seconds. The

overshoot behaviour is noticeable after 5nR/a. For these

experiments the overshoot in deformation thus occurs long after

the inertial effects have died away. Similar overshoot effects

have also been observed by Torza et al. (1972) for a viscosity

ratio of 25. These effects occur together with oscillations in the

orientation angle and are caused by the interaction between drop

deformation, drop rotation and energy storage at the interface. In

these situations the time scales related to drop rotation are

smaller than those related to drop deformation.

5.7.2 Triangle profile experiments

In these experiments it is observed (see figures 5.17-5.20) that

droplets will not breakup, even at very high shear rates (a few

times the critical shear rate), if the time in which the high

shear rate is experienced is very small. On the other hand, the

droplet will breakup at its critical shear rate if it gets enough

time to reach its critical deformation. Yhen a droplet was

subjected to a triangular shear rate profile with a maximum shear

rate of twice the critical value no breakup was observed for ramp

times of twice the characteristic drop deformation time nR/a. For

ramp times of 5 times the characteristic drop deformation time at

twice the critica! capillary number breakup was observed. Yhen a

droplet was subjected to a maximum shear rate of 4 times the

critical shear rate at a ramp time of 2.4 times the characteristic

drop deformation time breakup was also observed. Breakup thus

occurs faster when a droplet is subjected to higher shear rates.

The triangle profile experiments also indicate, just as was

observed for the step profile experiments, that the deformation

can be described by a simple first order model provided the

capillary number Q and the shear rate y are not too large. The

first order model response to a triangular shear rate profile has

been calculated from the step profile response using Fourier

transforms. The response of the droplet can be calculated from the

pulse response of the drop and the triangular pulse via

multiplication in the Fourier domain.

w(t) ~(t).s(t) [5.34]

where s(t) the triangular pulse and ~(t) the pulse response of the

drop and ~(t) the result of the convolution of the signals ~(t)

and s(t). In Fourier space this simplifies:

!(j) B(j).S(j) and ~(t) [5.35]

where the capitals denote the Fourier transforms of the signals.

In this case a simple multiplication between the

Fourier-transforms of the signal has to be carried out. It can be

seen from figure 5.18 that provided the capillary number and the

viscosity ratio are not too high, this first order model holds

well. This first order model can thus be used to calculate the

deformation in time of a droplet as a response to any kind of

shear rate profile. For small and medium droplet deformations the

actual drop deformation can thus be predicted for arbitrary shear

rate profiles using this first order model, without performing cpu

time intensive calculations.

The transient breakup experiments show that droplets tend to

breakup into 2,4,8,16,32 .•. fragments, even when the shear rates

are increased rapidly. Numbers of fragment in between this series

187

188

were rather rare. Yhen the shear rate is increased very slowly

this "binary" breakup behaviour can of course be expected, since

in quasi steady drop breakup in simple shear flow a drop generally

breaks up into two approximately equisized fragments. It is

however surprising that this "binary" breakup behaviour is also

observed in transient flows. Using this observation it is however

possible to predict the number of fragments, N, following a shear

rate profile with a maximum shear rate of Ymax·

3 ln(y /y .t)/ln2 N = 2 max cri [5.36]

Por the viscosity ratio 0.01 the "binary" breakup model does not

exactly hold, there must be a viscosity ratio effect which is not

included in the above description. This can be related to the

effect observed for low viscosity droplets that they not always

break into two almost identical fragments, but sometimes break

into three fragments.

5.7.3 Sinusoidal profile experiments

The results in figure 5.21 show some examples of the behaviour of

drops in oscillatory flows. It can be noticed that when the

droplets have been subjected to a sinusoidal shear rate profile

for a sustained period of time, the time averaged drop deformation

corresponds very well to the final deformation of a drop subjected

to a constant shear rate, equal to the time averaged shear rate of

the sinusoidal profile. This result was observed for all

frequencies of the oscillation. The maximum deformation, however,

was found to be strongly dependent on the frequency of the

oscillation. Por À ~ 1 and a period of the oscillation of •def'

the maximum deformation was found to be more than 12% higher than

the time averaged deformation. Yhen oscillatory shear rate

profiles with higher frequencies are applied the maximum

deformation is closer to the time averaged deformation, whereas

for lower frequencies the maKimum deformation will be larger.

Since breakup generally occurs when the drop deformation exceeds a

certain critica! value, it is thus expected that drops will

breakup in oscillatory flows with a time averaged shear rate well

below the shear rate required for breakup in quasi steady simple

shear flows, provided the period of the oscillation is smaller

than the characteristic drop deformation time \1Rla. For

oscillations with a much lower frequency the maximum shear rate

will become dominant. The results for a viscosity ratio of 5 show

a similar behaviour. These observations should be taken into

account when predicting drop breakup processes in flows in which

the droplet experiences rapidly oscillating flow conditions.

5.8 CONCLUSIONS

A numerical scheme has been developed, which allows prediction of

drop shapes in any transient linear flow field. The scheme is

based on the boundary integral method and uses 266 collocation

points to describe the drop surface. The scheme has been applied

to droplets with various viscosity ratios (0.5 < À< 5) subjected

to simple shear flows. Up to a drop deformation of 0.5 very good

agreement with experimental data was observed. At higher

deformations a larger number of collocation points is required,

although the predicted shape of the droplet prior to breakup is

already qualitatively correct.

In contrast to what has been suggested by other authors, the

critica! shear rate at which breakup occurs was experimentally

found not to differ between quasi steady shear situations and step

like shear rate profiles for viscosity ratios of 1 and smaller.

The deformation in these step profile experiments was always found

to be a monotonously increasing function of time. For viscosity

189

190

ratios greater than 1, overshoot behaviour in the deformation as a

function of time has been observed.

Yhen a drop is subject to a triangular shear rate profile with a

maximum shear rate which is higher than the critical shear rate in

quasi steady shear flows, breakup will only occur if this

triangular ramp stretches over a sufficiently long time. Yhen the

shear rate is applied for a very short time only, the drop will

relax to its spherical shape without breakup.

Yhen a droplet with À=l undergoes a shear rate which is much

higher than the critical shear rate in quasi steady si~ple shear,

the drop will break into a large number of fragments. The numbers

are very close the geometrical series with ratio 2. Lower

viscosity droplets generally breakup into a larger number of

fragments.

Yhen a droplet is subject to an oscillatory shear rate profile the

time averaged deformation was found to agree well with the final

deformation in a quasi steady shear flow with a shear rate equal

to the time averaged shear rate of the oscillatory profile.

Breakup in oscillatory flows is expected to be possible even when

the time averaged shear rate is smaller than the critical shear

rate, provided that the period of the oscillation is longer than

the characteristic drop deformation time.

5.9 REFERENCES

1 A. Acrivos and T.S. Lo, Deformation and break-up of a single

slender drop in an extensional flow, J. Fluid Mech. 86,

641-672, (1978)

2 D. Barthes-Biesel, Deformation and burst of liquid droplets

and non-Newtonian effects in dilute emulsions, Thesis Stanford

University, Michigan, USA, (1972)

3 D. Barthes-Biesel and A. Acrivos, Deformation and burst of a

liquid droplet freely suspended in a linear shear field, J.

Fluid Mech., 61, 1-21, (1973)

4 v. Bartok and S.G. Mason, Particle motion in sheared

suspensions. VIII singlets and doublets of fluid spheres, J.

Coll. Sci. 14, 13-26, (1959)

5 C.M. Bender, and S.A. Orszag, Advanced mathematica! methods

for scientists and engineers, McGraw-Hill, 1978

6 B.J. Bentley and L.G. Leal, A computer controlled four roll

mill for investigations of particle and drop dynamics in two

dimensional shear flows, J. Fluid Mech., 167, 219-240, (1986)

7 R.A. de Bruijn, Sealing laws for the flow of emulsions,

chapter 2 of this thesis

8 R.A. de Bruijn Newtonian drop break-up in quasi steady simple

shear flows, chapter 3 of this thesis

9 R.A. de Bruijn, Newtonian drop break-up in simple shear flows.

the tipstreaming phenomenon, chapter 6 of this thesis

10 J.D. Buckmaster, Pointed bubbles in slow viscous flow, J.

Fluid Mech. 55, 385-400, (1972)

11 J.D. Buckmaster, The bursting of slow viscous drops in slow

viscous flow, J. Appl. Mech.,40, 18-24, (1973)

12 R.G. Cox, The deformation of a drop in a general timedependent

fluid flow, J. Fluid Mech., 37, 601-623, (1969)

13 A. Einstein, Ann. Physik, 19, 289, (1906)

14 N.A. Frankel and A. Acrivos, The constitutive equation for a

dilute emulsion, J. Fluid Mech., 44,65-78, (1970)

15 H.P. Grace, Dispersion phenomena in high viscosity immiscible

fluid systems and application of statie mixers as dispersion

devices in such systems, Chem. Eng. Commun. 14, 225-277,

(1982)

16 E.J. Hinch, The evolution of slender inviscid drops in an

axisymmetric straining flow, J. Fluid Mech., 101, 545-553,

(1980)

192

17 D.V. Khak.kar and J.M. Ottino, Deformation and break-up of

slender drops in linear flows, J. Fluid Mech., 166, 265-285,

(1986)

18 O.A. Ladyzhenskaya, The mathematical theory of viscous

incompressible flow, Gordon and Breach, New York, (1963)

20 H. Lamb, Hydrodynamics, 6th ed. Dover press, NewYork, (1945)

21 J.M. Rallison, Note on the time dependent deformation of a

viscous drop which is almost spherical, J. Fluid Mech., 98,

625-633, (1980)

22 J.M. Rallison, A numerical study of the deformation and burst

of a viscous drop in general shear flows, J. Fluid Mech. 109,

465-482, (1981)

23 J.M. Rallison, The deformation of small viscous drops and

bubbles in shear flows, Ann Rev Fluid Mech 16, 45-66, (1984)

24 J.M. Rallison and A. Acrivos, A numerical study of the

deformation and burst of a viscous drop in an extensional

flow, J. Fluid Mech., 89, 191-200, (1978)

25 F.D. Rumscheidt and S.G. Mason, Particle motions in sheared

suspensions. XII deformation and burst of fluid drops in shear

and hyperbolic flow, J. Coll. Int. Sci. 16, 238-261, (1961)

26 H.A. Stone, B.J. Bentley and L.G. Leal, An experimental study

of transient effects in the break-up of viscous drops, J. Fluid Mech. 173, 131-158, (1986)

27 G.I. Taylor, The viscosity of a fluid containing small drops

of another fluid, Proc. Roy Soc. A 138, 41-48, (1932)

28 G.I. Taylor, The formation of emulsions in definable fields of

flow, Proc. Roy Soc. A 146, 501-523, (1934)

29 G.I. Taylor, Conical free surfaces and fluid interfaces

Proc. Int. Congr. Appl. Mech., llth, p790-796, Munich (1964)

30 S. Torza, R.G. Cox and S.G. Mason, Particle motions in sheared

suspensions. XXVII transient and steady deformation and burst

of liquid drops, J. Coll. Int. Sci. 38, 395-411, (1972)

31 G.K. Youngren and A. Acrivos, On the shape of a gas bubble in

a viscous extensional flow, J. Fluid Mech. 76, 433-442, (1975)

5.10 LIST OF SYMBOLS

B

D

terms of a series

set of coefficients describing a second order

curved surface around collocation point i

width of droplet

deformation: (L-B)/(L+B)

velocity gradient tensor

unit vector in k-th direction

f distribution function of external sources of

force

second order curved surface around point i

second order curved surface around point i with

respect to transformed coordinate system R1

~ rjrk

r r 3

3 r 1rjrk

4n r 5

curvature tensor

local curvature: tr ki

length of droplet

number of collocation points

3 dimensional space in neighbourhood of S

outward normal vector

pressure

scalar force (Fourier transform of q)

scalar force

drop radius

radii of curvature

orthogonal coordinate system in point i,

with one of the directions normal to the drop

surf ace

terms of Shanks series

shear rate ramp function

[-]

[m-2]

[m]

[-]

[s-1]

[-)

-1 [Pa m ]

[-]

[-]

[m-1)

[m-1]

[m-1]

[m]

[-]

[m]

(-] [Pa] [Pa] [Pa]

[m] [m-1]

[m]

[-)

[-]

193

194

LIST OF SYMBOLS (continued)

"

normal stress: ~.n

time

time constant

velocity (Fourier transform of ~)

disturbance velocity

velocity

posi tion

position of point source of force or position

at drop interface

Fourier transform of position ~

rate of shear

Dirac delta function

dynamic viscosity

viscosity ratio: nd/nc

kinematic viscosity

Fourier transform of t first order response function

interfacial tension

time

stress tensor

modified stress tensor

Fourier transform of ~

convolution of t and s

capillary number: nYR/a

subscripts

c continuous phase

crit critical

d droplet phase

dec decrease

i ,j ,k index of component

inc increase

[Pa]

[s)

[s) [m s-11 [m s-11

-1 [ms ]

[m]

[m]

[m]

[s-1] [-]

[Pa s]

[-]

[m2 s-11 [-]

[-] [N m-1]

[s]

[Pa]

[Pa]

l-1 [-]

[-]

LIST OF SYMBOLS (continued)

max maximum

s surf ace

S(e) limit approaching surf ace s from exterior

S(i) limit approaching surface s from interior

v volume

+ tangential direction

superscripts

in inside

inf at infinity

out outside

195

196

6. NEWTONIAN DROP BREAK-UP IN SIMPLE SHEAR FLOW: THE TIPSTREAMI

PHENOMENON

6.1 INTRODUCTION

Although most of the behaviour of small Newtonian drops in simple

shear flows is reasonably well established, the phenomenon of

tipstreaming is still poorly understood (Rallison, 1984).

Tipstreaming is an experimentally observed mode of drop break-up

(eg. Taylor, 1934 and Grace, 1982), in which the droplet takes upon

increasing the shear rate, a sigmoidal shape and a stream of very

small droplets is ruptured off the tips of the drop (figure 6.1).

This break-up behaviour is potentially very important since the

shear rates required for this type of break-up can be orders of

magnitudes lower than for the normal type of break-up, in which the

droplet is broken in two or three almost equally sized droplets

with a few tiny satellite drops in between (figure 6.1) and the

resulting droplets can be much smaller. Another area of potential

applicability is the emulsification of a droplet phase containing a

third phase (either solid or liquid) since the tipstreaming

phenomenon may be useful for separation processes (Srinivasan and

Stroeve, 1986 and Smith and van de Ven, 1985).

oo tipstreaming

/7 //000

C/ oook/

00c:P 0 •

fracture Fig. 6.1 Modes of drop breakup observed in simple shear

197

198

In this chapter an attempt is made to unravel the causes of the

tipstreaming phenomenon. In section 6.2 the literature on

tipstreaming will be surveyed and the suggested causes for the

phenomenon will be listed. In section 6.3 and 6.4 an experimental

and numerical investigation is reported, followed by a discussion

in section 6.5.

6.2 LITERATURE

Taylor (1934) was the first to report the tipstreaming phenomen in

simple shear flows. The materials he used and the operating

conditions are given in table 6.1. Taylor called it a transient

phenomenon since it disappeared when the shear rate was further

increased. Tipstreaming occurred at a capillary number of

0-0.71, whereas the expected critical capillary number for this

viscosity ratio (À=3•10-4) is 0-21 (chapter 3 of this thesis).

In agreement with Taylors observations Bartok and Mason (1959)

reported that when the shear rate was further increased a

tipstreaming droplet would resume an ellipsoidal shape. The

critica! capillary numbers for tipstreaming were reported to be

independent of drop sizes as for normal drop break-up.

Rumscheidt and Mason (1961) did observations of tipstreaming for

various fluid combinations (see table 6.1), all having in common

the low viscosity ratio l.3•lo-4 < À < 0.19. The critical

capillary numbers at which tipstreaming began were reported to be

almost constant: 0=0.5 with a standard deviation of 0.1. They

reported tip drops of about 50 µm, using mother drops of 0.5-2 mm

and inhibition of internal circulation for many of the fluid

combinations, including all those showing tipstreaming. When

inhibition occurs the observed phenomena can be expected to be

independent of the viscosity ratio. They observed that when a

droplet had been tipstreaming for some time at a certain shear rate

it would stop, but tipstreaming could restart when the shear rate

was further increased. Rumscheidt and Mason ascribed this effect to

a reduction of the drop volume, resulting in a drop of the

TABLE 6. 1 1.i terature observations o! tipstreaming

number reference drop pha.se cont; ~c ~d À r " 0

phase [Pas] [Pas] [-] (nm] CmN/ml [-]

Taylor (1934) CC14/parafin oil cs/w 11 3. 3E-3 3.0E-4 l. 57 23 0.71 1)

2 Bart.ok and CC14 es 9.4E-4 l.3E-4 0,332

Mason (1959} CC14 es 9.4E-4 l. 3E-4 0.698

CC14 es 7 9.4E-4 l. 3E-4 0.750

5 Rumscheidt and dibuthylphtalate 50 5.26 2E-2 4E-3 0.5-2 2.5 0.5+/-0.l

6 Mason (1961) ethylene glycol so 5.26 2E-2 4E-3 15. 0

7 distilled water so 5.26 lE-3 2E-4 38

water + O. 005% Tween 20 5.26 lE-3 2E-4 20

9 water + 0. 5% Tween 20 so 5.26 lE-3 2E-4 6.6

10 air so 5.26 1. 8E-5 3 .4E-6 20.9

11 silicone oil oco 6.00 1.1 0.19 4.5

12 glycerol oco 6.00 0.84 0 .14 10.4

13 silicone oil oco 6.00 2E-3 3E-4 2.7

14 air oco 6.00 l. 8E-5 3E-6 38. l

15 silicone oil g 0.8 3E-3 4E-3 26.3

16 CC14 es 9.0 1.2E-3 1.3E-4 38

17 Grace ( 1982) v 281.5 2.8E-3 1. OE-5 0.62

18 v 281.5 3. 7E-2 l. 3E-4 0. 55

19 4.55 1.0E-3 2.8E-4 0.58

20 oco 4. 55 1.3E-3 2.8E-4 0.65

21 oco 4. 55 3.0E-3 6. 6E-4 0.65

22 oco 4.55 5. 9E-3 1. 3E-3 0.69

23 281.5 0. 45 1. 6E-3 0. 93

24 v 281. 5 0.58 2.0E-3 0.69

25 oco 4.55 3.8E-2 8.3E-3 0.62

26 oco 4.55 7. 7E-2 1. 7E-2 0. 65

27 4.55 0.10 2.1E-Z 0.89

28 oco 4.55 0.15 3. 4E-2 0.58

29 oco 4. 55 0.25 5.4E-2 0.58

30 Smi tb and van water + TRS 1080 50 1.0 lE-3 lE-4 1.17 2.5 2.2 2 )

de Ven ( 1985)

abbreviations; cs/w corn sy:rup/water

corn sy:rup

sa silicone oil

oxidized castor oil

g glycerol

vorite 125

note: l) sbear ra te in this experiment: 0.95 .-1

2) 0. 49 .-1

199

200

capillary number below the critical one for tipstreaming. The

transition in droplet shape from rounded ends to pointed ends on

increasing the shear rate was observed to be sudden.

