The rheology of a bubbly liquid - · The rheology and ﬂow behaviour of suspensions of particles in liquids is of interest because of the enormous range and complexity of the phenomena

10.1098/rspa.2001.0924

The rheology of a bubbly liquidBy E. W. Llewellin

1, H. M. Mader

1and S. D. R. Wilson

2

1Department of Earth Sciences, University of Bristol,Wills Memorial Building, Queens Road, Bristol BS8 1RJ, UK

2Department of Mathematics, University of Manchester,Oxford Road, Manchester M13 9PL, UK

Received 26 April 2001; revised 8 August 2001; accepted 4 October 2001;published online 6 March 2002

A semiempirical constitutive model for the visco-elastic rheology of bubble suspen-sions with gas volume fractions φ < 0.5 and small deformations (Ca 1) is devel-oped. The model has its theoretical foundation in a physical analysis of dilute emul-sions. The constitutive equation takes the form of a linear Jeffreys model involvingobservable material parameters: the viscosity of the continuous phase, gas volumefraction, the relaxation time, bubble size distribution and an empirically determineddimensionless constant. The model is validated against observations of the defor-mation of suspensions of nitrogen bubbles in a Newtonian liquid (golden syrup)subjected to forced oscillations. The effect of φ and frequency of oscillation f onthe elastic and viscous components of the deformation are investigated. At low f ,increasing φ leads to an increase in viscosity, whereas, at high f , viscosity decreases asφ increases. This behaviour can be understood in terms of bubble deformation ratesand we propose a dimensionless quantity, the dynamic capillary number Cd, as theparameter which controls the behaviour of the system. Previously published consti-tutive equations and observations of the rheology of bubble suspensions are reviewed.Hitherto apparently contradictory findings can be explained as a result of Cd regime.A method for dealing with polydisperse bubble size distributions is also presented.

Keywords: rheology; bubble suspension; time-dependent flow;capillary number; relaxation time; visco-elasticity

1. Introduction

The rheology and flow behaviour of suspensions of particles in liquids is of interestbecause of the enormous range and complexity of the phenomena exhibited and alsobecause of the practical significance of this class of materials. The dispersed phasecan be solid particles, immiscible liquid droplets (an emulsion), or gaseous bubbles.Commercially important suspensions include all manner of foodstuffs and cosmeticsand industrial suspensions such as paint, ink and ceramics. Naturally occurring sus-pensions range from biological materials such as milk and blood to geophysical flowsof mud and magma.

Adding particles to a liquid does not just change the magnitude of the viscos-ity, it can introduce all known non-Newtonian phenomena, including shear and timedependence of the viscosity, and elastic effects such as stress relaxation and nor-mal stress differences. These phenomena arise because of a range of particle–liquid

Proc. R. Soc. Lond. A (2002) 458, 987–1016987

c© 2002 The Royal Society

988 E. W. Llewellin, H. M. Mader and S. D. R. Wilson

and particle–particle interactions, which are reviewed by Tadros (1985) and Macosko(1994). Previous research on suspensions has focused primarily on the dilute (vol-ume fractions of the dispersed phase φ < 0.05) and highly concentrated (φ > 0.75)endmembers and on suspensions of solid particles and emulsions (for reviews seeSchowalter 1978; Pal & Rhodes 1989; Pinkerton & Stevenson 1992; Stein & Spera1992; Macosko 1994).

Most of the reported research on the rheology of bubble suspensions is promptedby a desire to understand the dynamics of magmatic flows, although the results areclearly more widely applicable. Magmas are Newtonian liquids over a wide range ofstresses and strain rates (Webb & Dingwell 1990). The viscosity of pure magmascan be calculated as a function of temperature and composition and ranges over15 orders of magnitude (from about 10 to 1015 Pa s (Murase & McBirney 1973)).Bubbles occur in magmas due to exsolution of dissolved volatiles. The bubbles area crucial component of magmatic flows because it is the growth and expansion ofthe dispersed gaseous phase that drives many volcanic eruptions, especially the mostviolent. Gas volume fractions can range from zero to φ > 0.9 and frequently lie inthe intermediate range, in common with most suspensions of practical significance.

Our aim is to produce a general model for the visco-elastic rheology of bubblesuspensions that will allow the shear viscosity of the suspension to be calculated,given a knowledge of other material parameters, bubble content and the deformationrate. We seek a model that is applicable to magmatic systems and so it must coverintermediate gas volume fractions and bubble suspensions in Newtonian liquids. Adetermination of visco-elastic effects necessarily involves a consideration of time-dependent flows.

Rheological studies typically adopt one of several distinct approaches. A purelyempirical approach involves formulating a constitutive law for a material on thebasis of experimental data alone. Such laws can be very useful as they are couchedin terms of observable parameters of real systems, nevertheless, the form of theequation generally has no physical basis. An alternative approach is to producea physical theory in which explicit account is taken of the shape and mechan-ical properties of each phase to determine flow properties of the bulk. However,problems frequently arise in such theories because the simplifications necessary torender the mathematics tractable often result in a highly abstracted system thatbears little resemblance to any real material. We therefore favour a semiempiricalapproach, which has a physical basis but involves experimentally determined con-stants.