Torza, Cox and Mason (1972) only reported their tipstreaming

results qualitatively. They observed that the rate of increasing

the shear rate affected the break-up behaviour. For small values

of d 1/d t (2•10-4 s-2) the droplet did assume a sigmoidal shape,

but tipstreaming did not occur, whereas for higher values of

d ~/d t tip drops were released.

The results obtained by Grace (1982) have unfortunately not been

well documented. They however appear to confirm that tipstreaming

occurs at low viscosity ratios (À<<l) at almost constant capillary

numbers (0=0.65, standard deviation of 0.1). He probably also

observed that when the shear rate was further increased the droplet

resumed an ellipsoidal shape and could be broken at much higher

shear rates via a normal fracture mode.

Smith and van de Ven (1985) reported that an aqueous droplet

saturated with surface active material (see table 6.1) showed

tipstreaming behaviour with tip drops of about 10 µmin diameter,

although a droplet of distilled water did not show this behaviour.

Their main study concerned however the deformation and break-up of

droplets containing spherical solid particles. At low concentration

of these particles in the drop phase (3%) they observed that as the

shear rate was increased, the droplet became elongated and the

particles concentrated at the tips of the droplet. The particles

ruptured from the drops as singlets or doublets surrounded by

liquid of the drop phase. This phenomenon was observed at shear

rates that were too low to fracture the droplet itself. When given

sufficient time all particles were seen to leave the droplet from

the tips.

Tipstreaming related phenomena have also been observed in ether

type of flows, e.g. elongational flow (Taylor, 1934 and Sherwood,

1984) and the flow into a capillary (Carroll and Lucassen, 1976).

This literature survey shows that the tipstreaming phenomenon is

not understood although several factors have been shown to affect

it:

- the viscosity ratio

- the rate of increase of the shear rate

- the presence of surf actants

6.3 EXPERIMENTAL

6.3.1 Introduction

The experimental programme was aimed at unravelling the suggested

causes for the tipstreaming phenomenon, namely the viscosity ratio,

the rate of increase of the shear rate and the presence of

surfactants. Accordingly various sets of experiments were

performed. In the first set the viscosity ratio was varied from

2•10-4 to 0.1 for four different pairs of fluid combinations in

order to see the the effect of the viscosity ratio and the type of

fluid combination on the occurrence of tipstreaming (section

6.3.2). In the second set the tipstreaming phenomenon itself was

subject to a closer examination and the time dependency of the

phenomenon was studied (section 6.3.3). In the third set the effect

of the rate of increase of the rate of shear was studied (section

6.3.4). In the fourth set the effects of surfactants were studied

systematically by adding various levels of a surface active

material to a fluid combination that did originally not show any

tipstreaming.

The experiments were performed in the Couette device described in

chapter 3 of this thesis. The rheological properties of the model

liquids were measured with a Haake viscometer (type CVlOO) using a

concentric cylinder geometry (type ZClS). The interfacial

properties were either measured with a Wilhelmy plate or by the

201

202

method of drop deformation measurements in simple shear flow. The

latter method was generally applied when surfactants were added or

when the interfacial tension of a specific droplet in the Couette

device was to be measured. This method is based on the linear

dependency of the dimensionless drop deformation D = (L-B)/(L+B),

with L and B the length and the width of the droplet, on the

capillary number 0. This linear dependency is valid for small

drop deformations and constant interfacial tension. This method was

first described for viscosity ratios close to unity by Taylor

(1934), who derived a theoretical relation valid to the first order

in the deformation.

19 À + 16

16 À + 16 (6-1] D = 0

This relation was extended by Cox to be valid to the second order

in the deformation. Experimental data on the proportionality

constant have been obtained by Torza, Cox and Mason (1972) and

in chapter 5 of this thesis (see table 6.2).

TABLE 6.2 Steady drop deformations in simple shear flow

viscosity measured prediction by rat.ic proportionali ty Taylor

constant

8. 0 E-3 0.95 2)

8.0 E-2 1. 08 1). 1.01 0.25 1,02 1) 1. 04 0.55 1.00 1) l. 07 0. 82 1.20 1) 1.08 1. 0 1.12 2) 1.09 1.2 1. 02 1) 1.10 1. 7 1.20 1) 1.12 2. 6 0.93 ll l.14 3.6 1. 29 l) 10.2 0 .15 1)

18.6 0.085 1)

l) measurements from Torza et al, 1972 2) masurements from cba:pter S of this thesia

6.3.2 Viscosity ratio

To scan the effect of the viscosity ratio on the occurrence of

tipstreaming, a matrix of several types of fluids was used (see

table 6.3). The silicone oils were Rhodorsil oils type 47 (ex Rhone

Poulenc), the corn syrups were prepared by mixing corn syrups (type

Globe 01170 ex CPC Netherlands) with distilled water and the esters

were obtained by diluting Uraplast esters (ex DSM) with methyl­

ethyl-ketone. The experiments were done in a quasi steady way.

The results in table 6.3 show that tipstreaming was not observed

for the corn syrup and ester droplets in silicone oil, even though

the viscosity ratio was varied between 0.1 and 2•lo-4. For ester

droplets in corn syrup, however, tipstreaming was observed for each

of these viscosity ratios. The silicone oil droplets in corn syrup

showed generally no tipstreaming. Only for a 5 mPas silicone oil

tipstreaming was observed if the standard batch Rhodorsil 47VS

was used. If a 5 mPas silicone oil was made by mixing the standard

batches of 1 mPas and 10 mPas silicone oil, tipstreaming did not

occur.

These results indicate that the type of the fluids is very crucial

for tipstreaming to occur. The viscosity ratio is not very

important provided it is much smaller than unity. The S mPas

silicone oil drop phase results indicate that minor components in

the liquids can determine whether or not tipstreaming will occur.

6.3.3 Time dependency

In this section the tipstreaming phenomenon itself is subject to a

closer examination. Especially the time dependency and the history

effects involved in tipstreaming will be studied.

When a droplet was inserted in the Couette device and the shear

rate was slowly increased, the droplet at first assumed an

ellipsoidal shape. At higher shear rates a sigrnoidal shape was

assumed and at capillary numbers of around 0.5 tipstreaming

occurred. When this shear rate was maintained it was observed that 203

TAB LE 6.3 Occurrence of tipstreaming

6-3A: 5. 0 Pas silicone oil as 6-3C: 5. O Pas corn syrup as

continuous phase continuous phase

drop phase À tip- drop phase À tip-

streaming streaming

100 mPas corn syrup 2.0E-2 no 100 mPas silicone oil 2.0E-2 no

50 mPas corn syrup l.OE-2 no 50 mPas silicone oil 1.0E-2 no

10 mPas com syrup 2.0E-3 no 10 mPas silicone oil 2.0E-3 no

5 mPas corn syrup 1. OE-3 no 5 mPas silicone oil 1. OE-3 yes/no

1 mPas silicone oil 5. OE-4 no 100 mPas ester 2.0E-2 100 mPas ester 2. OE-2 yes

50 mPas ester l.OE-2 no 50 mPas ester 1.0E-2 yes

10 mPas ester 2. OE-3 no 10 mPas ester 2.0E-3 yes

5 mPas ester l.OE-3 no 5 mPas estér 1.0E-3 yes

1 mPas ester 2.0E-4 no 1 mPas ester 2. OE-4 yes

6-3B: 1. 0 Pas silicone oil as 6-30: 1.0 Pas corn syrup as

continuous phase continuous phase

100 mPas corn syrup 1. OE-1 no 100 mPas silicone oil 1. OE-l no

50 mPas corn syrup 5.0E-2 no 50 mPas silicone oil 5.0E-2 no

10 mPas corn syrup l.OE-2 no 10 mPas silicone oil l.OE-2 no

5 mPas corn syrup 5.0E-3 no 5 mPas silicone oil 5.0E-3 yes/no

l mPas silicone oil l. OE-3 no

100 mPas ester 1. OE-1 no 100 mPas ester 1. OE-1 yes

50 mPas ester 5.0E-2 50 mPas ester 5. OE-2 yes

10 mPas ester l.OE-2 10 mPas ester 1.0E-2 yes

5 mPas ester 5.0E-3 no 5 mPas estèr 5.0E-3 yes

1 mPas est.er 1. OE-3 no 1 mPas ester 1. OE-3 yes

204

tipstreaming ended after a certain period, varying between 10

seconds and a few minutes. However when the shear rate was further

increased tipstreaming often started again at a higher shear rate

and the same process could be repeated until the shear rate became

too high and the droplet assumed an ellipsoidal shape again. At

higher shear rates a normal mode of droplet fracture was observed.

Contrary to the statement by Rumscheidt and Mason (1961), who

ascribed the ending of tipstreaming to a decrease in drop volume

and a subsequent lower capillary number, the observed ending and

restarting of tipstreaming was not merely due to a reduction of the

capillary number. When ending of tipstreaming was observed, the

drop volume was decreased very little, the decrease was usually

hardly detectable, while the necessary capillary number to restart

tipstreaming was substantially larger.

When a droplet was subjected to a slowly increasing shear rate and

tipstreaming had started at a certain shear rate it generally ended

again at a higher shear rate. When the shear rate was then reduced

to zero and the experiment was repeated it was observed that

tipstreaming would start at significantly higher shear rates than

before, although the ending occurred at comparable shear rates. A

typical sequence of such experiments is given in Table 6.4. When a

droplet was left at rest after tipstreaming for a period of 10-30

minutes, tipstreaming would start again at the initia! critica!

shear rate.

It was further observed that tipstreaming never occurred when the

shear rate was slowly decreased from close to the critical shear

rate for normal fracture type drop break-up.

These observations of the tipstreaming phenomena indicate that

there appears to be some sort of depletion effect and the

occurrence of tipstreaming depends on the history of the droplet.

205

TABLE 6.4 History effects on tipstreaming

drop phase: 100 mPas esther

continuous phase: Pal$ corn syrup

interfacial tension: 26. 7 mNm-1

viscosity ratio: 0.024

drop radius shear rate capillary number remarks

[mm] [s-1] [-]

0.28 start of experiment

5. 4 0.25 start. af tipstre:aming

10. 4 0.48 end of tipstreaming

0.28 0 back to rest

7. 5 0 .34 start of tipstreaming

9.8 0. 45 end of tipstreaming

0.28 back to rest

9.0 0.41 start of tipstreaming

10. l 0.46 end of tipstreaming

13.6 0.62 fracture of droplet

0.35 0 start of experiment

4. 3 0,25 start of tipstreaming

a.z o. 47 end of tipstreaming

12.3 0.71 fracture of droplet

206

6.3.4 Acceleration

To examine the effects of the rate of increase of the shear rate,

experiments were performed with model liquids, that had shown

tipstreaming in the previous experiments, and with model liquids

that had not exhibited any tipstreaming in the previous quasi

steady experiments.

Several sets of the tipstreaming experiments are given in Table

6.5. These experiments were performed at various constant rates of

increase of the shear rate ~. In this table the shear rates at

which tipstreaming first occurred are given, together with the

corresponding capillary number and the ratio r of the

characteristic drop deformation time (tdef - ~c r/a) to the time of

increase of the shear rate up to the moment tipstreaming started

tramp· The results show that there is a strong tendency for the

critical shear rate to increase at higher accelerations.

The systems that did not show tipstreaming in the steady state

experiments did not show it in these acceleration experiments

either.

6.3.5 Surfactants

To study the effects of surfactants on the tipstreaming phenomenon

surfactants were added to the dispersed phase of a fluid

combination that did not show any tipstreaming, namely a 5 Pas corn

syrup/water mixture as the continuous phase and low viscosity

silicone oils (10 and 50 mPas) as dispersed phases. It was found

that glycerol-1-mono-oleate was a good surface active material for

these fluids. At saturation concentration the interfacial tension

was reduced to 2-3 mN/m. For both fluid combinations a set of

experiments was performed with a range of surfactant concentrations

added to the droplet phase. The results are given in Table 6.6. The

interfacial tension values given in this table for the non zero

levels of surfactant were obtained via the drop deformation

method. 207

TABLE 6.5 Effect of accéleration ( cont.inuous phae.e is corn syrup)

drop phase 'Id "• À q r 1 'Y tramp tdef T ll

[Pas! [Pas] [-] [mN/ml [mm] [l/sJ [l/s2J [•) [s] (-] [-]

ester 0.1 10 0.01 27 .9 0.51 3.9 l.O 3.9 0.18 O.Oli6 0.62

0.49 3.1 0.5 6.2 0.18 0.029 0.47

0.48 2.3 0.1 23 0.18 0. 0078 0.35

0.46 2.1 0.05 42 0.18 0. 0043 0.29

silicone oil 0.005 5 0 .0011 26.7 0.39 4.9 1.0 4.9 0. 07 0 .014 0.36

0.26 5. 7 0.5 11.4 0.05 0 .004 0.27

0.56 2.8 O.l 28 0.10 0. 004 0.28

0. 54 2. 7 0.05 54 0.10 0.002 0.26

ester 0.1 10 0 .01 33. 0 0.52 3.9 1. 0 3.9 0.16 0 .04 0 .49

0.63 3. 7 1.0 3. 7 0.19 0.05 0. 57

0.63 3.6 1.0 3.6 0 .19 0.05 0. 54

0.63 3.8 1.0 3.8 0 .19 0.05 0.57

0.59 2.9 0.5 5.8 0.16 0.03 0. 42

0 .60 3. 7 0.5 7 .4 0.18 0.03 0.52

0. 72 2.6 0.5 5.2 0.22 0. 04 0.46

0.69 2.6 0. 5 5.Z 0.21 0.04 0.43

0 .65 1.9 0.1 19 0.20 0.01 0.29

0.63 2.2 0.1 22 0 .19 0.009 0.33

208

TAB LE 6.6 Effect of surfactants on tipstreaming

TABLE 6,6A: drop phase 50 mPas silicone oil with various l&Vèls of

glycerol-l-mono-oleate

continuous phase: 5 Pas corn syrup

viscosi ty ratio : 0 .01

No. surfaotant " 1 n break-up rt.ip "tip [%] [mN/mJ Cnml [l/•l [-] mode [µm] [N/ml

30 0.391 30.1 2.0

0. 430 27.4 l. 7 f

0.560 21.7 1. 7 f

4 0. 0001 30 0 .361 >16.2 >1.0 n.t

5 0.435 >12.9 >1.0 n.t

6 0.360 >19.4 >1.2 n.t

7 0.554 19.9 1. 6 f

8 0.357 22.2 1.3 f

0.0005 30 0.305 >14.5 >0.8 n.t

10 0.374 >14.9 >1.0 n.t

11 0.586 5. 7 0.59 t 10

12 20.1 2.1 f after t

13 0.001 26 0.416 >17 .5 >1.2 n.t

14 0.483 7 .2 0.61 t 15 o. 736 5.6 0.69 t 10 5

16 0.854 3.6 0.52 t 20 2

17 0.550 6.7 o. 71 t

16 0.594 6.9 0.80 t

19 19.l 2.2 f aft.er t

20 0. 502 7 .l 0.69 t 6

21 0.573 6.3 0.69 t 6

22 0.565 6.0 0.68 t

23 20.5 2.3 f after t

24 30 0.392 29.1 1.9 f*

25 26 0.416 6.5 0 .64 t

26 0.463 10.0 0 .87 t

27 24.4 2.1 f after t

28 0.270 10.9 0.52 t

29 0 .431 6.0 0.45 t

30 0.567 5.6 0,55 t 10

Abbreviations: f =fracture t - tipstreaming n.t =no tipstreaming f af ter t fracture af ter

* drop is fragment of exp. 23 tipstreaming td - tipdropping t af ter r tipstreaming after rest

209

TABLE 6,6A: continued

No. surfactant " r ~ n break-up rtip "t..ip [%] CmN/mJ [Dl!l] [l/sJ [-] mode [/Jlll) [N/ml

31 0.005 22 0.386 6.81 0.58 t

32 0.370 6.5 0.53 t

33 0.349 7.2 0.55 t 8

34 0.197 >20.1 >0.84 n.t..