In the following section, we review the various constitutive equations that havebeen proposed to describe the rheology of bubble suspensions. The constitutive equa-tion for a dilute emulsion of Frankel & Acrivos (1970) is recast in terms of observ-able parameters in the form of a linear Jeffreys model. This model is compared withobservations of the visco-elastic deformation of bubble suspensions in a Newtonianliquid with 0 φ 0.5 subjected to forced oscillations. The experimental data pro-vide an empirical determination of just one of the parameters in the model. Usingthis parameter, the model describes our data closely. The visco-elastic behaviour ofthe bubble suspensions depends strongly on frequency of oscillation; viscosity canincrease or decrease with increasing φ depending on the conditions of strain. Theapparent contradictions in previously reported research are revisited and discussedin the light of these findings.

Proc. R. Soc. Lond. A (2002)

The rheology of a bubbly liquid 989

2. Constitutive equations for bubbly suspensions

The general differential equation for linear visco-elasticity relates stress τ and strainγ as follows,(

1+α1∂

∂t+α2

∂2

∂t2+ · · ·+αn

∂n

∂tn

)τ =

(β0 +β1

∂

∂t+β2

∂2

∂t2+ · · ·+βm

∂m

∂tm

)γ, (2.1)

where n = m or n = m − 1 (Barnes et al . 1989). The time derivatives are ordinarypartial derivatives restricting the applicability of the equation to cases where thereare only small changes in the variables (as in the present study). A constitutiveequation relates the constants α1, α2, β0, β1, etc., to the physical properties of thetwo-phase material.

For constant stress and steady shear (steady-state flow conditions), only terms upto n = 0, m = 1 are necessary. For a Newtonian fluid, β0 = 0, which leads to thefamiliar result τ = ηγ, where η ≡ β1. Many different constitutive equations existwhich relate β1 to the physical properties of suspensions experiencing steady-stateflow and these are reviewed below. It is conventional to normalize β1 to the fullydense viscosity η0 and present the relationship in terms of the ratio β1/η0 = ηr, therelative viscosity.

Steady-state conditions necessarily imply that the bubble deformation is constant;i.e. the bubbles are no longer actively changing shape and have reached what will becalled their ‘equilibrium deformation’ in this paper. The equilibrium deformation ofbubbles subjected to steady shear is a function of the capillary number Ca, which isthe ratio of viscous (deforming) to surface tension (restoring) forces,

Ca =η0rγ

Γ, (2.2)

where r is the radius of the undeformed bubble, η0 is the viscosity of the fully denseliquid and Γ is the liquid–bubble interfacial tension. If Ca is large, the restoringforce is small compared with the deforming force and the bubble is readily deformedunder shear. If Ca is small, surface tension forces dominate and the bubble attains itsequilibrium deformation rapidly after changes in strain rate. In general, the Taylordeformation D = kCan, where D is a dimensionless measure of the elongation ofthe bubble and k and n are dimensionless parameters (Taylor 1934; Loewenberg &Hinch 1996). The deformation can also be written as a dimensionless strain rate,with the strain rate normalized by a deformation time-scale λ called the ‘relaxationtime’. For small Ca, n = 1 (Taylor 1934; Loewenberg & Hinch 1996) and so

D = kCa = λγ, (2.3)

in which caseλ = k

η0r

Γ. (2.4)

The parameter k increases with the volume fraction of the disperse phase (Oldroyd1953; Oosterbroek & Mellema 1981; Loewenberg & Hinch 1996). In the limit ofa solitary bubble in an infinite, pure liquid, k ≈ 1 (Taylor 1934). In steady flow,constant shear conditions have existed for some time t λ.

Theory and observations that are directly applicable to bubbly suspensions arescarce. The flow of emulsions has been more thoroughly investigated. As a result,



models of bubbly flows frequently use constitutive laws derived for emulsions onthe grounds that bubbles and liquid droplets are both deformable inclusions. It isinstructive at this point to review the various models that have been proposed andto consider their validity for bubble suspensions.

(a) Constitutive equations for steady-state conditions

Einstein (1906, 1911) shows that, for a dilute suspension of solid particles,

ηr = 1 + 52φ. (2.5)

Taylor (1932) extends this theory to cover dilute emulsions. For the case of an invisciddispersed phase:

ηr = 1 + φ. (2.6)

The increase in viscosity with increasing φ predicted arises because flow lines aredistorted around the bubbles. The relationship assumes bubbles are approximatelyspherical (i.e. Ca 1). As Ca increases, bubbles become increasingly elongate andthe relationship is no longer valid.

The majority of workers find that equation (2.6) describes their experimental datapoorly. Various alternative formulations are proposed.

Stein & Spera (1992) propose a modified form of Taylor’s equation,

ηr = 1 + aφ, (2.7)

where the value of a must be experimentally determined. They found a = 13.1 basedon rotating rod viscometry of suspensions of air bubbles with 0.008 φ 0.055 inmolten glass with η0 ∼ 5 × 104 Pa s and 0.5 < Ca < 100. They suggest that thelarge difference between the value of a for their experiments and the value predictedby Taylor’s theory for emulsions (unity) is due to the violation of the deformationcriterion (Ca 1) in their experiments.

Barthes-Biesel & Chhim (1981) attempt to take account of higher values of Cawith their equation,

ηr = 1 + (2.5 − a(Ca)2)φ, (2.8)

where a 70. This equation is based on a mathematical treatment of the behaviourof liquid droplets with an elastic bounding membrane suspended in a viscous liquid(i.e. it is probably most applicable to particles such as blood cells). The constant ais a calculable parameter of the membrane material.

Several authors propose nonlinear relationships between ηr and φ.Roscoe (1952) presents a theoretical treatment of suspensions of solid spheres

which has been applied to micro-emulsions,

ηr = (1 − aφ)−2.5, (2.9)

where a = 1 for a polydisperse suspension and a = 1.35 for a monodisperse suspen-sion.

Eilers (1941, 1943) proposes a relationship based on experiments on bitumen emul-sions,

ηr =(

1 +1.25φ

1 − aφ

)2

, (2.10)

where 1.28 < a < 1.30 and is empirically determined.



Mooney (1951) proposes an exponential relationship for monodisperse suspensionsof solid spheres,

ln ηr =2.5φ

1 − aφ, (2.11)

where geometrical arguments give 1.35 < a < 1.91.Sibree (1934) presents high Ca measurements of the viscosity of ‘froths’ of a fungi-

cide (0.52 < φ < 0.73), which is stabilized by an organic colloid. He compares hisresults with the theoretical model for foam (φ > 0.74) rheology of Hatschek (1911),

ηr =1

1 − (aφ)1/3 , (2.12)

and obtains a good fit for a = 1.2. The liquid phase in his experiments is stronglynon-Newtonian (shear-thinning) due to the presence of the organic colloid.

(b) Investigations of porous glasses and field observations of magmas

All of the models presented so far predict an increase in viscosity with increasingφ and the experimental data (most of which derives from observations of emulsions)support this. By contrast, experiments relating to the behaviour of porous glasses(defined by Bagdassarov & Dingwell (1992) as having η0 > 109 Pa s) indicate adecrease in viscosity with increasing φ (Rahaman et al . 1987; Ducamp & Raj 1989;Sura & Panda 1990; Bagdassarov & Dingwell 1992, 1993; Lejeune et al . 1999). Exper-imental and theoretical studies predicting a decrease in viscosity with increasing φare presented below.

Mackenzie (1950) describes a theoretical treatment of the elastic constants of asolid containing a small volume fraction of identical spherical holes. He appeals toan analogy between viscous and elastic moduli to derive the following viscosity/gasvolume fraction relationship for a dilute, monodisperse bubble suspension:

ηr = 1 − 53φ. (2.13)

It is assumed that the viscous deformation of the bubbles is unopposed by surfacetension forces, i.e. λ → ∞. Mackenzie claims validity for slow shear for as long asthe bubbles remain approximately spherical.

Scherer (1979) uses an analogy between thermal and viscous stresses to derivea rheological model for highly viscous porous glasses. He assumes both phases arecontinuous and that the glass is arranged as a cubic framework of cylinders. He finds

ηr =3(1 − φ)

2(1 + 2φ) + (1 + φ − 2φ2)1/2 . (2.14)

Ducamp & Raj (1989) carried out dilatometric measurements on glasses (η0 ∼109 Pa s) with gas volume fractions in the range 0 φ 0.45. They propose arelationship of the form,

ln ηr = −a

(φ

1 − φ

). (2.15)

This equation has no theoretical basis but is constructed to satisfy three criteria: (i)the relationship must be nonlinear, (ii) it must reduce to the form of Mackenzie’s



equation at low φ, (iii) ηr = 1 when φ = 0. Ducamp & Raj obtain a good fit to theirdata when 2.5 a 4.

Rahaman et al . (1987) performed similar experiments to Ducamp & Raj on glassesbearing highly non-spherical pores and with 0.14 φ 0.34. They present a purelyempirical relationship based on their data,

ln ηr = −aφ, (2.16)

where they obtain a = 11.2.Bagdassarov & Dingwell (1992) also carried out dilatometric measurements on

porous glasses (η0 ∼ 1010 Pa s) over the range 0 φ 0.7. They propose an empiricalrelationship that satisfies two criteria: (i) the relationship reduces to the form ofMackenzie’s equation at low φ, (ii) it reduces to ηr = 1/φ at high φ (based onexperimental results of Sura & Panda 1990),

ηr =1

1 + aφ, (2.17)

where they obtain a = 22.4.Bagdassarov & Dingwell (1993) present, as far as we know, the only published

study of the time-dependent rheology of a bubbly material, performing oscillatoryrheometry on highly viscous magma (109 η0 1012 Pa s) with gas volume fraction0 < φ < 0.3. They state that the viscosity of the material decreases with increasinggas volume fraction but do not present a constitutive rheological equation.

Comparison of theory and behaviour of bubbly liquids and porous glasses is com-plicated for several reasons. The very high viscosity of glasses leads to very longrelaxation times and so steady-state conditions are unlikely to be attained. More-over, the high viscosity of the glasses prevents shear viscosity from being measureddirectly by, for example, rotating rod viscometry. Instead, samples are generallysubjected to uniaxial compression. Inferring shear viscosity from compressional mea-surements requires a knowledge of the bulk viscosity of the material, which has notbeen measured. The usual approach adopted is to appeal to an analogy with elasticstress analysis (Mackenzie 1950; Scherer 1979; Sura & Panda 1990; Bagdassarov &Dingwell 1992); however, it has been shown by Wilson (1997) that this analogy isinvalid. This is because in a compressible fluid the thermodynamic pressure enters theequations in an essential way via an equation of state. In the usual theory of elasticsolids, on the other hand, all the states considered are thermodynamic equilibriumstates.

Reported observations of magmatic flows in the field are reviewed by Manga etal . (1998) and are also contradictory, with some researchers inferring that vesicularmagmas have apparently higher viscosities and some lower viscosities than equivalentgas-free flows.