35 0 .214 13. 7 0.57 t

36 0.227 11. 8 o. 46 t

37 0.2419.00.56t

38 0.250 9.1 0.59 t

39 0.263 6.6 0.46 t

40 0.271 7 ,5· 0.53 t.. 10

41 0.276 10 .5 0.62 t

42 0.334 5.3 0 .49 t 10

43 0.338 7 .3 0. 53 5-10

44 o. 354 7 .6 o. 47 t

45 o. 516 5.1 0.46 t 10

46 0.620 3. 4 0.51 20

47 0.01 11 0.472 3.3 0. 70 t

48 0.449 3.0 0.62 t 18

49 0.05 0.359 1. 3 0.59 f

50 0.319 1. s 0,60 f

51 0.302 1.4 0.55 f

52 0.080 4.8 0. 49 f

53 0. 042 6.7 0 .36 f

54 0.237 1.9 0.57 f

55 0.221 3.1 0.87 f

56 0.1 3, 6 0. 440 1.3 0 .59 f

57 0.281 2.0 0.59 f

sa 0 .197 2.4 0.48 f

59 0.152 4. 3 0.67 f

60 o.s 2.~ 0.324 1.2 o. 75 f

61 0. 214 1.6 0 .66 f

62 0.250 1,4 0.68 f

63 0 .178 1. 7 0.61 f

64 0.119 2.4 0.56 f

65 0.197 1. 9 o. 73 f

210

TABLE 6.6B: drop phase 10 mPas" silicone oil wi th various levels of

glycerol-1-mono-oleate

continuous phase: 5 Pas corn syrup

viscosi ty ratio ; 0.002

No. surfactant <1 7 (l break-up rtip qtip [%] [mN/mJ [mm] [l/•l [-] mode [jllll] [N/m]

l 30 0 .341 >14.6 >1. 0 n.t

2 0.350 >19.2 >1.3 n.t

0.280 >18. l >1.0 n.t

0. 0001 30 0.398 >25.3 >1.8 n.t

0.411 >19.1 >l.8 n.t

0.430 >15.4 >1.2 n.t

0.484 >11.4 >l.'4 n.t

8 0.387 >23.8 >1.5 n.t

9 0.496 >15.0 >l.4 n.t

10 0. 0005 30 0.398 >17 .1 >1.2 n.t

ll 0.385 >17 .6 >1.2 n.t

12 0.318 >13.3 >0.8 n.t

13 0. 001 30 0.340 9. 0 0.58 td

14 9.5 0.61 t 24

15 0. 338 >13. 7 >0.74 n.t

16 0. 395 >ll.6 >0.73 n.t

17 0.335 10.2 0 .59 td

lB 0.334 10.2 0.59 td

19 11.1 0.64 t

20 0.005 22 0. 567 5.1 0.73 t 34 10

21 0. 606 4 .4 0.67 t 22 0 .537 4.6 0 .58 t

23 0. 718 5.0 0. 71 t 43 15

24 0.308 7. 0 0.54 t 10

25 0.322 7 .2 0,58 t

26 0.320 7. 0 0.56 t

27 0.319 7. 5 0.60 t

211

TABLE 6.6B: continued

No. surfact.ant Il' r :, Il break-up rtip "tip (%] [mN/mJ [mnJ [1/s] [-J mode [µml [N/ml

28 0.01 16 0.339 4.3 0.51 t 15

29 0.356 5.0 0.62

30 0.321 6.0 0.63 t 10

31 0.261 6.9 0.58 t 10

32 0.271 7 .0 0.62 t

33 0.510 3.6 0.59 t 8

34 0.218 8.1 0.58 t 35 0.407 4,6 0.60 t

36 0.299 6.3 0.61 t 37 0.450 4. 7 0.53 t 20 2.3

38 0.498 4.2 0.62 t 20 2.2

39 0.05 3.5 0.393 l.8 l.O t

40 0.393 2.0 1.1 t 10

41 0.359 2.4 l.3 t 10

42 0.312 2.5 l.1 t

43 0.1 2.2 o. 353 l.6 1.3 t 10 44 0.350 1.5 1.2 t aft.er r 10

45 1,8 1.5 t aft.er r 10

46 2.2 1. 8 t after r 10

47 0.5 1.7 0.343 1.8 2.1 f

48 0,078 2.9 o. 74 f

49 0.185 1.5 0.94 f

212

For bath the 10 mPas and the 50 mPas silicone oil drop phases the

results show the same trend. When no or extremely low levels of

glycerol-1-mono-oleate were added the systems do not show any

tipstreaming. At a certain level, however, tipstreaming is

observed, whereas at much higher levels of the surfactant no

tipstreaming was observed anymore and the droplets could only be

braken up via a normal fracture mode of break-up. For the 10 mPas

silicone oil drop phase, tipstreaming was not observed at

surfactant levels below 0.001%(wt). At 0.001%(wt) sometimes no

tipstreaming could be observed at all and only droplet fracture was

observed, whereas for other droplets tipstreaming did occur,

although sometimes the tip droplets were not emitted continuously

(tipstreaming) but more intermittently (tipdropping). For the

surfactant levels between 0.005%(wt) and 0.1%(wt) tipstreaming was

observed in all experiments. At the highest level of surfactant,

0.5%(wt), a normal fracture mode of drop break-up was observed,

without passing through a tipstreaming stage. Higher levels of

surfactants were not tried because of supersaturation. The radius

of the emitted tip droplets was found to vary between 8 µm and 25

µm. For the 50 mPas silicone oil drop phase a very similar trend

was observed although some variations were observed. At surfactant

level below 0.001%(wt) no tipstreaming was ever observed. At higher

levels tipstreaming was generally observed, although some droplets

exhibited drop fracture without passing through a tipstreaming

stage. At surfactant levels of 0.05%(wt) and higher tipstreaming

was never observed anymore.

For several emitted tip droplets the interfacial tension was

estimated from the deformation method. Even though the results are

not very accurate because the tip droplets were very small and the

droplet deformations could not be determined very accurately, these

experiments gave some striking results. The interfacial tension of

the tip droplets were invariably lower (usually much lower) than

the interfacial tension of the mother droplet. The interfacial

tension of the tip droplets was often close to the saturation

value.

213

214

The results described above indicate that tipstreaming can occur

if an interf acial tension gradient can be developed in the droplet

surface at a certain surfactant concentration range with a lower

interfacial tension near the tips of the droplet.

6.4 NUMERICAL

The numerical technique described in chapter 5 of this thesis has

been used to calculate the position of the rnaterial points on the

rnidplane of the drop surface, when the droplet is subjected to a

sirnple shear flow. The nurnerical calculations are based on a

surface integral rnethod, which rewrites the Stokes-equations in and

around the droplet, via Fourier transforrns, toa surface integral.

Thus one only needs values of quantities at the drop surface to

calculate the velocity of a point at the surface. This surface

integral was solved after discretizing the surface of the droplet.

The rnethod was slightly modified for the calculations under

discussion. The calculation of the velocity of each point of the

surface was identical, but the new position of the point was now

sirnply calculated by:

x x + v • dt -new -old

[6-2]

In chapter 5 of this thesis this sirnplest rnethod to calculate the

new point positions was not used, but a more sophisticated

redistribution technique to enhance numerical stability. Here,

however, it was desired to follow the position of a material point

on the drop surface, so that this basic method had to be applied.

Calculations were performed for droplets characterised by viscosity

ratios of 0.5, 1.0 and 2.0, that were subjected to a step profile

sirnple shear flow with capillary nurnbers ranging from 0.1 to 0.4.

The results are presented in figure 6.2 where the contours of the

rnidplane of the droplets are given at various time intervals after

the onset of the flow. The results clearly show that once the

TABLE 6.7 Concentration of material points for various viscosity ratios

Table 6.7A: Comparison at constant time, T-0.05

À D Dm in Dmax [ - l [ - l [ - l [ - l

2.0 0.046 0.96 1.04 1.0 0.067 0.92 1.07 0.5 0.082 0.82 1.16

Table 6. 7B: Comparison at constant drop deformation, D=0.05

>. T Droin Dmax [ - l [ l [ - ] [ l

2.0 0.55 0.96 1.04 1.0 0.36 0.94 1.05 0.5 0.29 0.90 1.09

TABLE 6.8 Viscous and interfacial stresses in tipstreaming experiments

'Ic "m "t l:!.u r Vstr intstr [Pas] [l/s] [mN/m] [mN/m] (mN/m] [mm] [Pa] [Pa]

5. 0.736 28. 28.5 5.0 3.6 26.0 2.0 24.0 0.854 18.0 28.1 5.0 3.4 22.0 2.0 20.0 0.620 17.0 32.2 5.0 5.1 22.0 10.0 12.0 0.567 25.5 21.2 5.0 5.0 22.0 15.0 7.0 0.718 25.0 9.7 5.0 4.7 16.0 2.3 13. 7 0.450 23.5 30.4 5.0 4.2 16.0 2.2 13.8 0.498 21.0 27.7

215

Time = 0.00 {dim.less) Time = 0.25 {dim.less) Time = 0.50 (dim.less)

Time == 0.75 (dim.lessl Time = 1.00 (dim.lessl Time == 0.00 (dim.less)

Fig. 6.2.a Droplet contours in simple shear flow as a function of the dimensionless time for a viscosity ratio of 0.5 and a Capillary number of 0.3

216

droplets obtain an ellipsoidal shape, the concentration of the

material points around the points of the ellipsoid increases,

indicating a contraction of the drop surface, whereas along the

long side of the disk the concentration decreases, indicating an

expansion of the drop surface. This effect is more pronounced for

lower viscosity ratios. Closer examination of the contours on

figure 6.2 reveals that the surface is most contracted just over

the pointed ends of the ellipse. This corresponds exactly with the

position of the tips during tipstreaming, These effect have been

quantified in figures 6.3 and 6.4. Figure 6.3 gives the

dimensionless drop deformation as a function of time for each of

the experiments. Figure 6.4 gives the distances between the two

neighbouring grid points at the midplane of the droplets that are

Time = 0.00 ldim.lessl Time = 0.50 (dlmJessl Time = 1.00 (dlm.lessl

Time = t50 (dimJessl Time = 2.00 (dîm.less) Time = 3.00 (dim.lessl

Fig. 6.2.b Droplet contours in simple shear flow as a function of the dimensionless time for a viscosity ratio of 1.0 and a Capillary number of 0.3

closest to one another and that are the farthest apart. These

distances have been normalized with the initial spacing between the

grid points at the midplane. These results show that for a

capillary number of 0.3 and a dimensionless time of 0.5 the

concentration effects are about four times more pronounced for the

viscosity ratio of 0.5 than for 2.0. (see Table 6.7). These

calculations were performed assuming a constant interfacial

tension. This assumption is only valid for small times.

217

218

1 ~'

( \ ' · .TT; \ . /

'

Time = 0 00 (dim.lessl Time = 0.50 (dimJess) Time == 1.00 (dim.less)

Time = 2.00 (dim.less) Time " 3.00 (dim.less) Time = 5.00 (dimJessl

Fig. 6.2.c Droplet contours in simple shear flow as a function of the dimensionless time for a viscosity ratio of 2.0 and a Gapillary number of 0.3

For larger deformations the numerical calculations are not anymore

appropriate, since interfacial tension gradients may than become

important and these effects are not accounted for in these

calculations.

6.5 DISGUSSION

The experimental and numerical results seem to indicate that

tipstreaming is a result of the built up of interfacial tension

gradients over a droplet resulting in low interfacial tensions

À= 1.00, Time= 0.50 À -= 2.00, Time = 0.50

À -= 0.50, Time-= t.00 À -= 1.00, Time -= 1.00 À = 2.00, Time = 1.00

Fig. 6.2.d Droplet contours in simple shear flow as a function of the viscosity ratio for dimensionless times of 0.5 and 1.0 and a Capillary number of 0.3

close to the points of the ellipsoids. Such interfacial tension

gradients make the drop surface less mobile allowing the shear

stresses exerted by the continuous phase to pull out a stream of

tip droplets. This explanation however can only hold when the shear

stresses exerted by the continuous phase are large enough to

maintain such an interfacial tension gradient and when the

diffusion of the surface active material from the droplet to the

surface is slow enough not to interfere with the build up of the

interfacial tension gradients. The restarting of tipstreaming after

a droplet was brought back to rest and a new experiment was begun,

219

Ul IJ)"' (1) • __,o E _ _, n

vc::i

c QN

jc::i 0 E L-Oo

<+-(j) Do

tf)

0

({)

({)"' ID •

__,o

c _g ~ _,..)

0 E L Oo

<+-(])

Do

ei+'-----,.---,-----,----,-----, D-4f:'---~---.------~--~

0.0 1.0 2.0 3.0 4.0 Eme !&m.Less)

LEGEND ei =Curve l t;, =Curve 2 x =Curve 3 o •Curve 4

n - 0.10 & >. - o.5o 0 - 0.20 &. À = 0.50

0 - 0.30 &. À = 0.50

tl - 0.40 & /, - 0.50

5.0 0.0 1.0 2.0 3.0 4.0 T~me (&m. Less l

LEGEND @=Curve o- 0.10 Il/.. - LOO

"_ 2 0 = 0.20 &: ;\ • 1,00

"'""' 3 0 = 0.30 & À= 1.00

•= 4 n• o.40 & /, - 1.00

•= 5 o- 0.30 & 1.- 2.00

5.0

Fig. 6.3 Drop deformations as a function of time for various Capillary numbers and viscosity ratios

220

indicate that in about a minute the concentration of surface active

material can be restored.

An order of magnitude calculation to see whether or not the shear

stresses exerted by the continuous phase will be able to maintain

an interfacial tension gradient on the droplet can be performed by

comparing both stresses for actual experiments. The viscous

(j) u c

Cl

t'Î

0 "' :;; à . -' D

c; Ü-J-~~~~~~~~~~~~T·~--,

0.0 1.0 2.0 3.0 TL.me [ dlm. Less l

LEGEND o Curve l li =Curve 2 x = 3

4

o- 0.10 tl.,.._ 0.60

n = 0.20 &. >. = 0.60

0 = 0.30 e. 'A - 0.50

0: - 0.40 &. /\. - 0.50

Cl

N

(IJ (IJ tl)

ID_: -'

E .J u c;

ID u c 0 tf)

(;i ci . -' D

Eme (di..m.Lessl

LEGEND ©-= Cut""ve 1 11 =Curve 2 "=Curve 3 •=Curve 4 •=Curve 5

0 - 0.10 " À - 1.00 0 - 0,20 6 À - 1.00

(l - 0.30 & À= 1.00

o - 0.40 & À - 1.00 a-o.3o a >.-a.oo

5.0

Fig. 6.4 Minimum and maximum distances between material points at the drop surface as a function of time for various Capillary numbers and viscosity ratios

stresses will be of the order of: ~c7• whereas the interfacial

gradient stresses will be of the order of Aa/r, in which Aa is

the difference in interfacial tension between the mother droplet

and the tip droplet. In table 6.8 these order of magnitude

calculations are given for all the experiments for which the

interfacial tension of the tip droplet was determined. The results

show that the two stresses are reasonably well of the same order of

magnitude. This was already expected beforehand since the critica!

capillary number was of the order 1 and Aa was of the order of a. 221

222

Whether or not surfactant diffusion occurs on these time scales is

somewhat more difficult to estimate, since it involves an

estimation of the diffusion coefficient of glycerol-1-mono-oleate

in silicone oils. Tipstreaming was found to occur typically at a

concentration, w, of 0.005%(wt) in the droplet phase. The number of

surfactant molecules in a droplet is thus given by

(6-3)

where p is the density of the drop phase, p - 1000 kg m·3, Nav the

Avogadro nurnber and M the molecular weight of glycerol-1-mono­

oleate, M-348, resulting in Nd - 4.5•1012 fora 0.5 mm droplet.

The number of molecules needed to cover the undeformed drop surface

completely is given by

(6-4]

where Am is the area of a glycerol-1-mono-oleate molecule in the

drop interface, Am-38A2, resulting in Ns - 8.3•1012. Thus

approximately a f if th of the total number of surf actant molecules

in the droplet suffices to cover the droplet surface completely. To

estimate the time scales involved in surfactant diffusion in the

droplet, one can use the standard solutions to transient diffusion

in a sphere. These show that complete equilibrium is obtained when

Fo- IDt/r2 - 0.5 [6-5)

where Fo is the Fourier number. At a Fourier number of about 0.01

already a substantial amount of diffusion from a layer of thickness

r/10 has taken place (such a layer contains already enough

surfactant molecules to cover the drop surface completely).

A reasonable guess of the diffusion coefficient, taking into

account the high viscosity, seems to be

ID [6-6]

This would imply that complete equilibrium by diffusion from a

thin layer near the drop surface can, for a droplet with r-0.5 mm,

already occur after 10 minutes. The time scale for equilibrium

corresponds reasonably well to experimental time scales showing

that when the droplet has been brought to rest after a tipstreaming

experiment and an experiment was started again after about a minute

that tipstreaming would take place, but at higher capillary numbers

and that when a droplet was brought back to rest after a

tipstreaming experiment and an experiment was started after 15

minutes, tipstreaming occurred again at the original capillary

number.

If the hypothesis is correct that tipstreaming can only occur when

interfacial tension gradients are present, the tipstreaming

conditions must also fulfil the rapidly varying flow criteria

(chapter 2 at this thesis). These criteria state that interfacial

tension gradients can only be present when the Peclet number based

on a layer from which diffusion must occur to equilibrate the

interfacial tension, is smaller than 1.

Pe Ud/ ~ [6-7]

An estimation of the layer thickness d for low concentrations of

surfactants can be obtained from the lower limit of the Langmuir

adsorption isotherm, such that there are as many surfactant

molecules present in this layer as there are needed for equilibrium

adsorption:

d [6-8]

223

224

with r 00 the maximum surface concentration of the surfactant and

r~/C1 the slope at low concentrations of the Langmuir adsorption

isotherm. r 00 can be estimated from the size of the surfactant

molecules:

[6-9)

with N the Avogadro number and Am the area of the adsorbed

molecule. A typical value for Am is 4•lo-19 m2, resulting in

r~ = 4.10-6 molesm·2. The order of magnitude for C1 follows from

the interfacial tension measurements at several surfactant

concentration levels (see table 6.6). A typical value for C1 is

given by 0.005 wt% - 0.1 moles/kg. This implies that Pe = 1 is

reached for a shear rate of

[6-10]

Since tipstreaming is only thought to be possible if Pe>>l, which

is the rapidly varying flow condition and tipstreaming was observed

for shear rates typically between 1 and 10, (see section 2.3.2 and

2.3.3), the above order of magnitude calculations confirm the

hypothesis that tipstreaming in simple shear flows is due to

interfacial tension gradients near the tip of the droplet.

Using this hypothesis it is also possible to predict the presence

of tipstreaming under different emulsification conditions. The

main differences are the smaller initial droplets, higher shear

rates and lower viscosities, resulting in higher diffusivities.

When a typical value ofID-5•lO·lO is taken for the diffusivity of

surfactants in a low viscosity liquids, say 50 mPas, the shear

rate that corresponds with Pe=l becomes 100 s-1. Since the applied

shear rates applied during emulsification in liquids of about 50

mPas are generally in excess of this shear rate, the flow

conditions can be termed rapidly varying. It is thus likely that

tipstreaming can also occur during emulsification in lower

viscosity liquids. When however higher surfactant concentrations

are present, the layer thickness d becomes much smaller than

described by r 00/C1, resulting in much higher shear rates required

for tipstreaming.

6.6 CONCLUSIONS

Tipstreaming can occur when interfacial tension gradients can

develop, resulting in low interfacial tension at the tips and a

higher tension elsewhere.

Tipstreaming will not occur at extremely low surface active

material levels, when the interfacial tension can not even be

lowered locally, nor at high levels where there is so much surface

active material present that the interfacial tension will be low

all over the droplet.

Sealing laws based on the above hypothesis have been formulated.

These allow prediction of the occurrence of tipstreaming under

different emulsification conditions.

6.7 REFERENCES

1. W. Bartok and S.G. Mason, Particle motion in sheared

suspensions. VIII singlets and doublets of fluid spheres, J.