Evidently, there are contradictions present in the literature and confusion as toeven the direction of the effect of adding bubbles to a liquid. These issues maybe partly explained by recent numerical work by Manga & Loewenberg (2001) andexperimental work on corn syrup bubble suspensions by Rust and Manga (M. Manga2001, personal communication), which suggests that bubble suspensions are shearthinning and that viscosity can increase with gas volume fraction when Ca < O(1)and decrease with gas volume fraction at high when Ca > O(1). We revisit this issuelater in the light of our data and show that the different results can be explained for



constant, low Ca if the unsteady nature of the measurements on ultra-high viscositysuspensions (i.e. the porous glasses) is taken into account.

(c) Time-dependent flow

In unsteady flows visco-elastic effects become important and more terms fromequation (2.1) are required to describe the behaviour. Constitutive equations thatinclude the visco-elasticity of a suspension allow systems in which shear stress andstrain rate change with time to be modelled. One such equation, commonly used todescribe the visco-elastic behaviour of polymers, is the Jeffreys model, a specific caseof equation (2.1), where n = 1, m = 2 and β0 = 0, giving

τ + α1τ = β1γ + β2γ. (2.18)

Frankel & Acrivos (1970) obtain a constitutive equation for a dilute, monodis-perse emulsion in which the droplets remain approximately spherical. Their modelprovides one of the most comprehensive and rigorous physical analyses presented todate for such a flow. For the case of inviscid inclusions, such as bubbles, and forsmall amplitude oscillatory shear such that Ca 1 as in the present experimentalprogramme, their model can be recast in terms of observable parameters in the formof a Jeffreys model (see Appendix A)

τij + 65λτij = η0(1 + φ)γij + η0

65λ(1 − 5

3φ)γij . (2.19)

This model forms the basis for the analysis of our measurements of visco-elasticityin bubble suspensions.

(d) Emulsions versus bubble suspensions

It is reasonable to expect that models derived for emulsions will be relevant tobubble suspensions to some extent because both liquid droplets and bubbles aredeformable. However, bubble suspensions differ from emulsions in a number of impor-tant respects.

(i) Liquid droplets are viscous whereas bubbles are inviscid and so internal flowand viscous dissipation within bubbles will be negligible.

(ii) Bubble suspensions are compressible whereas emulsions are incompressible.

(iii) Bubbles have much higher internal energies than liquid droplets. Surface ten-sion is important to aggregative and breaking stability in bubble suspensionsand emulsions and has been discussed by Babak (1994) and Princen & Kiss(1986, 1989) with reference to highly concentrated emulsions. Surface tensionenters nonlinearly into the equation for shear viscosity presented by Princen& Kiss (1989). Moreover, the lower surface tensions of liquid droplets meanthat they will deform more easily than bubbles and so bubbly liquids might beexpected to behave more like suspensions of solid particles than emulsions for agiven particle size. However, it is suggested that bubbles have a stronger influ-ence on rheological properties than similar concentrations of crystals (Stein &Spera 1992; Pinkerton & Norton 1995).



COOLINGCOOLINGJACKETJACKET

SYRUPSYRUPINLETINLET

OUTLETOUTLETSTATORSTATOR

ROTORROTORGASGAS

INLETINLET

Figure 1. The aerator. Cutaway of the mixing head of the Mondomix aerator showing theinterlocking teeth of the stator and the rotor. Nitrogen is injected at about 4 bar into themixing head. The rotor turns at about 1000 RPM. The cooling jacket is maintained at −10 C.The diameter of the mixing head is 15 cm.

(iv) The density contrast between the dispersed phase and the liquid phase is muchhigher in bubble suspensions than in emulsions. Indeed, whereas it is possibleto generate neutrally buoyant emulsions, bubbles are always inherently buoyantwith respect to the liquid phase. This also affects the aggregative properties.

For these reasons, it is unlikely that any model of emulsions can be applied withoutamendment to bubble suspensions. We therefore use experimental observations ofthe viscoelastic rheology of bubble suspensions to investigate how well the modelproposed in equation (2.19) describes the data and what adjustments are necessary.

3. The bubble suspensions

Suspensions of nitrogen bubbles in golden syrup are created using a commercialaerator (figure 1). Golden syrup was chosen because it has a Newtonian rheology, iswater soluble and non-hazardous. Physical properties of golden syrup are given infigure 2. The suspension produced by the aerator typically has a gas volume fraction



0

100

200

15 20 25 30 35 40

1410

1420

1430

1440

1450

10 20 30 40 50temperature (ºC)

dens

ity (

kg m

−3)

visc

osity

,

(Pa

s)η

(a)

(b)

Figure 2. Properties of golden syrup. (a) Viscosity of golden syrup as a function of temperature.The viscosity measurements were made using a Haake RV20 controlled-rate, rotational viscome-ter with a concentric-cylinder sensor. (b) Density of golden syrup as a function of temperature.The density was determined by weighing a known volume of syrup at a range of temperatures.

in the range 0.3 < φ < 0.4. This is poured into a tall box (10 cm×10 cm×100 cm) andleft standing for about one week at room temperature until bubble rise produces astratified column with φ increasing from φ = 0 at the base to φ > 0.6 at the top. Thegas volume fraction of samples taken from different heights in the box is measuredto an accuracy of 2% by weighing a known volume. Samples are then transferredto the sensor system of the rheometer, described in § 4 b. Some alteration of the gasvolume fraction of the samples may occur during transfer due to trapped or lost air.The total uncertainty on φ in the test material is estimated to be less than 5%.