Coll. Sci. 14, 13-26, {1959)

2. R.A. de Bruijn, Newtonian drop break-up in quasi steady simple

shear flows, Chapter 3 of this thesis.

3. R.A. de Bruijn, Sealing laws for the flow of emulsions, Chapter

2 of this thesis.

4. R.A. de Bruijn, Deformation and break-up of newtonian drops in

transient simple flows, Chapter 5 of this thesis.

5. B.J. Carroll and J. Lucassen, in Theory and practica of

emulsion technology, Chap 1 Smith, A.L. ed. Academie, London,

(1976)

225

226

6, R.P. Grace, Dispersion phenomena in high viscosity immiscible

fluid systems and application of statie mixers as dispersion

devices in such systems Ghem. Eng. Commun. 14, 225-277, (1982)

7. J.M. Rallison, The deformation of small viscous drops and

bubbles in shear flows Ann Rev Fluid Mech 16, 45-66, (1984)

8. F.D. Rumscheidt and S.G. Mason, Particle motions in sheared

suspensions XII deformation and burst of fluid drops in shear

and hyperbolic flow, J. Coll. Int. Sci. 16, 238-261, (1961)

9. J.D. Sherwood, Tipstreaming from slender drops in a non-linear

extensional flow, J. Fluid Mech. 144, 281-295, (1984)

10. P.G. Smith and T.G.M. van de Ven, Shear induced deformation and

rupture of suspended solid/liquid clusters, Coll. and Surf. 15,

191- 210, (1985)

ll. M.P. Srinivasan and P. Stroeve, Subdrop ejection from double

emulsion drops in shear flow, J. of Membrane Sci. 26,231-236,

(1986)

12. G.I. Taylor, The formation of emulsions in definable fields of

flow, Proc. Roy Soc. A 146, 501-523, (1934)

13. S. Torza, R.G. Cox and S.G. Mason, Particle motions in sheared

suspensions. XXVII transient and steady deformation and burst

of liquid drops, J. Coll. Int. Sci. 38, 395-411, (1972)

6.8 LIST OF SYMBOLS

~

B

C1

D

ID d

Fo

L

M

Nd

area of adsorbed molecule

width of droplet

material property in Langmuir adsorption

isotherm

drop deformation

diffusion coefficient

[ l

diffusional layer thickness [m]

Fourier number [-)

length of droplet [m]

molecular weight [kg]

number of surfactant molecules in drop [-]

LIST OF SYMBOLS (continued)

Pe

r

t

v

x

w

p

a

T

0

Avogadro number, 6.0231 1023

number of molecules required to cover

the undeformed droplet with a monolayer

Peclet nurnber

drop radius

time

characteristic drop deformation time

ramp time of shear rate profile

velocity

position

weight concentration

material constant in Langmuir adsorption

isotherm

shear rate

rate change of shear rate

dynamic viscosity

viscosity ratio, Àd/Àc

density

interfacial tension

time ratio, tdef/tramp

capillary nurnber

[ J [ - l [m]

[s]

[s]

[s]

[m·s-1]

[m]

[ - 1

[moles•m-2]

[m·s-1]

[m·s-2]

[Pa•s]

[ - ]

[kg·m-3]

[N•ml]

[ -1 [ - ]

227

228

SUMMARY

This thesis is part of a long term investigation aimed at modelling

the operation of emulsifying devices in the food industry. The

approach is to model on the one hand the flow and mixing in the

devices concerned and on the other hand to model drop breakup and

coalescence as local processes. The work in this thesis aims to

describe some aspects of the viscous breakup of drops in simple

shear flows that are not well understood, namely the breakup of

non-Newtonian drops, the deformation and breakup of drops in

transient shear flows and the origin of the tipstreaming phenomenon

observed in simple shear flow.

In chapter 2 the basic equations and sealing laws for the flow of

emulsions are presented for use in experimental and numerical

investigations. Fora number of situations the practical

limitations of application of sealing laws to the local processes

of drop deformation and break-up are considered. It is concluded

that sealing laws can be applied without great difficulties to

emulsions with constant interfacial tension. Sealing of emulsions

with surfactant adsorption is usually only possible in "slowly"

varying flows in which surfactant diffusion dominates convection,

resulting in an equilibrium surface tension all over the droplet,

or in dilute "rapidly" varying flows in which adsorption and

desorption processes may be neglected. Sealing of emulsions of non­

Newtonian liquids is usually only possible when a homologous series

of liquids is available with the same type of constitutive equation

(e.g. shear thinning liquids and Boger liquids).

In Chapter 3 a Couette device, which was developed for experimental

investigation of drop break-up in simple shear flow, is described.

The Couette device operates on the principle of two counter

rotating cylinders. A stagnant layer is thus created between the

two cylinders which permits statie observations of droplets in

quasi steady and transient simple shear flows. This device has been

successfully tested by measurement of Newtonian drop breakup in

229

230

quasi simple shear flow and in the absence of surface active

materials. Reliable breakup criteria have been formulated in terms

of the critical capillary number, which only depends on the

viscosity ratio.

Chapter 4 describes the use of the Couette device to study the

breakup of non-Newtonian droplets in quasi steady simple shear

flows. Two particular types of non-Newtonian behaviour were

studied: shear thinning liquids with viscosities obeying the power

law equation but with negligible fluid elasticity and viscoelastic

liquids with substantial elasticity combined with a shear rate

independent viscosity. These particular types of non-Newtonian drop

phases were chosen in order to separate the eff ects of shear rate

dependent viscosities and fluid elasticity.

For shear thinning drops it is concluded that below a viscosity

ratio of 0.1 the drop breakup criteria can be described by the

breakup criteria for Newtonian drops, provided a modified viscosity

ratio is used to account for the internal shear rate in the

droplet, which is higher than the applied shear rate. For higher

viscosity ratios shear thinning drops are more difficult to break

up than Newtonian drops. This is probably due to the fact that the

internal shear rate then becomes considerably lower than the

applied shear rate.

For viscoelastic drops it is concluded (in contrast with certain

statements in the literature) that drop elasticity impedes breakup.

This effect increases with drop elasticity and is rnuch stronger for

viscosity ratios of order unity than for low viscosity ratios. The

deformation of viscoelastic drops prior to breakup was often

observed to be much larger than for Newtonian drops. It was

observed further that critical drop sizes exist, below which

breakup is irnpossible in quasi steady sirnple shear, irrespective of

the magnitude of the shear rate. This probably sterns frorn the more

rapid increase with shear rate of the elastic forces inside the

droplet than of the shear stresses exerted by the continuous

phase.

In chapter 5 of this thesis the deformation and breakup of

Newtonian droplets in transient simple shear flows is studied. This

investigation is mainly of numerical nature but is supported by

experimental work. A computer programme is developed which

calculates the evolution of Newtonian droplets in any transient

shear flow. The programma is based on the boundary integral method

by which the creeping flow equations inside and outside the droplet

are transformed into a form that only involves quantities at the

drop interface. This programma is used to calculate the shape of

droplets as a function of time with viscosity ratios ranging

between 0.5 and 5 and various shear rate profiles: step profiles,

triangular profiles and sinusoidal profiles. The computed results

correspond very well with experimental data up to drop deformations

of 0.5. For equiviscous drops it was observed that the critical

shear rate at which breakup occurs was the same for quasi steady

shear situations and step like shear rate profiles. For viscosity

ratios greater than 1 "overshoot" in the deformation as a function

of time bas been observed in the step profile experiments. When a

droplet is subject to a triangular shear rate profile with a

maximum shear rate which is larger than the critica! shear rate,

breakup will only occur if this triangular ramp stretches over a

long enough time. When a droplet suddenly undergoes a shear rate

which is much larger than the critical shear rate, it will breakup

into many fragments. The number of fragments has been determined

for two viscosity ratios as a function of the maximum shear rate

and the duration of the triangular ramp. When droplets are subject

to oscillatory flows it was observed that the time averaged

deformation corresponds well with the final deformation in a quasi

steady shear flow with a shear rate equal to the time averaged

shear rate of the oscillatory profile. Breakup in oscillatory flows

is possible when the time averaged shear rate is smaller than the

critical shear rate, provided that the period of the oscillation is

longer than the characteristic drop deformation time.

In the final chapter a special mode of drop breakup, tipstreaming,

is investigated. Tipstreaming is an experimentally observed mode of

231

232

drop breakup in which the droplet takes, upon increasing the shear

rate a sigmoidal shape and a stream of very small droplets is

ruptured off the tips of the drop. The investigations in this

chapter are aimed at unravelling the causes of this phenomenon by

use of both experimental and numerical techniques. It is concluded

that tipstreaming can only occur when interfacial tension gradients

(associated with surface active contaminants) can develop,

resulting in reduced interfacial tension at the tips. Tipstreaming

will not occur at extremely low surfant concentrations, nor at high

concentrations where the interfacial tension is low all over the

droplet.

SAMENVATTING

Dit proefschrift maakt deel uit van een langlopend onderzoek

gericht op de modellering van de werking van emulgeerapparatuur in

de levensmiddelenindustrie. Hierbij wordt enerzijds de stroming en

menging in de betreffende apparaten gemodelleerd en anderzijds

worden het opbreken en coalesceren van druppels als locale

processen gemodelleerd. Het werk in dit proefschrift heeft tot doel

enige aspecten van het visceuze opbreken van druppels in

enkelvoudige afschuifstromingen te besèhrijven, die nog niet goed

begrepen zijn, namelijk het opbreken van niet-Newtonse druppels, de

deformatie en het opbreken van druppels in tijdsafhankelijke

afschuifstromingen en de reden van het verschijnsel "tipstreaming"

dat optreedt in enkelvoudige afschuifstromingen.

In hoofdstuk 2 worden de basisvergelijkingen en schalingsregels

voor de stroming van emulsies gepresenteerd voor gebruik in

experimenteel en numeriek onderzoek. Voor een aantal situaties

worden de praktische begrenzingen voor het toepassen van schalings­

regels op de locale processen van deformeren en opbreken van

druppels beschouwd. Er wordt geconcludeerd dat schalingsregels

zonder al te grote moeilijkheden toegepast kunnen worden op

emulsies met constante oppervlaktespanning. Schaling van emulsies

met adsorptie van surfactants is over het algemeen alleen mogelijk

in "langzaam" variërende stromingen, waarin diffusie van surfactant

domineert over convectie, resulterend in een evenwicht van de

oppervlaktespanning over de gehele druppel, of in verdunde "snel"

varierende stromingen, waarin adsorptie- en desorptie-processen

verwaarloosd kunnen worden. Schaling van emulsies van niet-Newtonse

vloeistoffen is over het algemeen alleen mogelijk wanneer een

homologe reeks vloeistoffen beschikbaar is met eenzelfde type

constitutieve vergelijking (b.v. afschuifsnelheid-verdunnende

vloeistoffen en Boger-vloeistoffen).

In hoofdstuk 3 wordt een Couette-apparaat beschreven dat ontwikkeld

is voor experimenteel onderzoek naar het opbreken van druppels in

233

234

enkelvoudige afschuifstromingen. Het Couette-apparaat werkt volgens

het principe van twee tegen elkaar indraaiende cilinders. Zo

ontstaat er een stilstaande laag vloeistof, waardoor het mogelijk

is druppels in quasi-stationaire en tijdsafhankelijke afschuif­

stromingen stilstaand waar te nemen. Het apparaat is met succes

getest door het opbreekgedrag van Newtonse druppels te meten in

quasi-stationaire afschuifstromingen en in de afwezigheid van

oppervlakte-actieve materialen. Betrouwbare criteria voor opbreken

zijn geformuleerd in de vorm van een kritisch capillair getal dat

alleen afhankelijk is van de viscositeitsverhouding.

Hoofdstuk 4 beschrijft het gebruik van het Couette-apparaat om het

opbreek gedrag van niet-Newtonse druppels te bestuderen in quasi­

stationaire enkelvoudige af~chuifstromingen. Twee bijzondere typen

niet Newtons gedrag zijn onderzocht: afschuifsnelheid-verdunnende

vloeistoffen, met viscositeiten die zich gedragen volgens het

machtwet-model maar met verwaarloosbare elasticiteit in de

vloeistof en viscoelastische vloeistoffen met een aanmerkelijke

elasticiteit in de vloeistof, gecombineerd met een viscositeit die

onafhankelijk is van de afschuifsnelheid. Deze bijzondere typen van

niet-Newtons gedrag zijn gekozen om onderscheid te kunnen maken

tussen effecten veroorzaakt door afschuif snelheid-afhankelijke

viscositeit en elasticiteit van de vloeistof. Voor afschuif­

snelheid-verdunnende druppels wordt geconcludeerd dat beneden een

viscositeitsverhouding van 0.1 de criteria voor het opbreken van

druppels beschreven kunnen worden door die voor Newtonse druppels,

vooropgesteld dat een aangepaste viscositeits-verhouding gebruikt

wordt om de af schuifsnelheid in de druppel in rekening te brengen,

die hoger is dan de opgelegde afschuifsnelheid. Bij hogere

viscositeitsverhoudingen zijn afschuifsnelheid-verdunnende druppels

moeilijker op te breken dan Newtonse druppels. De oorzaak hiervan

ligt waarschijnlijk in het feit dat de afschuifsnelheid in de

druppel dan aanmerkelijk lager is dan de opgelegde afschuif­

snelheid.

Voor viscoelastische druppels wordt geconcludeerd (in tegenstelling

tot enige uitspraken in de literatuur) dat druppelelasticiteit het

opbreken bemoeilijkt. Dit effect neemt toe met elasticiteit van de

druppel en is veel sterker voor viscositeitsverhoudingen van orde 1

dan voor kleine viscositeitsverhoudingen. De deformatie van

viscoelastische druppels vlak voor het moment van opbreken bleek

vaak veel groter te zijn dan voor Newtonse druppels. Verder is

waargenomen dat er kritische druppelgroottes bestaan, waar beneden

opbreken van druppels in quasi-stationaire enkelvoudige afschuif­

stromingen onmogelijk is, onafhankelijk van de grootte van de

afschuifsnelheid. Dit is er waarschijnlijk de oorzaak van dat de

elastische spanningen in de druppel sneller toenemen met de

afschuifsnelheid dan de afschuifspanningen uitgeoefend door de

continue fase.

In hoofdstuk 5 van dit proefschrift wordt het deformatie- en

opbreek-gedrag van tijdsafhankelijke enkelvoudige afschuif­

stromingen bestudeerd. Dit onderzoek heeft een overwegend numeriek

karakter, maar wordt ondersteund door experimenteel werk. Er is een

computer-programma ontwikkeld dat de ontwikkeling van de vorm van

Newtonse druppels in een willekeurige tijdsafhankelijke

afschuifstroming berekent. Dit programma is gebaseerd op de

oppervlakte-integraal-methode door welke de vergelijkingen voor de

kruipstroming in en om de druppel getransformeerd worden tot een

vorm die alleen grootheden op het druppel-oppervlak bevat. Dit

programma is gebruikt om de vorm van druppels als een functie van

de tijd te berekenen voor viscositeitsverhoudingen varierend tussen

0.5 en 5 en voor diverse profielen van de afschuifsnelheid: stap­

profielen, driehoek-profielen en sinusvormige profielen. De

berekende resultaten komen erg goed overeen met de experimentele

data tot aan deformaties van 0.5. Voor equivisceuze druppels werd

waargenomen dat de kritische afschuif snelheid waarbij opbreken

plaats vond gelijk is voor quasi-stationaire en stap-profiel­

experimenten. Voor viscositeitsverhoudingen groter dan 1 is

"overshoot" in de deformatie als functie van tijd waargenomen in de

stap-profiel"experimenten. Wanneer een druppel onderworpen is aan

een driehoeksvormig afschuifsnelheidsprofiel met een maximale

afschuifsnelheid welke groter is dan de kritische afschuifsnelheid,

dan zal opbreken alleen plaatsvinden wanneer dit driehoeksvormig 235

236

profiel voldoende lang aangehouden wordt. Wanneer een druppel

plotseling een afschuifsnelheid ondergaat die veel groter is dan de

kritische afschuifsnelheid, zal deze in veel fragmenten opbreken.

Het aantal fragmenten is voor een tweetal viscositeitsverhoudingen

bepaald als functie van de maximale afschuifsnelheid en de

tijdsduur van het driehoeksvormige afschuifsnelheidsprofiel.

Wanneer druppels onderworpen zijn aan een oscillerende stroming,

wordt waargenomen dat de tijdsgemiddelde deformatie goed overeen

komt met de eind-deformatie in een quasi-stationaire afschuifstro­

ming met een afschuifsnelheid gelijk aan de tijdsgemiddelde af­

schuifsnelheid van het oscillerende profiel. Opbreken in oscil­

lerende stromingen is mogelijk wanneer de tijdsgemiddelde afschuif­

snelheid kleiner is dan de kritische afschuifsnelheid, vooropge­

steld dat de periode van de oscillatie langer is dan de druppel­

deformatietijd.

In het laatste hoofdstuk van dit proefschrift wordt een bijzondere

vorm van opbreken, "tipstreamlng", onderzocht. Tipstreaming is een

vorm van druppel opbreken, die experimenteel waargenomen is.

Hierbij neemt de druppel, als deze vervormt, een sigmoidale vorm

aan en breekt een stroom zeer kleine druppels aan de uiteinden van

de druppel af. Het onderzoek in dit hoofdstuk is erop gericht

de oorzaken van dit verschijnsel op te sporen met gebruikmaking

van zowel experimentele als numerieke methoden. Er wordt

geconcludeerd dat tipstreaming alleen plaats kan vinden wanneer

oppervlaktespanningsgradienten (verbonden met oppervlakteactieve

verontreiniging) ontstaan, resulterend in verlaagde oppervlakte­

spanning aan de uiteinden. Tipstreaming zal niet plaats vinden bij

zeer lage concentraties van surfactant en ook niet bij hoge

concentraties, waarbij de oppervlaktespanning over de gehele

druppel laag is.

ACKNOWLEDGEMENTS

The work described in this thesis was carried out in the period

1984-1988 while I was employed in the section surface chemistry and

rheology of the Unilever Research Laboratory in Vlaardingen (NL).