To measure the bubble size distribution, samples are pressed between two glassplates 0.5 mm apart to separate the bubbles. A digital image is taken through amicroscope at 4× magnification. A typical image contains around 1000 bubbles withradii in the range 0.001–0.1 mm. Figure 3 shows that the distributions of bubblenumber, radius, surface area and volume are distinctly different.



0

0.1

0.2

0.3

0.4

0

0.1

0.2

0.3

0.4

0

0.2

0.4

0.6

0.8

−1 0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1.0

−1 0 1 2 3 4 5 6 7 8 9 10Φbubble size,

bubb

le n

umbe

r fr

actio

n

Φbubble size,

bubb

le s

urfa

ce a

rea

frac

tion

(a)

(b) (c)

(d ) (e)

bubb

le r

adiu

s fr

actio

nbu

bble

vol

ume

frac

tion

Figure 3. Bubble size distributions. (a) A typical image of a sample with φ = 0.140 pressedbetween plates 0.5 mm apart. The radii of the bubbles are measured and sorted into bins accord-ing to the standard Φ scale, where Φ = − log2(diameter in mm). Image is 3 mm across. Fourdifferent bubble size distributions are calculated: (b) number, (c) radius, (d) surface area, (e) vol-ume. In each case, the distributions are given in terms of the fraction of the total which is presentin each Φ class.



viscous component

elastic component

total stress

strain rate

strain

stra

in

or

stre

ss

tω

tω

δ

γγ

τ

.

Figure 4. Idealized waveforms for forced oscillations of a viscoelastic medium. The imposedstress is given by τ = τ0 cos(ωt + δ) = τ0 cos δ cos ωt − τ0 sin δ sin ωt, which can be viewed as asuperposition of two components: the cos ωt term is in-phase with the strain, γ = γ0 cos ωt, and sodescribes the elastic component; the − sin ωt term is in-phase with the strain rate, γ = −γ0 sin ωt,and so gives the viscous component. Viscoelastic materials therefore have phase-shifts in therange 0 < δ < π/2.

4. Experimental method

(a) The method of forced oscillations

A standard technique used to investigate the visco-elastic properties of liquids is themethod of ‘forced oscillations’ in which a sinusoidally varying stress τ is imposedon the sample and the resultant deformation is observed. Within the linear visco-elastic region the structure of the material is stable, the observed deformation issimilarly sinusoidal and measurements are repeatable. The total stress consists oftwo components: an elastic component in-phase with the strain γ and a viscouscomponent in-phase with the strain rate γ (and so π/2 out-of-phase with the strainand the elastic deformation). The superposition of these components generates asinusoidal signal that is phase-shifted with respect to the applied stress by someamount δ. Idealized wave forms are shown in figure 4 and can be described by thefollowing system of equations,

τ = τ0 cos(ωt + δ) = Re(τ0eiδeiωt), (4.1)

γ = γ0 cos(ωt) = Re(γ0eiωt), (4.2)

γ = −γ0 sin(ωt) = Re(iγ0eiωt), (4.3)



where ω is the angular frequency of oscillation and γ0 = ωγ0. It is convenient to definea ‘complex viscosity’ η∗ in terms of the moduli of amplitudes of the oscillations suchthat

η∗ =τ0eiδ

iγ0= η′ − iη′′, (4.4)

hence

η′ =τ0 sin δ

γ0(4.5)

and

η′′ =τ0 cos δ

γ0. (4.6)

η′ is called the ‘dynamic viscosity’ and describes the viscous deformation which is in-phase with the strain rate. Conversely, η′′ is called the ‘loss viscosity’ and describesthe elastic deformation which is in-phase with the strain. The phase shift δ is givenby

tan δ =η′

η′′ . (4.7)

It is immediately apparent that the phase shift is a measure of the relative contri-butions of elasticity and viscosity to the deformation. A purely viscous material willhave no loss modulus and δ = π/2, whereas a purely elastic material will have nodynamic modulus and δ = 0. In general, for a visco-elastic material 0 < δ < π/2.

The linearized Frankel & Acrivos (1970) model, as expressed in equation (2.19),has real and imaginary parts of the complex viscosity of the sample, η′ and η′′,respectively, given by (see Appendix A)

η′ =β1 + α1β2ω

2

1 + α21ω

2 , (4.8)

η′′ =(β1α1 − β2)ω

1 + α21ω

2 , (4.9)

where ω is the angular frequency of oscillation and the coefficients are related tomeasurable physical properties of the suspension as follows:

α1 = 65λ, (4.10)

β1 = η0(1 + φ), (4.11)

β2 = η0α1(1 − 53φ). (4.12)

Note that α1 and β2 only enter into unsteady flow problems (see equation (2.18))and that (4.11) is the Taylor equation for steady flow conditions (see equation (2.6)).

(b) Experimental constraints and rheological tests

The response of the bubble suspensions to forced oscillations is observed usinga Haake RS100 rheometer fitted with a specially designed wide-gap sensor system(figure 5). All rheological tests are performed at 25.0 ± 0.1 C. Viscous heating is



9 mm

8 cm

Figure 5. The wide-gap, parallel-plate sensor system. The sensor system has a circular cross-sec-tion. The lower ‘plate’ is provided by a shallow dish which contains the sample. The gap is9 mm, which is more than 30× the maximum bubble radius. The sensor is enclosed in a waterjacket that holds the whole system at 25.0 ± 0.1 C (not shown). The parallel plate geometrywas chosen as it is non-intrusive and so its use does not modify the bubble suspension in anyway.