Part of this work was subsidlsed by the European Community under

the BRITE project RIIB.0085.UK(H) on computational fluid dynarnics,

narned Development of experirnentally tested 3-D computer codes for

fundarnental design of process equipment involving non-Newtonian

multi-phase turbulent fields. This was a joint project which

involved the New Science group of ICI, based in Runcorn (UK), the

Department of Mechanical Engineering of Imperial College, London

(UK) and Unilever Research in Vlaardingen (NL). I would like to

acknowledge the management of Unilever Research for perrnission to

use this part of my research for my thesis. Thanks are also due for

the support and facilities I received during the completion of this

thesis.

I would also like to thank a number of people without whom this

could not have been cornpleted. First of all I would like to thank

my promotor, Allan Chesters, who initiated this work when he was

section manager of the section surface chemistry and rheology of

the Unilever Research Laboratory in Vlaardingen. Thanks for all

the valuable suggestions on the direction of this work, for all the

things I learned from you and for the pleasant cooperation. Thanks

are also due to Nico Hoekstra for his contribution in the

development of the Couette device and his continuous, friendly and

helpful support on the experimental parts of this thesis. Further I

would like to thank Reinier Baker, Ab Boon and Quint van Voorst

Vader for their valuable contributions to various parts of this

thesis. Finally I would like to thank Wim Tjaberinga for his

competent contribution to the investigation into transient drop

deformation and break-up, first as a student at Delft University of

Technology and later as a colleague at the Unilever Research

Laboratory in Vlaardingen.

237

238

CURRICULUM VITAE

De schrijver van dit proefschrift is op 1 augustus 1960 te

Heerhugowaard geboren. De middelbare school heeft hij deels aan het

Christelijk Lyceum te Alphen aan de Rijn en deels aan de

Christelijke Scholengemeenschap te Aalten gevolgd. Na het behalen

van het VWO-diploma begon hij in 1978 met de studie Technische

Natuurkunde aan de Universiteit Twente. In 1983 studeerde hij af

als natuurkundig ingenieur bij prof. dr. P.F. van der Wallen

Mijnlieff. Het afstudeerwerk, betreffende het lineair

viscoelastisch gedrag van vesicles, werd uitgevoerd onder

begeleiding van dr. J. Mellema. Van september 1983 tot maart 1988

was de auteur werkzaam bij het Unilever Research Laboratorium in

Vlaardingen in de sectie Oppervlakte Chemie en Reologie. Dit

proefschrift geeft een verslag van een deel van het onderzoek dat

de auteur tijdens zijn werk aldaar heeft verricht. Dit onderzoek

werd begeleid door prof. dr. A.K. Chesters, eerst als sectieleider

en later als hoogleraar aan de Technische Universiteit Eindhoven.

Vanaf maart 1988 is de auteur werkzaam bij het Unilever Research

Laboratory Golworth House te Sharnbrook, UK.

239

240

APPENDIX A: VISCOEIASTIC DROP BREAK-UP IN LITERATURE

TABLE A.l Model liquids used by Gauthier et al 1971

Drop pbases

No. Description K n m

Gl Poly propylene glycol 0.024 1

G2 Carbopol in PPG 2.29 0.71 ? ?

G3 Carbopol in PPG 0.11 0.97 ? ?

G4 Water 0.001 1

GS 1.5% PAA in water ? ? ? ?

G6 4.0% PAA in water ? ? ? ?

Continuous Phase

No. Description Viscosity [Pa.s]

G7 Silicone oil 5.31

241

242

TABLE A.2 Drop break-up experiments performed by Gauthier et al 1971

Drop Cont. a

phase phase [m N m-1]

Gl G7 8.6

G2 G7 8.6

G3 G7 8.6

G4 G7 21.2

G5 G7 19.4

G6 G7 16.5

R

[mm]

0.28 1.06

0.42-1.16

0.31-0.80

0.43-1.14

0.33-0.83

0.54-0.83

0.96-4.11

0.87-2.48

1.02-2.81

1.79-5.18

3.04-8.20

N.o.

À

[ - l

0.004

0.41-0.56

0.021

0.0002

0.22

2.84

0

?

?

0

?

?

flcrit

[ - J

0.59±0.12

0.62±0.03

0.57±0.07

0.54±0.05

0.71±0.03

TABLE A.3 Model liquids used by Tavgac

Drop phases

No. Description K n m

Tl 0.1% PAA in 20% glyc 0.082 0.65 0.021 1.16

T2 0.4% PAA in 20% glyc 1.4 0.43 3.5 0.82

T3 0.75% PAA in 20% glyc 3.7 0.36 13 0.61

T4 1.0% PAA in 20% glyc 5.6 0.33 20 0.67

T5 2.0% PAA in 20% glyc 3.3 0.25 190 0 49

T6 2.6% PAA in 20% glyc 56 0.3 330 0.61

T7 1.0% HEC in water 6.5 0.79 0.24 1. 62

T8 1. 5% Kelzan in water 14 0.19 16 0.47

Continuous phase

No. Description Viscosity [Pas]

T9 Silicone oil 30.8

243

TABLE A.4 Drop break-up experiments performed by Tavgac

Drop Cont. (}' R .y À SR 0 phase phase [mN/m] [mm] [l/s] [ - J [ - J [ - ]

Tl T9 23.5 0.63 58.00 6.4E-04 2.03 48.00 0.66 57.00 6.5E-04 2.01 49.00 1.01 34.00 7.7E-04 1.55 45.00 1.24 24.00 8.8E-04 1.30 39.00 1.49 21.00 9.2E-04 1.21 41.00 2.11 13.00 l. lE-03 0.95 36.00

T2 T9 20 1.00 4.50 1. 9E-02 4.49 7.45 0.94 4.20 2.0E-02 4.38 6.54 0.89 4.00 2.lE-02 4.29 5.90 1.10 3.80 2.lE-02 4.21 6.92 0.84 4.00 2.lE-02 4.29 5.56 1.10 3.30 2.3E-02 3.98 6.01 1. 30 3.20 2.3E-02 3.94 6.89 0.89 4.00 2.lE-02 4.29 5.90 1.10 2.40 2.8E-02 3.52 4.37 1.20 2.10 3.0E-02 3.34 4.17 0.92 2.00 3.lE-02 3.28 3.05

T3 T9 18.6 0.66 3.51 5.4E-02 4.81 3.82 0. 77 1.67 8.7E-02 3.99 2.13 1.03 0.91 1 3E-Ol 3.43 1.55 0.49 3.11 5.8E-02 4.67 2.54 1.23 0.74 1. 5E-Ol 3.26 1.51 0. 71 1. 72 8.5E-02 4.03 2.01 0.49 5.45 4.lE-02 5.37 4.42 0.39 11.54 2.5E-02 6.48 7.53 0.72 1.65 8.7E-02 3.98 1. 97 0.43 7.95 3.2E-02 5.90 5.63 0.51 4.81 4.4E-02 5.20 4.05 1.40 0.59 1. 7E-Ol 3.08 1. 36 0.65 3.81 5.lE-02 4.91 4.13 0.42 8.69 3.0E-02 6.03 6.10 1.19 0.76 l.4E-Ol 3.28 1.49 0.82 1.29 l.OE-01 3.75 1. 76 0.58 2.36 6.9E-02 4.36 2.29 0.62 2.84 6.2E-02 4.56 2.92 0.42 6.62 3.6E-02 5.64 4.58

T4 T9 14.9 0.40 7.00 4.9E-02 6.92 5.80 0.41 6.20 5.4E-02 6.64 5.20 0.42 5.50 5.8E-02 6.38 4.80 0.43 4.70 6.4E-02 6.04 4.20 0.47 3.40 8.0E-02 5.41 3.30 0.47 3.00 8.7E-02 5.19 2.90 0.48 1.90 1. 2E-Ol 4.44 1.90 0.48 1. 80 1.2E-Ol 4.36 1. 80 0.55 1.50 l.4E-Ol 4.10 1. 70

244

TABLE A.4 Continued

Drop Cant. (J R 'Î' À SR n phase phase [mN/m] [mm] [l/s] [ - 1 [ -1 [ - 1

T4 T9 14.9 0.82 1.00 1. 8E-Ol 3.57 1.70 1.00 0.63 2.5E-Ol 3.05 1. 30 1.27 0.42 3.3E-01 2.66 1.10

T5 T9 9.1 0.37 4.10 3.7E-02 80.78 5.10 0.38 3.50 4.2E-02 77. 77 4.50 0.45 2.90 4.8E-02 74.34 4.40 0.62 2.20 5.9E-02 69.57 4.60 0.75 1.50 7.9E-02 63.46 3.80 0.86 1.20 9.3E-02 60.15 3.50

T6 T9 6.6 0.29 6.40 5.0E-01 10.48 8.70 0.34 5.10 5.8E-01 9. 77 8.20 0.56 2.70 9.lE-01 8.02 7.00 0.56 2.60 9.3E-01 7.92 6.80 0.67 2.10 l.lE+OO 7.42 6.60

T7 T9 38.1 0.28 3.25 1. 6E-Ol 0.10 0.75 1.28 0.55 2.4E-01 0.02 0.56 1.04 0.58 2.4E-Ol 0.02 0.49 0.81 0.71 2.3E-01 0.03 0.46 0.41 1.43 2.0E-01 0.05 0.47 0.65 0.89 2.2E-01 0.03 0.47 0.53 1.12 2.lE-01 0.04 0.48 1.23 0.59 2.4E-01 0.02 0.58 0.35 1.85 1. 9E-01 0.06 0.52 0.67 0.78 2.2E-Ol 0.03 0.42 0.99 0.62 2.3E-Ol 0.02 0.50 0.52 1.09 2.lE-01 0.04 0.45 0.30 2.27 1. 8E-Ol 0.07 0.55 0.28 2.86 1. 7E-01 0.09 0.64 0.47 1.31 2.0E-01 0.05 0.50 0.38 1. 78 1. 9E-Ol 0.06 0.54 1.37 0.52 2.4E-Ol 0.02 0.58 1. 95 0.38 2.6E-Ol 0.02 0.60

T8 T9 20.9 0.41 9.81 7.2E-02 2.17 6.58 1.05 1. 69 3.0E-01 1. 32 2.94 0.79 1. 89 2.7E-01 1. 37 2.47 0.75 2.22 2.4E-Ol 1.43 2.75 1.16 1.41 3.4E-01 1.26 2.69 0.45 5.43 1. 2E-Ol 1.83 4.04 0.41 8.21 8.3E-02 2.06 5.63 0.50 4.74 1. 3E-01 1. 77 3.96 0.50 4.61 1. 3E-Ol 1. 75 3.82 0. 77 7.62 8.8E-02 2.02 9.68 0.35 8.39 8.lE-02 2.07 4.86

245

246

TABLE A.5 Critica! drop size predictions by Tavgac

drop

phase

Tl

T2

T3

T4

TS

T6

T7

T8

3.5

0.37

0.094

0.088

0.022

0.012

0.24

0.05

Àl

[s]

0.08

3.9

7.6

8.5

26

41

0.33

6.3

a

[mNm]

23.5

20

18.6

14.9

9.1

6.6

38.l

20.9

f'/c

[Pas]

30.8

30.8

30.8

30.8

30.8

30.8

30.8

30.8

Rcrit

[mm]

0.21

0.94

0.43

0.36

0.17

0.11

0.10

0.21

TABLE A.6 Model liquids used by Elmendorp

Drop phases

No. Descript ion

El PAA in water

(1% Separan AP 273)

E2 PAA in water

(l.5% Separan AP 30)

E3 PAA in water

0.5% Separan AP 273)

E4 PAA in water

(0.75% Separan AP 30)

Gontinuous phase

No. Description

ES Silicone oil

K n

7.9 0.3

7.9 0.4

1.2 0.4

1.4 0.5

Viscosity

[Pa.s]

5.5

m

5.4 0.8

1. 7 1.5

1.0 1.1

0.6 1.0

247

248

TABLE A.7 Drop break-up experirnents perforrned by Elrnendorp

Drop Cont.

phase phase

El ES

E2 ES

E3 ES

E4 ES

22

22

22

22

R

[mm)

1. 7

1.0

0.7

1. s 1.0

0.7

1.4

1.0

0.7

0.6

0.5

1.4

0.9

0.7

0.4

1. 9

3.5

S.l

1.5

1. 9

2.9

1. 3

1. 9

2.5

3.1

3.6

1.0

1. 5

2.0

3.1

À

[ - J [ - )

0.78 0.99 0.81

0.Sl 1.3 0.88

0.36 1. 7 0.89

0.96 0.33 0.54

0.78 0.47 0.48

0.64 0.60 O.Sl

0.18 1.0 0.46

0.14 1. 3 0.48

0.11 1. 6 0.44

0.096 2.0 0.47

0.084 2.3 0.45

0.22 0.44 0.35

0.18 0.54 0.34

0.16 0.63 0.35

0.13 0. 77 0.31

TABLE A.8 Model liquids used by Prabodh and Stroeve

Drop phase

No.

PSl

Description

PAA in corn syrup

(0.0275% Separan 30)

Continuous phases

No. Description

PS2 Silicone oil

PS3 Silicone oil

PS4 Silicone oil

PS5 Silicone oil

PS6 Indopol H25

PS7 Silicone oil

k n

2.7 1.0

Viscosity

[Pa.s]

57.7

25.5

12.0

6.2

2.3

1. 7

m

4.1 1.06

249

TABLE A.9 Drop break-up experiments performed by Prabodh and Stroeve

Drop Cont. a R 1' À SR flcrit phase phase [mNm-1] [µm] [s-1] [ - J [ - l [ - J

PSl PS2 31.8 37 24 0.47 1. 8 1.6 37 12 1.8 0.8 30 104 1. 9 5.6 25 85 2.0 3.8 24 146 1.6 6.4 22 104 1. 9 4.1 21 265 1. 3 9.9 21 43 1. 9 1.8 21 170 1.5 6.5 21 130 1. 7 5.0 21 93 2.0 3.5 21 75 2.0 2.6 21 58 1. 9 1.8 21 40 1. 9 1. 5 20 67 2.0 2.4 20 49 1. 9 1. 8 19 265 1.3 9.0 19 104 1.9 3.6 19 104 1. 9 3.5 18 61 1. 9 2.0 17 186 1. 5 5.7 16 104 1. 9 3.0 15 104 1. 9 2.8 15 61 1. 9 1. 7 15 43 1. 9 1. 8 14 104 1. 9 2.7 14 43 1. 9 1. 7 11 40 1.9 0.8

8 40 1. 9 0.6 PSl PS3 31. 7 34 37 0.11 1. 9 1.0

31 58 1. 9 1.2 31 40 1. 9 1.0 28 265 1. 3 5.9 27 46 1. 9 1.0 27 210 1.4 4.5 27 170 0.11 1. 5 3.7 27 130 1. 7 2.8 27 75 2.0 1.5 26 104 1. 9 2.2 26 67 2.0 1.8 26 93 2.0 1.9 24 104 1. 9 2.2 23 265 1.3 1.0

250

TABLE A.9 Continued

Drop Cont. a R ,y À SR Ocrit phase phase [mNm-1] [,um] [s-1] [ - J [ - l [ - l

PSl PS4 35.8 73 37 0.23 1. 9 0.9 71 55 1. 9 1. 3 68 61 1. 9 1.4 60 40 1.9 0.8 57 104 1.9 2.0 57 75 2.0 1.3 56 58 1. 9 0.9 51 93 1. 9 1.6 50 130 1. 7 2.2 47 170 1.5 2.7 46 156 1.6 2.4 44 217 1.4 3.2 43 210 1.5 3.0

PSl PS5 17 .2 58 58 0.44 1. 9 1.0 45 75 2.0 1.1 45 61 1.9 1.0 40 61 1. 9 0.9 39 93 1. 9 1. 3 37 104 1. 9 1.4 35 104 1. 9 1.3 30 128 1. 7 1.4 30 130 1. 7 1.4 28 128 1. 7 1. 3 27 104 1.9 1.0 24 170 1. 5 1. 5 22 210 1. 5 1. 7 22 180 1. 5 1. 5 20 217 0.44 1.4 1.6 17 259 1.3 1.6

PSl PS6 33.3 142 104 1. 2 1. 9 1.0 77 104 1. 9 0.6 75 150 1. 6 0.8 64 140 1.6 0.6 58 147 1. 6 0.6 57 217 1.4 0.9 54 104 1. 9 0.4 44 311 1. 2 0.9 39 217 1.4 0.6 34 315 1.2 0.8 27 320 1. 2 0.6

PSl PS? 35.5 107 220 1.6 1.4 1. 2 88 223 1.4 1.0 70 210 1. 5 0.7 70 223 1.4 0.8 49 323 1. 2 0.8

251

252

TABLE A.10 Model liquids used by Mirmohammad-Sadeghi

Drop phases

No. Description

MSl eorn syrup (CS)

MS2 0.001% PAA in es

MS3 0.002% PAA in es

MS4 0.005% PAA in es

MSS 0.01% PAA in es

MS6 0.02% PAA in es

Continuous phases

No.

MS7

MS8

Description

Silicone oil

Silicone oil

K

3.4

9.2

9.7

9.7

11.2

10.3

ri [Pa.s]

12.1

12.9

n

1

0.96

0.98

0.89

0.93

0.89

m

0.48 0.97

1.0 1.06

0.89 1.27

5.1 0.97

3.3 1. 26

TABLE A.11 Drop break-up experiments performed by Mirmo-hammad-Sadeghi

Drop Cont. a R 'Y >. SR Ocrit phase phase [mNm-1] [µm] [s-1] [ - l [ -1 [ - l

MSl MS8 50 154 14 0.26 0.56 127 23 0.75 126 18 0.57 123 22 0.70 107 24 0.66

91 29 0.68 85 31 0.68 74 39 0.75 68 34 0.59 56 47 0.68

MS2 MS7 33 109 17 0.67 0.05 0.68 63 25 0.05 0.58 59 27 0.05 0.58 47 36 0.05 0.62 30 62 0.05 0.68 29 64 0.05 0.67

MS3 MS7 33 53 11 0. 77 0.1 0.21 46 20 0.1 0.34 35 29 0.1 0.37 34 46 0.1 0.57 30 57 0.1 0.62 30 57 0.1 0.63 28 76 0.1 0.78

MS4 MS7 33 75 9 0.55 0.2 0.24 49 23 0.3 0.41 38 29 0.3 0.41 31 52 0.4 0.59 28 66 0.5 0.68 27 71 0.5 0.69 22 80 0.5 0.65

MS5 MS7 33.5 54 9 0.73 0.5 0.18 52 10 0.5 0.19 48 17 0.5 0.29 43 37 0.5 0.57 43 30 0.5 0.46

253

TABLE A.11 Continued

Drop Cont. a

phase phase [mNm-1]

MS5 MS7 33.5

MS6 MS7 33.5

254

R

[µm]

41

41

41

39

38

38

35

32

23

23

136

113

91

82

76

68

67

63

61

42

À SR Ücrit

[ - l [ - J [ - l

36 0.73 0.5 0.57

31 0.5 0.46

30 0.5 0.44

39 0.5 0.55

36 0.5 0.49

34 0.5 0.47

44 0.5 0.55

41 0.5 0.48

72 0.5 0.60

72 0.5 0.60

6 0.61 0.6 0.31

8 0.7 0.34

15 0.9 0.48

19 0.9 0.55

21 1.0 0.58

27 1.1 0.66

29 1.1 0.70

22 1. 0 0.50

26 1.1 0.57

52 1.4 0.79

APPENDIX B DROP BREAK-UP EXPERIMENTS

TABLE B.l Viscoelastic drop break-up.