negligible at the strain rates attained during the experiments. The RS100 applies atorque M to the top plate of the sensor and measures the resultant angular displace-ment Ω. For all of our experiments M = 8.4 × 10−5 N m and Ω varies in the range1.7 × 10−4 Ω 0.2 rad depending on frequency of oscillation and viscosity of thesample. Torque and angular displacement are converted to stress and strain using twocalibration factors, A and D, such that τ = AM and γ = DΩ. A and D are deter-mined for the wide-gap sensor system against calibration oils with 5 < η < 100 Pa sto an accuracy of 3.1%. The spatial variation of stress and strain rate within thesensor system cannot be calculated without a detailed a priori knowledge of therheology of the fluid being tested. The calibration procedure means that the ratioof stress to strain rate gives the ‘apparent viscosity’ of the sample; i.e. a Newtonianfluid of this viscosity would strain at the same rate as the sample for the imposedstress.

The maximum angular displacement of the sensor which we allow is π/10 rad,ensuring that all deformations are small compared with the sensor gap. We estimatethe maximum strain rate attained to be of order 0.01 s−1 for all experiments at allfrequencies. The calculated relaxation times λ (equation (2.4) with k = 1) are oforder 0.1 s for the largest bubbles giving a maximum bubble capillary number Caof order 10−3. This means that the bubbles remain essentially spherical, in keepingwith the theoretical assumptions made earlier (§ 2 c and § 4 a). However, Ca can onlystrictly be calculated for steady flow conditions and so has little physical meaning forexperiments where oscillations are occurring around a point of zero strain, exceptwhere the period of oscillation is much longer than the relaxation time. We willreturn to this point later (§ 6 b).

It is essential for the structure of the sample to remain as constant as possible forthe duration of a test. The most significant problem in this regard is that of bubble



rise. An estimate for the rate at which bubbles rise u can be calculated from theHadamard–Rybczynski equation (Hadamard 1911; Rybczynski 1911) u = r2ρg/3η0,where r is the bubble radius, ρ is the density of the suspending medium and g isthe acceleration due to gravity. This equation is a generalization of Stokes’s equationto arbitrary viscosity ratios. It applies for bubble rise in clear liquids. Bubble riseis likely to be impeded by the presence of other bubbles in a suspension and so theequation tends to overestimate the rise velocity. To minimize bubble rise we rejectsamples containing bubbles larger than Φ = 1 (r = 0.25 mm). This means that eventhe largest bubbles take more than 20 min to rise through the sample. Since themaximum test duration is about 30 min, only the largest bubbles rise significantlyduring an experimental run.

Samples are first subjected to a ‘stress sweep’ in which the stress amplitude is var-ied at constant frequency. The range of stresses over which the complex viscosity |η∗|is constant defines the region of linear visco-elastic response. This range is typically0.1 < τ0 < 200 Pa for our samples.

The visco-elasticity of the suspensions is then determined by conducting a ‘fre-quency sweep’. In this test, the stress amplitude τ0 is fixed well within the linearvisco-elastic region (as determined from the stress sweep) and the strain responseis observed as a function of the frequency of oscillation f . The range of frequenciesinvestigated is limited at the low end to f > 0.01 Hz by the need to keep the totaltime taken for the experiment to under 30 min and at the high end to f < 10 Hz bythe sensitivity of the instrument to the very small strains produced at high frequen-cies.

Bubble rise is minimized by restricting the bubble size and the duration of thesweeps but cannot be completely eliminated. The effect is also progressive duringthe course of a frequency sweep. To even out the residual effect of bubble rise, eachfrequency sweep is performed from low f to high f and then back down to low f .The data presented here are the averaged data from the up and down sweeps.

The complex viscosity |η∗| of pure syrup (φ = 0) is expected to be independent off . Thus the scatter in the frequency sweep data for pure syrup gives a measure ofthe random error on |η∗|. The 2σ error is 2%.

5. The rheological data

(a) Monodisperse model

The linearized Frankel & Acrivos model (equation (2.19)) describes the behaviourof a bubbly liquid in terms of three material properties: the gas volume fraction φ,the viscosity of the fully dense liquid η0 and the relaxation time λ. φ is measureddirectly for each sample and is known to within 5%. η0 for golden syrup variesover a small range from batch to batch. It is also likely to change slightly duringthe aeration process. It is not, therefore, possible to know η0 for each sample toa high degree of accuracy and so we assume the true value lies somewhere in therange 40 η0 70 Pa s for all samples. The relaxation time λ involves η0, theliquid–bubble interfacial tension Γ , the bubble radius r and the parameter k (seeequation (2.4)). Strictly, the analysis of Frankel & Acrivos assumes k = 1 (case fordilute limit), however, since we are investigating non-dilute suspensions, we allowk to vary. For the surface tension of golden syrup we use Γ = 0.08 N m−1. This isthe value measured by P. Heath (Carl Stuart Ltd 2001, personal communication)



40

60

80

100

120

140

1 1010−2 10−1

frequency (Hz)

| ∗ |

(Pa

s)

η

Figure 6. Monodisperse model curve fits. Best fits to data for φ = 0.140 using a monodispersemodel as described in § 5 a. Solid line assumes β1 = η0(1 + φ), dashed line assumes β1 = bη0,where b is a free parameter that can take any positive value. For the dashed line, η0 = 33.8 Pa sand b = 2.71.