Drop Cont. " R .y ), 6& 0 crit phase phase [mNm-1 J [mmJ (•-ll [-J r-1 r-1

VE,D,l VE.C.1 29 0.648 23. 75 0 .13 0.48

0.512 32. 93 0 .13 0.52

VE.D.1 VE.C.2 29 0.461 13.12 0.020 1.3

0.357 16. 11 0.020 1.2

0.281 18.05 0.020 1.1

0.217 23.35 0.020 1.1

0.173 31. 70 0.020 1.1

VE.D.1 VE.C.3 29 0. 533 8,33 0.011 1. 7

0.366 12.08 0 '011 1. 7

0.295 15.90 0.011 1.8

0.227 21. 15 0.011 1.9

VE.D.l VE.C.5 29 0.402 8.41 0.0031 - 4.4

0.322 9.09 0.0031 - 4.2

0.251 11.90 0.0031 - 3.9

VE.D.2 VE.C.l 29 0.59 21 0.20 0.23 0.39

0.47 31 0.20 0.28 0.45

0.37 45 0.20 0.33 0.51

0,29 58 0.20 0,37 0 .53

0.22 51 0.20 0.35 0 .20

VE.D.2 VE.C.2 29 0. 50 9. 87 0.030 0.16 l. 0

0. 40 12.32 0.030 0.18 1,0

0.32 15. 73 0.030 0.20 1.0

0.24 20.34 0 .030 0.23 1.0

0.18 30.62 0 .030 0.28 1.1

0, 14 53. 79 0.030 0.36 1.6

VE.D.2 VE.C.3 29 0.53 4.11 0.016 0.11 0. 85

0.40 6. 05 0.016 0.13 0.95

0.29 12.22 0.016 0.18 l.4

0.22 20.09 0. 016 0.23 1. 7

0.16 28.77 0 .016 0.27 1.8

VE.0.2 VE.C. 5 29 0,48 4. 77 0. 0047 0.12 3.0

0.37 7 .12 0.0047 0.14 3.5

0.26 9.80 0.0047 0.16 3 .4

0,20 12.59 0.0047 0.18 3 .3

0' 15 15. 87 0.0047 0,20 3.1

VE.0.2 VE.C.6 29 0.625 4.02 0.0042 0.11 3. 7

0 .470 4.95 0.0042 0.12 3.5

0.381 5.89 0.0042 0.13 3.2

0.289 8. 74 0.0042 0.15 3.8

0.208 10 .27 0.0042 0.17 3 .2

0 .154 14. 63 0,0042 0.20 3. 4

0 .108 18.99 0.0042 0.22 3.1

255

TABLE B.l (Continued)

Drop Cont. 17 R 7 À 8R °'crit. phase phase CmNm- 11 (mm] [s-11 (-1 c-1 [-]

VE,D.3 VE.C.1 29 0. 476 >64 0.54 2.8 >LO

0.609 >64 0.54 2.8 >1.2

0.428 >64 0.54 2.8 >0.9

VE.D.3 VE.C.2 29 0.515 8.16 0.043 0.77 0.87

0.395 9.90 0.043 0.86 0.81

0.308 13.60 0.042 1.1 0. 87

0.236 22.33 0,042 1.4 1.1

0.491 8.35 0.043 0. 78 0.85

0.383 11.24 0.043 0.94 0.89

0.293 14.62 0.042 1.1 0.89

0.26 26.09 0.042 1.6 1.2

0.172 39.93 0.040 2.1 1. 4

VE.D.3 VE.C.3 29 0.686 4.34 0.024 0.52 1.2

0 .527 5.84 0.024 0.62 1.2

o. 414 8.12 0.023 o. 76 1. 3

0.316 11.46 0.023 0.95 1. 4

0.217 16.31 0.022 l.2 1.4

0.166 22.72 0.022 1.5 1.5

VE.D.3 VE.C.5 29 0.536 3.58 0.0071 0.46 2.5

0.407 4. 71 0. 0071 0.55 2.5

0.312 6.65 o. 0068 0.68 2.7

0.233 S.86 0. 0068 0.81 2.7

VE.D.3 VE.C.6 29 0.678 2. 76 0.0065 o. 39 2.8

0.536 3.84 0.0063 0.48 3 .1

0.407 5.37 0. 0063 0.59 3.3

0.293 6.63 0. 0060 0.67 2.9

0.223 8.81 0.0060 0.80 2.9

0.169 11.92 0.0060 0.97 3.0

0.136 13.99 0.0058 1.1 2.8

VE.D.4 VE.C.2 28 0.678 5.27 0.11 1.8 0. 77

o. 434 11.99 0.095 2.8 1.1

0.308 19. 30 0.088 3.7 1.3

VE.D.4 VE.C.3 28 0.559 3.86 0 .060 1. 5 0.88

0.434 5.07 0.058 1. 8 0.89

0.342 6.91 0.055 1. 9 0.96

0.259 13.96 0.049 3.1 l. 5

0.199 18.26 o. 047 3.6 1. 5

0.628 3.54 0. 061 1. 4 0.90

0. 464 5.62 0.056 1. 9 l.1

0. 316 8. 43 0.053 2.3 1.1

0.226 17 .15 0. 048 3.5 1. 6

0.176 20.45 0 .046 3.8 1.5

256

TABLE B.l (Continued)

Drop Cont. " R ~ À ~ 0crit phase phase (mNm- 1 J (lllll) (s-11 (-] (-] [-)

VE.D.4 VE.C.5 28 0.609 2.34 0.019 1.1 1.9

0.431 3.41 0,018 1. 4 2.0

0.310 5.02 0.017 1. 7 2.1

0.211 8.10 0.016 2.3 2.3

0.157 13. 75 0.015 3.1 2.9

VE.D.4 VE.C.6 28 0.452 2.84 0.017 1. 3 2.0

0.354 3.90 0.016 1.5 2.13

0.261 5,33 0.015 1.8 2.2

0.190 7 .30 0.014 2.2 2.1

0.139 12.53 0 .013 2.9 2.7

VE.D.5 VE.C.2 27 0.640 >16 0.26 8.8 >2.3

0.455 >16 0.26 8.8 >1.6

VE.D.5 VE.C.3 27 0.613 3. 70 0.23 4.0 0.95

0.448 5.79 0.20 5.1 1.1

0.398 6.10 0.19 5.2 1.0

0.446 1.81 0.088 2.7 1.1

0.256 3.60 0.070 3.9 1.3

0,193 >9 0.051 6.3 >2.4

0,613 1.23 0.089 2.2 1.2

0.298 1.99 0.075 2.8 0,95

0.218 3.92 0.059 4 .l 1. 4

0.166 >14 0.039 7 .9 >3.5

0.253 >19 0.13 9.6 >2,0

VE.D.5 VE.C.5 27 0.582 1. 72 0.090 2.6 1. 4

0.352 2.09 0.084 2.9 1. 0

0.211 3.20 0,072 3. 7 0. 95

0. 704 0.72 0.12 1.6 0.72

0.829 0.72 0.12 1.6 0.84

VE.D.6 VE.C.2 31 0.536 6.90 0.048 0.71

0.431 8.67 0.048 0.72

0.339 11.06 0.048 0. 73

0.262 13.92 0.048 0.71

0.205 17.26 0.048 0.68

VE.D.6 VE,C,3 31 0.548 6.18 0.026 l. 2

0. 383 9.32 0.026 1. 3

0.261 12. 31 0.026 1.2

VE.D.6 VE.C.5 31 o. 383 4. 75 0.0076 - 2.2

0.295 6. 47 0.0076 - 2.3

0.215 7. 75 0.0076 - 2.1

VE.D.6 VE.C.7 31 0.655 2.59 0. 0045 - 3.5

0.429 3.21 0. 0045 - 2.1

0.322 3.94 0.0045 - 2.6

0.190 6.54 0.0045 - 2.6

257

TABLE B.l (Continued)

Drop Cant. (f R ; 8R ncrit phase phaae [rnllm-l] [lllll] [a-l] [-] [-] [-]

VE.D.7 VE.C.2 31 0.569 6.21 0.072 1.1 0.68

0.400 8.69 0.068 1.3 0. 67

0.312 18. 76 0,082 1.1

0.253 >41.65 o. 057 3.4 >2.0

VE.D.7 VE.C.3 31 0.621 3.98 0. 040 0.82 0.9

0.455 5.37 0.038 0.98 0.9

0 .343 8.04 0.036 1. 3 1.0

0 .238 14 .91 0.033 l. 8 1.2

0.182 20.20 0.033 2.2 1.3

VE.D.7 VE.C.5 31 0.443 3 .67 0.012 o. 78 2.0

0.301 4 .96 0.012 0.93 1.8

0.215 7 .30 0.011 1.2 l.9

0.148 9.93 0.010 1. 4 1.8

0.042 31.59 0.0092 2.9 1.6

0.449 3.25 0.012 0.73 1.8

0.355 5.04 0.012 0.95 2.2

0.241 6.52 0.011 1.1 l. g

0.175 8.05 0.011 l.3 1.3

0.137 10.01 o. 010 1. 4 1. 7

0.175 17.82 0.069 3.2 1.1

0.69 1. 94 0.020 0.94 1. 7

0.50 2. 76 0.020 1.1 1. 7

0.36 3.99 0.020 1.4 1. 8

0.25 4.93 0.020 1.6 1.5

0.20 6.69 0.020 1.9 1.6

0.15 8.97 0.020 2.2 1. 7

0.13 11.09 0.020 2.5 1.8

VE.D.8 VE.C.2 31 0.551 s. 79 0.13 1. 7 0.62

0.440 9. 74 0.13 2.3 0.83

0.328 15.52 0 .13 3.0 0.99

0.262 16. 47 0 .13 3.1 0.84

0.202 >54 0.13 5.6 >2.l

VE.D.8 VE.C.3 31 0.640 3.45 0.069 1.3 0 .81

o. 443 5.31 0.069 1.6 0.86

0.299 6.89 0.069 1. 9 0. 75

0.230 9.66 0.069 2.3 0.81

VE.D.8 VE.C.6 31 0,694 1. 76 0.018 0.89 l. 7

0.503 2.87 0. 018 1.2 2.0

0 .335 4.34 0.018 1. 5 2.0

0.229 5.22 0.018 l.6 1. 7

0.172 7 .10 0.018 l. 9 1. 7

VE.D.9 VE.C.2 30 0.622 4.99 0.22 3.5 0.62

0.482 6.82 0.21 4.2 0.66

0.347 14.11 0.18 6. 7 0.98

0.266 >17 .43 0 .18 7 .6 >0.93

0.202 >25.95 0.17 9. 8 >1.1

0.461 >14 0.18 6.6 >l.3

258

TABLE B.1 (Continued)

Drop Cont, " R 7 ~ ~ 0cri~ phase phase [mNm- 11 [nm] [$-1] [-] [-] [-]

VE.D.9 VE.C.3 30 o. 554 4. 84 0.11 3.4 1. 0

0. 417 6.42 0.11 4 .1 1.0

VE.D.9 VE,C.3 30 0. 343 8.46 0.10 4. 8 1.1

VE.D.9 VE.C.5 30 0.584 1. 74 0.040 l.B 1.3

0.335 3.38 o. 036 2.7 1.4

0.194 11.19 0.030 5.6 2.8

VE.D.9 VE.C.6 30 0.586 1.58 0.036 1. 7 1. 3

0.370 2.63 0.033 2.3 1.4

0.232 8.89 0.027 5.0 3.0

VE.D.10 VE.C.3 29 0.555 10. 39 0.21 10.5 2.3

VE.D.10 VE.C.5 29 0. 781 0. 74 0, 16 2.5 0. 76

0. 563 1.55 0.12 3. 7 1.2

0. 374 2.26 0.11 4. 6 l. l

0.267 2.95 0.10 5,3 1.0

0.163 >11 0.063 10.4 >2.2

VE.D.10 VE.G.6 29 0.686 1. 05 0.13 3.0 l.1

0. 390 2.19 0.096 4. 5 1.3

0.226 5.39 0.069 7. 4 l.8

0.167 >19 0. 044 14.6 >4. 7

VE.D.11 VE.C.l 33 0.464 36. 57 0.87 0 .46

0. 367 48.11 0.87 0. 48

VE.D.11 VE.C.2 33 0. 628 4.67 0.13 0.53

0.506 6.17 0 .13 0.57

0.386 7 .36 0 .13 0.51

0 .303 9.27 0.13 0, 51

0' 235 11.95 0.13 0. 51

0 .185 14. 78 0.13 0.50

VE.D.11 VE.C.2 33 0. 708 2.93 0.069 0.71

0.530 3. 79 0.069 0.69

0.412 4. 97 0.069 0. 70

o. 310 6. 95 0.069 0. 74

0.220 9.18 0.069 0. 70

VE.D.ll VE.C.5 33 0.563 2.19 0.020 1.4

0.366 2.80 0.020 1.2

0.285 3. 87 0.020 1. 3

o. 479 4.28 0.11 0.22 0. 71

0. 347 5.52 0.11 0 .24 0.66

0 .245 8.20 0.11 0.27 0.69

0.178 10 .10 0.11 0.29 0.62

VE.D.11 VE.C,5 33 0.209 5. 02 0.020 1.2

VE.0.11 VE.C.7 33 0. 355 1.45 0,012 1. 5

0.280 1.51 0.012 l. 5

0.175 1.34 0.012 1.3

0.140 1.36 0.012 1.4

259

TAB LE B.1 (Continued)

Drop Cent. q R :., 3R Ocrit phase phaSEt [mNm-1] [nm] [s-1] [-] [-] [-]

VE.D.12 VE.C.2 33 0. 410 7 .88 0.21 0.2 0.59

0.320 11.06 0.22 0.3 0. 64

0.224 31. 57 0.23 0.4 1.3

0.670 4.lt9 0.21 0.2 0,55

0.524 5.53 0,21 0.2 0.52

o. 414 7.45 0.21 0.2 0.56

0.322 9.31 0.22 0.2 0.55

0.248 11.86 0.22 0.3 0.53

0.193 15. 70 0.22 0.3 0.55

0.154 19.86 0.22 0.36 0.56

VE.D.12 VE.C.3 33 0.675 2.90 0.11 0.20 0.67

VE.D.12 VE.C.5 33 0.605 1.96 0.031 0 .17 1.4

0.402 2. 78 0.032 0.19 1.3

0.291 3.57 0.032 0.21 1.2

0.230 4.42 0.033 0.22 1.2

0.191 6.61 0 .033 0,25 1. 5

VE.D.12 VE.C.6 33 0.524 2.36 0.028 0.18 1. 6

0.332 3 .42 0.028 0.21 1. 5

0.239 4.11 0.028 0.22 1. 3

0.184 5.25 0.029 0.24 1. 3

0 .145 6.54 0,029 0.25 1.2

VE.D.12 VE.C.7 33 0 .386 3.52 0.019 0.19 1.9

0.247 3. 37 0.019 0.21 1.6

VE.D.12 VE.C.7 33 0.176 4. 39 0.019 O.Z2 1.5

0.136 6.15 o. 020 0.25 1.6

0.089 7 .86 0.020 0.27 1.4

VE.D.13 VE.C.2 33 0.571 5.18 0.22 0. 71 0.54

0.446 6.64 0.22 0. 78 o. 54

0.352 9.39 0.22 0.88 0.60

0.272 >30 0.24 1.3 >1.5

VE.D.13 VE.C.3 33 0.448 4.50 0,11 0.68 0.69

0.328 6.34 0 .12 0. 76 0.72

0.230 8. 70 0.12 0.85 0.60

0.167 13 .01 0.12 O.S8 0. 75

VE.D.13 VE.C.5 33 0.723 1. 59 0.032 0.47 1.3

0.44S 2.40 0 .033 0.54 1.2

0.2S7 3.S2 0. 034 0.65 1.3

0.187 5.38 0.034 0. 72 1.2

0 .134 6.S2 0.035 0. 79 l, l

VE.D.13 VE.C.6 33 0.678 1. 68 0.028 0.48 1. 5

0.452 2.58 0.029 0.56 1.5

0.297 3.72 0.030 0. 63 1.5

0.196 5.10 0.030 0. 71 1. 3

0.140 6.91 0.031 0. 79 1. 3

260

TAB LE B.l (Continued)

Drop Cont. " R 7 À 8R 0crit. phase pbase [mNm-1 J [Dili] r.- 11 1-1 [-] (-)

VE.D.13 VE.C.7 33 0.554 l.90 0.019 0.50 2. l

o. 301 3.15 0.020 0. 60 l.9

0.169 4.14 0.020 0.66 1.4

0.126 8.49 0.021 0.65 2.1

0.087 8. 44 0.021 0.84 1.4

VE.D.14 VE.C.2 32 0.567 >18. 66 0.32 3.3 >2.0

0.422 >28.81 0.32 4.2 >2.3

VE.D.14 VE.C.3 32 0.506 3.93 0 .17 1.3 0. 71

0.347 7.03 0.17 1.9 0. 87

0.253 9.05 0.17 2.2 0. 81

0.205 17 .69 0.17 3.2 1.3

VE.D.14 VE.C.5 32 0.590 1. 43 0.052 o. 74 1.0

0.325 2.90 0.052 1.1 1.1

0.598 3. 08 0.246 0.60

0.390 4. 37 0.246 0.55

0.265 5.84 0.246 0.50

VE.D.14 VE.C.5 32 0.196 4.29 0.052 1.4 1.0

0.131 6.33 0 .051 1.8 0.99

VE.D.14 VE.C.6 32 0.652 l.42 0 .046 0. 74 l.3

0.443 2.06 0.046 0.91 1.2

0.281 2.84 0.046 l.l 1.1

0.205 3.67 0.046 1.3 1.0

0 .149 5.02 0.046 1. 5 1.0

0.114 8. 79 0 .045 2.1 1.4

VE.D.14 VE.C.7 32 0.494 1.50 0 .031 0. 76 l.5

0.308 1.98 0.031 0.89 1.2

0.215 2.94 0.031 1.1 l. 3

0.161 3.52 0.030 1.2 1.3

VE.D.15 VE.C.5 31 0,417 2.92 0.22 8.4 1.5

VE.D.15 VE.C.6 31 0.69'i 1.36 0.24 5.6 1.3

0.424 2.27 0 .21 7 .3 1.3

0.308 3.53 0 .18 9.3 l.5

0.214 >10. 46 0.12 16.7 >3.1

0.154 >28.16 0.090 28.5 >6.0

VE.D.16 VE.C.2 35 0.617 4. 65 0.47 0. 49

0. 452 5.93 0.47 0. 46

0,352 8.06 0.47 O.'i9

VE.D.16 VE.C.3 35 0. 684 1.96 0.25 0.44

0.506 3.14 0.25 0.52

0.345 4.63 0.25 0.52

0.233 6.21 0.25 0,47

VE.D.16 VE.C.4 35 o. 410 2.17 0.105 0.68

0.265 3.24 0.105 0.65

VE.D.16 VE.C.5 35 0.571 1.59 0.073 0.99

0.299 3.01 0.073 0.98

261

TAB LE B.l (Continued)

Drop Cont. (f R 7 À SR Ocrit phase pbase [mNm-1 J [nm] [s-1] [-) [-] [-)

VE.D.16 VE.C.6 35 0.632 1.29 0.065 1.0

VE.D.16 VE.C.6 35 0,330 1.89 0.065 0. 77

VE.D.16 VE.C.7 35 0. 578 1.30 0. 043 1.4

0.266 l. 81 0.043 0.89

VE.D.17 VE.C.2 35 0.446 6.11 0.66 o. 79 0.47

0.325 >19 0.65 1.6 >1.l

0,383 17 .25 0.65 1. 5 1.1

0.320 .>27 0.64 1. 9 >1.5

0.521 6.58 0.66 0.83 0.59

0.417 13. 50 0.65 1.3 0. 97

0.371 13. 70 0.65 1. 3 0.87

0.297 >25 0. 64 1.8 >l. 3

0.655 11.47 0.65 1.2 1.3

VE.D.17 VE.C.3 35 0.663 2.64 0.35 0.48 0.57

o. 449 4.40 0.35 0.65 0.64

0,355 >20 0.34 1.6 >2.3

VE.D.17 VE.C.3 35 0 .667 2.16 0.36 0.43 0.47

0.497 3.16 0.35 0.54 0. 51

0.383 6.27 0.35 0.81 0. 78

0.253 >17 .13 0.34 1. 5 >l.4

0.542 2.85 0,35 0.50 0 .50

0. 414 4. 07 0.35 0.62 0.55

0.312 >19 0. 34 1. 6 >2,0

0,364 >14 o. 34 1. 3 >l. 7

0. 586 2. 76 0.35 0. 49 0.52

0.451 3.97 0.35 0.61 0.58

0.337 >19 o. 34 1.0 >2.1

VE.D.17 VE.C.4 35 0.439 1. 58 0.15 o. 35 0.53

0.536 1.48 0.15 0.34 0.60

0.188 4 .28 0.15 0.64 0. 61

0 .134 6.11 0 .15 0. 79 0.62

0.111 >16 0 .15 1.4 >l.3

0. 361 2.30 0.15 0. 44 0. 63

0.221 3 .61 0 .15 0. 58 0.61

0.151 6.96 0 .15 0.85 0.80

0.091 >21 0.14 1. 7 >l,6

262

TABLE B.l (Continued)

Drop Cont. R ; À 8R 0crit phase phase [mNm- 1 ] [mml [s-1] (-] [-] (-)

VE.D.17 VE.C.5 35 0. 540 1.23 O.ll 0.30 0. 72

0. 343 2.08 0 .10 0.42 0. 78

0, 196 3.27 0.10 0.55 0.70

0.133 4.85 0.10 0.69 0. 70

0.096 >17 0.10 1.5 >1.8

0.563 1.47 O.ll 0.34 0.90

0.297 2.00 0.10 0.41 0.65

0.206 3.44 0.10 0.56 0.77

0.126 >13 0.10 1.2 >l.8

0,530 1.50 0.11 0 .34 0,87

0.278 2. 78 0.10 0.50 0.84

0.154 3. 77 0.10 0.59 0.63

0 .117 >17 0.10 1.4 >2.1

0 .500 1.59 0.11 0.35 0. 87

0.281 3.08 0.10 0. 53 0. 94

0.142 5.43 0.10 0. 74 0 .84

VE.D.17 VE.C. 6 35 0.648 o.97 0.095 0.26 0. 78

0.436 l. 31 0. 094 0.32 0.70

0.306 1. 88 o. 094 0.39 0.71

0.194 2.29 0.093 0.44 0. 55

0.157 >9 0.091 0.99 >1.8

0. 440 l. 28 0.094 0. 31 0. 70

0. 316 2.31 0 .093 0. 44 0.90

0.160 3.34 0. 093 0.55 0.66

0.128 >8 0.091 0.92 >1.3

0.625 1.13 0.095 0.29 0.87

0.342 1.59 0.094 0,35 0. 67

VE.D.17 VE.C.6 35 0 .241 2.23 0.093 0 .43 0. 66

0.166 3.73 0.092 0.59 0. 76

0.130 >18 0.090 l. 5 >2.8

0.640 1.32 0.094 0.32 1.0

0.328 2. 46 0.093 0.46 1.0

0.175 3.89 0,092 0.61 0 .84

0.137 >14. 28 0.090 1. 3 >2.4

VE.D.17 VE.C.7 35 o. 512 1. 01 0.063 0.27 0.95

0.322 1.37 0.063 0.32 0.81

0.211 2.21 0 .063 0. 43 0.86

0.131 3.29 0.062 0.55 0.80

0.479 1.16 0. 063 0.29 1.0

0.166 Z.68 0.062 0.48 0.82

0. 078 4. 77 0.061 0,69 0.69

0.063 >64 0.058 3. 3 >7 .5

263

TAB LE B.l (Continued)

Drop Cont. 17 R '1 À SR 11cr1t phase phase [mNm-11 [Olll] [s-1] [-] [-] [-]

Vl!.0.18 VE.C.3 34 0.586 l0.06 0.57 4.7 2.0

0.663 9.22 0 .57 4.5 2.0

0.451 4.59 0.62 3.2 0.69

0.345 13 .40 0.55 5.4 1.5

VE.D.18 VE.C.4 34 0.417 2.26 0.28 2.2 o. 74

0.272 >8 0.25 4.1 1. 7

0.542 2.03 0.29 2.l 0.86

0.314 3. 73 0.27 2.8 0.92

0.220 >12 0.24 5.1 2.1

VE.0.18 VE.C.5 34 0.586 l.27 0.21 1.7 0.83

0,369 2.48 0.20 2.3 1.0

0.194 >14 0.16 5.4 >3.0

0.536 1. 50 0.21 1.8 0.90

0.301 >11 0 .17 4. 9 >3.8

0.305 3.82 0.19 2.9 1.3

0.223 >20 0.16 6.5 >4.9

VE.D.18 VE.C.6 34 0.515 1.18 0.19 1.6 0. 77

0.326 2.06 0.18 2.1 0.85

0 .205 >10 0.15 4 .5 >2.5

0.566 1.13 0.19 1.6 0.81

0.332 2.03 0.18 2.1 0.86

0.232 3.30 0.17 2. 7 0.97

0 .154 >15 0.14 5. 7 >2.9

0.527 1.17 0.19 1. 6 o. 78

o. 314 2.24 0.17 2.2 0. 89

0.694 2.95 0.48 l.6 0. 70

0,488 >9 0.47 3.6 >l.4

VE.D.18 VE.C.6 34 0.200 3.87 0.16 2.9 0.98

0.140 >14 0.15 5.4 >2.5

0.571 l.22 0.18 1.6 0.89

0.328 2.29 0.17 2.2 0.95

0. 232 >ll. 36 0.15 5.0 >3.3

0.181 >20.73 0.14 6.7 >4.8

VE.D.18 VE.C. 7 34 0.460 o. 93 0.13 l.4 0.81

0.295 1. 72 0.12 1.9 0.96

0,188 >8 0.10 4.2 >2.9

0.318 l.47 0.12 l.8 0.89

0.190 3. 97 0.11 2.9 1.4

0.448 1.03 0.13 1. 5 0.88

0 .273 >7 0.10 3' 7 >3.4

o. 572 1.01 0.13 1.5 1.1

0.301 1. 64 0.12 1.9 0.94

0.173 >7 0.10 3' 8 >2.2

264

TAB LE B.1 (Continued)

Drop Cont. " R .; À 8R ocrit phase phase [mNm-1J Cmml (s-ll (-] (-] [-]

VE.D.19 VE.C.2 33 0.448 >20 0.88 7 .o >1.8

0.509 >12 0.89 4.8 >1.2

VE.D.19 VE.C.3 33 0.594 2.84 0 .48 1.6 0.58

O.li49 3. 85 0.48 z.o 0 .60

0.343 >19 0.46 6. 7 >2.3

0.509 6.83 0.47 3.1 1.2

0.689 2.52 0.48 1.4 0.60

0.482 4.10 0.48 2.1 0.68

0.369 >11 0.47 4. 3 >l.4

VE.D.19 VE.C.4 33 0.463 1. 85 0.21 1.1 0.69

0.289 2.91 0.21 1.6 0.68

0 .203 >14 0.20 5.3 >2.3

0.366 2.27 0.21 1.3 0.67

0.232 3 .51 0.21 1.8 0.66

VE.D.19 VE.C.5 33 0.414 l.77 O.H 1.1 0.85

0.235 3.47 0.14 1.8 0 .94

0.139 >24 0.14 7.8 >3.8

0.544 1.31 0.15 0.86 0.82

0.312 1.81 0.15 1.1 0 .65

0.226 2.95 0 .14 1.6 0.77

0.173 >14 0 .14 5. 3 >2.8

0.512 1.52 0.15 0.96 0.90

0.303 2.35 0.14 1.3 0.82

0.188 3. 79 0 .14 l. 9 0.82

0.157 >16 0 .14 5.8 >2.9

VE.D.19 VE.C.6 33 0.322 1. 78 0.13 1.1 0. 75

0.203 2.87 0.13 1.6 0. 76

0.137 >8 0.12 3.2 >l.3

o.539 1.24 0.13 0.82 0.87

0 .318 2.27 0.13 1. 3 0.94

0.172 3. 77 0.13 1. 9 0.85

0.133 >12 0.12 4.6 >2.1

0.563 1.19 0.13 0.80 0.88

0.347 2.23 0.13 1.3 1.0

0.178 >11 0.12 4.3 >2.5

VE.D.19 VE.C.7 33 0.601 0.97 0.087 0.52 1.14

0.305 1. 92 0.086 1.2 1.2

0 .494 1.06 0.087 0.73 1.0

0.288 1. 68 0.086 1.0 0.95

0.163 2.58 0.085 1.4 0.82

0.402 1. 47 0.086 0.94 1.2

0.145 2.91 0.085 1.6 0. 83

0.102 >9 0.083 3.5 >L7

265

TABLE B.2 Shear thinning drop break-up

drop

phase

cont.

phase [mN/ml

R

[mm]

' 7

[l/•l Àm,l [-]

\n,2 [-]

ST.D.l ST.C.l 26.5 0.298 63.83 0.65 4.9E-Ol l.4E-01 2.9E-01

ST.D.l Sî.C.l 26.5 0.560 25.68 0.49 9.7E-Ol 2.9E-Ol 7.0E-01

Sî.D.l Sî.C,1 26.5 0.448 32.58 0.50 8.2E-Ol 2.4E·Ol 5.4E-Ol

ST.D.l ST.C.1 26.5 0.364 40.62 0.50 6.9E-Ol 2.0E-01 4.4E-01

Sî.D.l Sî.C.3 26.5 0.347 8.45 0.63 3.6E-Ol l.lE-01 2.0E-01

ST.D.1 ST.C.3 26.5 0.214 14.90 0.68 2.3E-Ol 6.SE-02 1.2E·Ol

ST.D.l ST.C.5 26.5 0.464 2.18 0.61 3.SE-01 1.0E-01 l.9E-Ol

ST.D.1 ST.C.5 26.5 0.371 2.88 0.65 2.SE-01 8.2E-02 1.5E-Ol

ST.D.l ST.C.5 26.5 0.295 3.65 0.65 2.4E-Ol 7.lE-02 l.2E-Ol

ST.D.l ST.C.5 26.5 0.176 7.40 0.79 l.4E-Ol 4.lE-02 7.lE-02

ST.D.l Sî.C.5 26.5 0.139 10.02 0.84 l.lE-01 3.2E-02 5.SE-02

ST.D.l ST.C.5 26,5 0.108 13.67 0,90 8.SE-02 2.SE-02 4.4E-02

ST.D.l Sî.C.6 26.5 0.405 1.15 0.70 2.3E-Ol 6.8E-02 1.2E-Ol

ST.D.2 Sî.C.l 28

ST.D.2 ST.C.l 28

Sî.D.2 ST.C.l 28

ST.D.2 ST.C.l 28

ST.D.2 ST.C.1 28

ST.D.2 ST.C.1 28

ST.D.2 ST.C.l 28

ST .D.2 ST .C.1 28

ST.D.2 ST.C.3 28

ST.D.2 ST.C.3 28

ST.D.2 ST.C.3 28

ST .D.2 ST ,C. 3 28

ST.D.2 ST.C.3 28

ST.D.2 ST.C.5 28

ST.D.2 ST,C.5 28

ST.D.2 ST.C.5 28

ST .D.2 ST .C.5 28

ST.D.2 ST.C.6 28

ST.D.2 ST.C.6 28

ST.D.2 ST.C.6 28

ST.D.2 ST.C.6 28

ST.D.2 ST.C.6 28

ST.0.3 ST.C.l 29

ST.D.3 ST.C.3 29

ST.D.3 ST.C.3 29

ST.D.4 ST.C.l 29

ST.D.4 ST.C.1 29

ST.D.4 ST.C.1 29

266

0.566 26.93 0.49 5.2E-Ol l.6E-Ol 3.1E-Ol

0.460 35.05 0.52 4.3E-Ol l.4E-Ol 2.5E-Ol

0,366 45.63 0.54 3.5E-Ol 1.lE-01 2.0E-01

0. 286 58. 80 0. 54 3. OE-01 9. 4E-02 l. 7E-01

0. 554 28. 54 0. 51 4. BE-01 l. SE-01 3. OE-01

0.442 34.25 0.48 4.2E-01 1.3E-Ol 2.6E-01

0.333 47.07 0.50 3.4E-01 1.lE-01 2.0E-01

0.268 65.88 0.57 2.7E-Ol 8.SE-02 1.5E-Ol

0,536 5.02 0.54 2.7E-Ol 8.SE-02 l.SE-01

0.434 6.60 0.58 2.2E-Ol 6.9E-02 l,2E-Ol

0.347 9.24 0.65 LSE-01 5.7E-02 9.3E-02

0.271 11.86 o.65 1.5E-Ol 4.7E-02 7.7E-02

0.220 14.93 0.66 l.2E-Ol 3.8E-02 6.5E-02

0.500 2.52 0.72 1.4E-Ol 4.4E-02 8,3E-02

0.400 3.56 0.81 1.lE-01 3.SE-02 6.4E-02

0.398 4.34 0.99 9.5E-02 3.0E-02 5.5E-02

0.215 24.81 3.10 2.BE-03 8.SE-03 1.6E-02

0.571 1.05 0.86 l.lE-01 3.5E-02 6.2E-02

0.452 1.48 0.95 8.9E-02 2.8E-02 4.7E-02

0.383 2.06 1.10 7.0E-02 2.2E-02 3.7E-02

0.287 2.85 1.20 5.6E-02 1.8E-02 2.9E-02

0. 360 2. 29 1. 20 6. 7E-02 2. lE-02 3. 4E-02

0.496 34.28 0.53 2.0E-01 6.8E-02 1.2E-Ol

0.582 9.52 1.10 5.5E-02 1.9E-02 4.2E-02

0.533 12.03 1.30 6,5E-02 Z.2E-02 3.6E-02

0.476 31.15 0.50 5.6E-Ol 2.2E-Ol 4.2E-01

0.386 39.45 0.51 4.8E-Ol 1.9E-Ol 3.BE-01

0. 305 49. 90 0. 51 4. 2E-01 1. 6E-Ol 3. lE-01

TABLE B.2 Continued

drop cont q R ,Y Ocrit >-a >.", 1 (-]

\n,2 [-] phase phasa [mN/mJ [mm] [l/sJ [-] (-)