for golden syrup at 21 C. The value is in broad agreement with the relationshipgiven in Sinat-Radchenko (1982) and with values measured for the surface tensionof other concentrated sugar solutions, such as corn syrup (0.07 N m−1 (Manga &Stone 1995); 0.08 N m−1 (Borhan & Pallinti 1999)). There is no evidence to suggestthat the surface tension of such solutions is a strong function of water content ortemperature and we expect the true value to be within 10% of Γ = 0.08 N m−1.Defining a single bubble radius is problematic as all of the samples are significantlypolydisperse, with bubble radii varying over four orders of magnitude. We thereforeallow the bubble radius to take any value within the range of radii present in thesample. The statistical analysis and model fitting package, Simfit (see Bardsley(1993) for a description), is used to determine the values of η0, r and k that providethe best fit of the model (equation (2.19)) to each sample of known φ. η0 and r areconstrained within the ranges specified above and k can take any value. An exampleof a curve fit produced using this method is shown in figure 6. Although the shapeof the model curve reflects the trend in the data, the fit is extremely poor.

It was pointed out earlier that β1 as given in equation (4.11) is the same as Taylor’sexpression, equation (2.6). Both Taylor’s expression and the Frankel & Acrivos theorywere derived for emulsions and so we might reasonably expect that this expressionmay need adjustment if it is to be applied to bubble suspensions. To this end, westart by generalizing the functional form of equation (4.11) to

β1 = bη0, (5.1)

where b is now a free parameter. We will use our data to determine the functionalform that describes how b depends on gas volume fraction φ. Figure 6 also shows acurve fit using the same procedure described above, but assuming the functional formgiven in equation (5.1). The fit is much improved but still shows marked deviationfrom the data. This is primarily a consequence of the fact that our highly polydisperse



40

60

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120

140

40

60

80

100

120

140

1 1010−2 10−1

frequency (Hz)

| ∗ |

(Pa

s)

η

1 1010−2 10−1

frequency (Hz)

| ∗ |

(Pa

s)

η

(a) (b)

(c) (d )

Figure 7. Polydisperse model curve fits. Best fits to data for φ = 0.140 using a polydispersemodel (as described in § 5 b) based on the four bubble size distributions shown in figure 3. For(a) bubble number distribution fit η0 = 48.9 Pa s and b = 2.63, for (b) bubble radius distributionfit η0 = 44.8 Pa s and b = 2.91, for (c) bubble area distribution fit η0 = 51.2 Pa s and b = 2.45,for (d) bubble volume distribution fit η0 = 54.2 Pa s and b = 2.28.

samples cannot be accurately modelled by a single bubble size and a single relaxationtime.

(b) Polydisperse model

The monodisperse model described in the previous section can be accuratelyapplied to each Φ size class (where Φ = − log2(diameter in mm) as described inthe caption to figure 3) to give a viscosity for that fraction of the sample. The gasvolume fraction for each Φ class is taken to be the same as that of the bulk (whicheffectively means dividing up the liquid among the Φ classes according to the bub-ble volume distribution). The viscosity of the bulk is then taken to be the weightedsum of the individual monodisperse viscosities for the different Φ classes, where theweightings are provided by the fractional distribution of the Φ classes. This pro-cess leads to four potential polydisperse models, depending on whether the bubblenumber, radius, area or volume distribution is used to provide the weightings. Thecurve-fitting process involves finding the optimal values of b, η0 and k that are thesame for all Φ classes within a given distribution and lie within the ranges specifiedin § 5 a.



Table 1. Values of WSSQ for fits of the monodisperse model and the polydisperse model basedon the four bubble size distributions (number, radius, area and volume) for all data

(Low WSSQ indicates a better fit.)

WSSQ︷︸︸︷

φ number radius area volume monodisperse

0.000 — — — — 4.42 × 100

0.036 1.87 × 100 1.86 × 100 1.89 × 100 1.91 × 100 2.07 × 100

0.057 2.66 × 100 2.68 × 100 2.58 × 100 2.68 × 100 9.53 × 100

0.063 5.48 × 100 4.85 × 100 5.12 × 100 6.95 × 100 1.46 × 101

0.085 2.80 × 101 1.84 × 101 2.72 × 101 5.12 × 101 9.45 × 101

0.103 8.04 × 101 5.34 × 101 6.24 × 101 1.26 × 102 2.10 × 102

0.140 4.06 × 102 1.40 × 102 7.59 × 102 1.02 × 103 1.42 × 103

0.235 4.78 × 102 2.54 × 102 1.21 × 103 1.49 × 103 2.16 × 103

0.302 1.35 × 103 4.84 × 102 2.55 × 103 2.98 × 103 4.27 × 103

0.461 3.96 × 102 3.96 × 103 5.82 × 103 6.02 × 103 7.93 × 103

Figure 7 shows the best-fit curves produced by Simfit using the polydispersemodel for each of the four distributions (based on bubble number, radius, area andvolume) for a specific sample. The fits are all significantly better than that obtainedby applying the monodisperse model as in figure 6. Values of the weighted sum ofsquares (WSSQ) of the curve fits for all datasets are given in table 1. A lower WSSQvalue indicates a better fit. For all but two of the samples tested, the best fit wasobtained for the model based on the bubble radius distribution, suggesting that thisis the property that controls the response of a bubble to strain. This is to be expected,since the relaxation time λ (equation (2.4)) contains a first-order radius term.

Figure 8 shows the complete dataset with fits obtained for each sample using thepolydisperse model based on the bubble radius distribution. The values of η0, kand b found for each sample are given in table 2. Note that the general trend ink increases with increasing φ, as expected (Oldroyd 1953; Oosterbroek & Mellema1981; Loewenberg & Hinch 1996). However, k is significantly greater than the valuesobtained in the numerical calculations of Loewenberg & Hinch (1996) for emulsionsin which the disperse and continuous phases have the same viscosity.