ST.D.4 ST.C.3 29 0,578 5.47 0.62 2.6E-01 l.OE-01 l.6E-Ol

ST.D.4 ST.C.3 29 0,463 7.77 0.70 2.2E-01 8.SE-02 l.3E-Ol

ST.0.4 ST.C.3 29 0.353 11.33 0.78 l.7E-Ol 6.SE-02 l.OE-01

0.297 15.67 0.91 l.4E-Ol 5.4E-02 8.SE-02

0.236 22.23 1.00 1.21!-0l 4.7E-02 6.9E-02

0. 184 34 .18 1. 20 9. lE-02 3. 5E-02 5. 3E-02

0 .152 50. 34 1. 50 7. 3E-02 2. 8E-02 4. 2E-02

ST.D.4 ST.C.3 29

ST.D.4 ST.C.3 29

ST.D.4 ST.C.3 29

ST.D.4 ST.C.3 29

ST.D.4 ST.C.5 29 0.508 3.18 0.98 1.3E-Ol 5.lE-02 7.SE-02

ST.D.4 ST.C.5 29 0.407 4.65 1.20 1.0E-01 3.9E-02 6.0E-02

ST.D.4 ST.C.5 29 0,315 7.05 1.40 8.lE-02 3.11!-0Z 4,7E-02

ST.D.4 ST.C.5 29 0.248 10.56 1.60 6.4E-02 2,5E-02 3.7E-02

0.194 15.61 1.80 5.lE-02 2.0E-02 2.9E-02

0.536 1.69 1.30 7 .4E-02 2.9E-02 4.3E-02

0.402 2.31 1.30 6.2E-02 2.4E-02 3.SE-02

ST.D.4 ST.C.5 29

ST.D.4 ST.C.6 29

ST.D.4 ST.C.6 29

ST.D.4 ST.C.6

ST.D.4 ST.C.6

ST.D.4 ST.C.6

29

29

29

0.322

0.247

0.193

3.56 1.60

5.66 1.9

9.08 2.4

4, SE-02 1. SE-02 2. BE-02

3.6E-02 l.4E-02 2.lE-02

2. 8E-02 l. lE-02 1. 6E-02

ST.D.4 ST.C.6 29 0.155 14.33 3.1 2.lE-02. 6.2E-03 1.2E-02

ST.D.5 ST.C.l

ST.D.5 Sî.C.l

SI.D.5 SI.C.l

ST.D.5 Sî.C.l

ST.D.5 SI.C.l

ST.D.5 ST.C.3

ST.D.5 ST.C.3

ST.D.5 ST.C.3

Sî .D. 5 ST .C. 3

ST.D ST.C.5

ST.D.5 ST.C.5

ST.D.5 ST.C.5

29.5 0.472 33.21 0.48 3,0E-01 1.2E-01 l.9E-Ol

2.9.5 0.376 53.44 0.61 2.3E-Ol 9.2E-02 l,4E-Ol

29.5 0.2.95 63.12 0.57 2.lE-01 B.4E-02 1.3E-Ol

29.5 0.414 41.71 0.57 2.4E-Ol 9,6E-02 1.7E-01

29. 5 o. 330 61.14 0. 67 1. 9E-Ol 7. 6E-02 1. 3E-02

2.9.5 0.504 9.43 0,91 l.OE-01 4.0E-02. 5.7E-OZ

2.9. 5 0. 403 13, 36 1. 00 7. 8E-02 3. lE-02 4. 7E-02

29.5 0.310 18.07 1.10 6.6E-02 2.6E-02 3.91!-02

29.5 0.248 31.41 1.50 4.8E-02 l.9E-02 2..9E-02

29.5 0.523 3.78 1.10 5.6E-02 2.2E-02 3.3E-02

29.5 0,411

29.5 0,285

5. 74

9,05

1,30 4.5E-02 1.BE-02 2.6E-02

1. 40 3. 5E-02 1. 4E-02 2. OE-02

ST.D.5 ST.C.5 29.5 0.226 18.20 2.20 2.3E-02 9.2E-03 l.4E-02

ST.D.5 ST.C.5 29.5 0.205 19.67 2.20 2.2E-02 6.8E-03 l.3E-02

ST.D.5 ST.C 6 29.5 0.485 3.64 2.40 2.3E-02 9.2.E-03 l.3E-02

ST.D.5 ST.C.6 2.9,5 0,328 6,74 3,00 l,6E-02 6.4E-03 9.SE-03

ST.D.5 ST.C.6 29.S 0.330 6,83 3.10 l.6E-02 6.4E-03 9.4E-03

ST.D.5 ST.C.6

ST.D.6 ST.C.l

ST,D.6 ST.C.l

ST.D.6 ST.C.1

ST.D.6 S!.C.3

ST.D.6 Sî.C.3

29.S 0,245 10,14 3.40 l.3E-02 5.2E-03 7.6E-03

30 0.578 35.08 0.61 1.0E-01 4.lE-02 6.lE-02

30 0.472 47.26 0.67 B.SE-02 3.5E-02 5.1E-02

30 0.378 59.62 0.68 7.5E-02 3.lE-02 4.5E-02

30 0. 611 ll. 09 1. 30 3. OE-02 1. ZE-02 l. BE-02

30 0. 496 15. 32 1. 40 2. SE-02 1. OE-02 1. 5E-02

ST.D.6 ST.C.3 30 0. 432 21. 05 1. 70 2. lE-02 8. 6E-03 1. 2E-02

267

TABLE B.2 Continued

drop

phas&

cont.

phase !mN/mJ

ST.D.6 ST.C.3 30

ST.0.6 ST.C.3 30

ST.D.6 ST.C.5 30

ST.D.6 ST.C.5 30

ST.D.6 ST.C.5 30

ST.D.6 ST.C.5 30

ST.0.6 ST.C.5 30

ST.D.6 ST.C.5 30

ST.0,6 ST.C.6 30

ST.D.6 ST.C.6 30

ST.D.6 S'.l'.C.6 30

ST.D.6 ST.C.6 30

ST.D.6 ST.C.6 30

R

[mm]

7

{l/s] Àm,1 [-]

Àm,2 {-]

0.304 37.56 2.20 1.5E-02 6.lE-03 9.0E-03

0.267 48.67 2.50 1.3E-02. 5.3E-03 7.SE-03

0.307 20.55 3.40 7.6E-03 3.lE-03 4.5E-03

o .194 46. 11 5.10 4. n:-03 l. 9E-o3 2. aE-03

0.419 12.58 2.80 9.9E-03 4.0E-03 5.81!-03

0.376 14.60 2.90 9.2E-03 3.SE-03 5.4E-03

0.262 25.56 3.60 6,7E-03 2.7E-03 3.9E-03

0.203 42.23 4.60 5.lE-03 2.lE-03 3.0E-03

0. 303 15 .15 6 .10 3. SE-03 l. 5E-03 2. lE-03

0. 196 23. 12 6. 20 2. SE-03 1. lE-03 l. SE-03

0.381 12.43 6.30 4.0E-03 1.6E-03 2.3E-03

0.310 18.79 7.80 3.2.E-03 1.3E-03 l.9E-03

0.223 28.14 7.60 2.6E-03 l.lE-03 1.5E-03

ST.D.7 ST.C.l 30.5 0.528 54.91 0.85 4.8E-02 2.lE-02 3.0E-02

ST.0.7 ST.C.l 30,5 0.523 54,84 0.94 4.4E-02 1.9E-02 3.0E-02

ST.D.7 ST.C.3 30.5 0.590 24.68 2.70 l.2.E-02 5.lE-03 7.lE-03

ST.D.7 ST.C.3 30.5 0.426 40.40 3.20 9.0E-03 3.9E-03 5.SE-03

ST.D.7 ST.C.3 30.5 0.345 61.55 4.00 7.2E-03 3.lE-03 4.4E-03

ST.0.7 ST.c.5 30.5 0.455 15.04 3.60 5.3E-03 2.3E-03 3,2.E-03

ST.D.7 ST.C.5 30.5 0.449 27.42 6.50 3.9E-03 l.7E-03 2.4E-03

ST.D.7 ST.C.5 30.5 0.461 20.62 5.00 4,SE-03 l.9E-03 2.7E-03

ST.0.7 ST.C.5 30.5 0.355 33.76 6.30 3.5E-03 l.5E-03 2.lE-03

ST.D.7 ST.C.5 30.5 0.288 52,02 7.90 2.BE-03 l.2E-03 l.7E-03

ST.0.7 ST.C.6 30,5 0.442 20.13 11.70 l.SE-03 7.7E-04 1.l!l-03

ST.D.7 ST.C.6 30.5 0.533 13.55 9.50 2.3E-03 9.9E-04 1.4E-03

ST.D.7 ST.C.6 30.5 0,352 42.05 19.40 l.3E-03 5.6E-04 7.6E-04

ST.0.11 ST.c.4 38

ST.0.11 ST.C.4 38

ST.0.11 ST.C.2 38

ST .lL ll ST ,C.2 38

ST .0.11 ST .C.2 38

ST.D.11 ST.C.2 38

ST.D.12 ST.C.4 35

ST.0.12 ST.C.4 35

ST.D.12 ST.C.4 35

ST.D.12 ST.C.4 35

ST.0.13 ST.C.4 35

ST.D.13 ST.C.4 35

ST.D.13 ST.C.2 35

ST.D.13 ST.C.2 35

268

0,355 15.12 0.89 7.5E-Ol 2.9E-Ol 4.7E-Ol

0.401 10.68 0.71 9.lE-01 3.5E-Ol 6.0E-01

0,448 53,50 1.64 8.BE-01 3.liE-01 5.6E-Ol

0. 464 39. 37 1. 25 l, lE+OO 4. lE-01 7. lE-Ol

0.405 38.25 1.06 l.4E+OO 5.6E-Ol 7.3E-Ol

0,467 44.75 1.43 9.8E-Ol 3.8E-Ol 6.4E-Ol

0.447 19.64 l.58 8.8E-01 3.7E-Ol 7.7E-Ol

0.424 18.47 1.41 9.3E-Ol 3.9E-Ol 8.0E-01

0.458 17.35 1.43 9.5E-Ol 4.0E-01 8.4E-Ol

0.331 28.70 1.71 7.2E-Ol 3.0E-01 5.9E-Ol

0.361 22.93 1.49 9.9E-Ol 4.2E-Ol 7.5E-Ol

0.506 20.97 1.91 8.8E-Ol 3.8E-Ol 8.0E-01

0.482 44.41 1.59 l,2E+OO 5.0E-01 l.6E-OO

0.405 64.15 1.93 l.OE+OO 4.4E-Ol l.2E-OO

APPENDIX C

APPENDIX Cl

EVALUATION OF SINGULARITIES IN BOUNDARY INTEGRAL

METHOD

SINGULARITY IN THE J-INTEGRAL

To calculate the J-integrals in Eq. [5.18] numerically, the

integrand should have a finite value in the singularity E = O.

Since this is not the case the J-integral will be linearised in a

triangular surface t:.S, composed by the vectors ~· z and ~ (see

Fig. Cl.1), and the linearised integral will be evaluated

analytically. The linearisation was done as follows:

1 I - g(,::) dS

r

The area of this triangle is:

óS

0

B

" " ' ' ' ' ', I ', I

- - - -'1- - - .llo,:: - - -' ,, ' ' I ', I '

---~'' ,, ___ ::::+--- 11+6µ

' ~~,, ' / II / ', ' I ' - - - -'t, - - - - - - - -'- - -', ,,, 1', ', 1,, .... , ' '

- - - - - - - - ~ - - - :" - - - .::- '!; - - -'- - - -

' I ', ' ',

• ' ' ' ' ' ... '

À

[Cl.1]

[Cl.2]

A

Fig. Cl.l Discretisation of the triangular surface element àS

269

270

For any vector E e àS the function g(E) can be linearised as

follows:

[Cl.3]

in which A and B can be determined by substituting r=x and r-z:

[Cl. 4]

[Cl. 5]

The integral is evaluated by subdividing the triangle àS in

similar subtriangles, using the parameterisation , E = ~ + µ~.

The surface of such a subtriangle, enclosed by À, À+àÀ, µ and

µ+6µ, is given by:

àStr = àS1 + àS2 6À 6µ 6S + 6À 6µ àS " 2 6À 6µ 6S [Cl.6]

Now the integral I can be calculated:

[Cl. 7]

Rearranging the terms with respect to À and µ leaves us two

integrals which can be solved separately:

I

---------- 26S dµdÀ [Cl.8] LL J 2 2 2 2 A x + 2Àll(~·~> + µ y

To evaluate the first integral we first calculate:

2 2 + JJ y

dµ 1 - ln y (

x+y+z )

x-y+z

The first integral can now be evaluated:

2g0 + (A+B)x2

+ B(~·:O ( x+y+z ) I 1 = ln -- llS

y x-y+z

To evaluate the second integral we first calculate:

[Cl. 9]

[Cl.10]

2 2 + JJ y

dµ = ~ ( :.:: - (~·;> ln ( :::.:.:. J ) y y y x-y+z

[Cl.11]

The second integral can now be evaluated:

y ( :.:: - <~· l> ln ( x+y+z ) ) llS

y x-y+z [Cl.12]

The total linearised J-integrai can now be written as:

I (( 2lls

2(A+B)) (:+y+z) 1 [ zZ+X)

g0+ Y2 ln(;_y+z + ~ (B-A)y(z-x)+(B+A)(z-x) -;-

2llS

y

[Cl.13)

271

272

APPENDIX C2 SINGULARITY IN THE K-INTEGRAL

To calculate the K-integral in Eq. [5.18] an approach similar to

the calculation of the J-integral (see Appendix Cl) was adopted.

This approach involves linearisation of the integrand close to the

singular point and calculation of the linearised integral by

an analytica! method. The following linearisation was used:

I'

with

I l h(f) dS LIS

(C2.l]

[C2.2]

[C2.3]

[C2.4]

After applying the same discretisation of the triangular surface

LIS as in Appendix Bl the integral I' can be rewritten:

I' 1 IÀ 2A'(x.r) + 2B'(z.r) 2 2 - - - -2 2 2llS dµdÀ

>-=O µ=0 X x + 2Àµ(~·l) + µ y [C2.5]

Rearranging the terms with respect to À and µ leaves us two

integrals which can be evaluated separately:

I' = I'1 + I'2 = ï fÀ JÀ=O µ=0

---------- 2llS dµdÀ +

~ 2 2 2 2 À x + 2Àµ(~·l) + µ y

r fÀ µA'(~·l) + µB'(~·l) + µB'y 2

----------- 2LIS dµdÀ

>-=o µ=0 Jx2x2 + 2Àµ(~·l> + µ2y2

(C2.6]

To evaluate the first integral we first calculate:

r (/+(x.z::)J ((x.z::)J) larctan

2llS - arctan

2àS

1

2ÀllS

The first integral 1 1

1 can now be evaluated:

To solve the second integral 1 1

2 we first calculate:

(C2.7)

[C2.8]

IÀ 2 2 µdµ 2 2 " 1 2 ln r:) µ"o >- x + 2Àµ(~·z::> + µ y y l;

-- (A'+B')x2+B'(!!;·~) ~·V ( ) 2/às

[C2.9)

The second integral r• 2 can now be evaluated:

( 2àS ~) (~·Z::) ( (y2+(~·Z::)) I' = ln - - -- arctan

2 / 2llS ((x.z::))) arctan ZllS

(C2.10]

The total integral can now be written as:

273

l' "' [ ((A'+B')(~·;t:)+B'l) lnË) +

2(A'+B')6s ( arctanl2

:~·:t:>) - arctan(<~~~>) ) J 2; 1c2.111

274

STELLINGEN

behorende bij het proefschrift

DEFORMATION AND BREAK.UP OF DROPS IN SIMPLE SHEAR FLOWS

van Robert Antonie de Bruijn

1. De opbreekcriteria voor afschuifsnelheidsverdunnende druppels kunnen beneden een viscositeitsverhouding van 0.1 goed beschreven worden door de criteria voor Newtonse druppels, wanneer men een gemodificeerde viscositeitsverhouding gebruikt, die rekening houdt met een afschuifsnelheid in de druppel; die hoger is dan de opgelegde afschuifsnelheid. (Hoofdstuk 4 van dit proefschrift)

2. In tegenstelling tot wat beweert wordt in een aantal publikaties bemoeilijkt elasticiteit van de vloeistof in de druppelfase het opbreken van druppels. Dit effect is sterker voor viscositeits­verhoudingen van orde 1 dan voor kleine viscositeitsverhoudingen. (Hoofdstuk 4 van dit proefschrift)

3. Voor druppelfasen met een sterke mate van elasticiteit in de vloeistof bestaat een kritische druppelgrootte, waar beneden opbreken in enkelvoudige afschuifstr0mingen niet mogelijk is. (Hoofdstuk 4 van dit proefschrift)

4. In tegenstelling tot beweringen van Rallison, blijkt het opbreek­cri terium voor druppels met een viscositeitsverhouding van 1 niet afhankelijk te zijn van de snelheid waarmee de kritische afschuif­snelheid wordt bereikt. (Hoofdstuk 5 van dit proefschrift)

5. Vanneer een druppel een driehoeksvormig afschuifsnelheidsprofiel ondergaat met een maximale afschuifsnelheid die hoger is dan de kritische afschuifsnelheid voor quasi-stationaire afschuif­stromingen, zal opbreken alleen plaatsvinden wanneer het afschuif­snelheidsprofiel voldoende lang wordt aangehouden. (Hoofdstuk 5 van dit proefschrift)

6. Het verschijnsel "tipstreaming" kan in enkelvoudige afschuif­stromingen optreden wanneer oppervlakte-actieve stoffen aanwezig zijn in concentraties waarbij gradiënten in de oppervlaktespanning kunnen ontstaan, resulterend in een lage oppervlaktespanning nabij de uiteinden van de druppel en een hogere oppervlaktespanning elders. (Hoofdstuk 6 van dit proefschrift)

7. De relaxatie tijden die het lineair visco-elastisch gedrag van een dispersie van hydrodynamisch interacterende capsules beschrijven, kunnen sterk beïnvloed worden door de dikte van de schil van de capsules. (R.A. de Bruijn en J. Kellema, Rheol. Acta 24:159-174 (1985)

8. De toepasbaarheid van het model van Thomas et al. voor de beschrijving van warmte- en massatransportproblemen in poreuze media, is sterk beperkt omdat de thermofysische eigenschappen niet afhankelijk mogen zijn van zowel de temperatuur als het vochtgehalte. (B.R. Thomas, K. Morgan en R.W. Lewis, Int. J. Num. Meth. Eng. 15:1381-1393 (1980) )

9. De toepasbaarheid van het numerieke model van Ohlsson en Bengtsson voor de verwarming van levensmiddelen door microgolven is sterk beperkt in toepasbaarheid door de verwaarlozing van het golfkarakter van deze electromagnetische golven. (T. Ohlsson en N.E. Bengtsson, Microw. Energy Appl. Newsletter 4:3-8 (1979) )

10. De conclusie van Davies dat alleen turbulentie de goede emulgeren­de werking van hoge druk homogenisatoren kan verklaren is onjuist. (J.T. Davies, Chem. Engng. Sci. 40:839-842 (1985) )

11. De variatie van de gemiddelde druppelgrootte met het toerental van een geroerd vat, welke Stamatoudis en Tavlarides hebben waar­genomen, is afhankelijk van de viscositeit van de continue fase. Deze afhankelijkheid kan verklaard worden door de overgang van laminaire naar volledig turbulente stroming in het vat. (M. Stamatoudis en L. Tavlarides, Ind. Eng. Chem. Process Des. Dev. 24:1175-1181 (1985) )

12. De hedendaagse teruggang in het aantal diersoorten en plante­soorten ondersteunt eerder een scheppingstheorie dan een evolutie­theorie.

13. Acceptatie van kleine criminaliteit is een grotere bedreiging voor de maatschappij dan grote criminaliteit.

Wymington, september 1989