6. Semiempirical model for the rheology of a bubbly liquid

We can use the data given in table 2 to determine how the free parameter b of equation(5.1) depends on the gas volume fraction φ. Figure 9 shows that the relationship isof the form b = 1 + aφ, where 7.5 a 9.5 with a best-fit curve of a = 9.0. Thus, bis a modified form of the Taylor expression. The scatter in the data is thought to beprimarily a result of residual bubble rise. The parallel plate geometry is particularlysensitive to bubble rise effects because the rising bubbles accumulate beneath theupper plate on which the torque is measured.

The model curve fits to the data in figure 8 are sensitive to the exact value ofthe parameter a. Figure 10 illustrates the effect of varying the parameter across theuncertainty band shown in figure 9, i.e. from a = 7.5 to a = 9.5, for a particularsample.



49

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51

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53

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0

100

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10−2 10−1 1 10

(a) (b)

(c) (d )

(e) ( f )

(g) (h)

(i) ( j )

frequency (Hz)10−2 10−1 1 10

frequency (Hz)

| ∗ |

(Pa

s)

η|

∗ | (

Pa s

)η

| ∗ |

(Pa

s)

η|

∗ | (

Pa s

)η

| ∗ |

(Pa

s)

η

Figure 8. Complex viscosity data. Solid lines show the best fits using the polydisperse model (asdescribed in § 5 b) based on the bubble radius distribution. Note the large variation in verticalscale between datasets. Values of φ are (a) 0.000, (b) 0.036, (c) 0.057, (d) 0.063, (e) 0.085,(f) 0.103, (g) 0.140, (h) 0.235, (i) 0.302, (j) 0.461.



Table 2. Parameter values of best fit of polydisperse model based on bubble radius distribution

φ η0 (Pa s) k b

0.000 50.7 — 1.000.036 52.3 10.6 1.010.057 53.2 22.0 1.130.063 52.5 18.7 1.140.085 58.5 22.2 1.500.103 53.8 29.2 1.550.140 44.8 39.2 2.900.235 46.0 34.6 3.350.302 57.8 37.7 3.610.461 70.0 17.3 5.08

0

1

2

3

4

5

6

0.1 0.2 0.3 0.4 0.5gas volume fraction,

b

φ

Figure 9. Values of b from polydisperse fit based on bubble radius distribution against φ. Bestfit (solid line) has equation b = 1 + 9.0φ. Upper and lower dashed lines (representing limits ofacceptable fit) have equations b = 1 + 9.5φ and b = 1 + 7.5φ, respectively.

For a steady flow, consideration of equations (2.18), (4.11) and (5.1) and thediscussion at the start of § 2 allows us to conclude that b = ηr. We can thereforeeasily compare the functional form of b as determined from our data with the formsproposed by other researchers as reviewed in § 3 a, and this is done in figure 11 andthe top half of table 3.

Our data are best described by the relationships proposed by Sibree (1934) andStein & Spera (1992), who are the only other researchers, to the best of our knowl-edge, to propose constitutive equations specifically for bubble suspensions that areunderpinned by observation. However, the Sibree data consist of five measurementsof bubble suspensions in a strongly shear-thinning liquid with high gas volume frac-tions (0.5 < φ < 0.73). Sibree’s data points lie outside the graph shown in figure 11,in the region where our constitutive law would diverge strongly from his. In addi-



50

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150

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1 1010−2 10−1

frequency (Hz)

| ∗ |

(Pa

s)

η

Figure 10. Data and polydisperse model based on bubble radius distribution for φ = 0.302. Solidline shows model prediction assuming β1 = η0(1 + 9.0φ). Upper and lower dashed lines showlimits of uncertainty taken from figure 9 (assuming β1 = η0(1 + 9.5φ) and β1 = η0(1 + 7.5φ),respectively).

tion, Sibree’s equation is derived from a model specifically for highly concentratedsuspensions (Hatschek 1911) and so it is likely that the agreement at low φ (welloutside his data range) is entirely coincidental. The other data of interest are thethree values measured in the low gas volume fraction range 0.008 φ 0.055 byStein & Spera (1992). These points are shown superimposed on our data in figure 11and are well-described by the equation ηr = 1+aφ using our best-fit value of a = 9.0.

We can now propose the following constitutive rheological equation for a bubblyliquid:

τij + 65λτij = η0(1 + 9φ)γij + η0

65λ(1 − 5

3φ)γij . (6.1)

This is a semiempirical model. Its theoretical basis is the linear Jeffreys model,parametrized according to the physical analysis of Frankel & Acrivos, with the depen-dence of the parameter β1 on the gas volume fraction φ determined empirically.

(a) Time-dependent rheology

Time-dependent effects emerge in our experimental data due to the increasinglyunsteady nature of the flow as frequency of oscillation increases. These effects man-ifest themselves as a frequency dependence of absolute and relative magnitudes ofthe viscous and elastic components of deformation.

The relative importance of the viscous and elastic components is encapsulated inthe phase shift δ (equation (4.7)) and is dependent on both φ and f . The randomerror on tan δ is conservatively estimated to be 10% from the scatter in η′ and η′′.Figure 12 shows δ data for a particular sample and compares the monodisperse andpolydisperse model curve fits. The polydisperse model based on the bubble radiusdistribution provides the best fit to the data. The pronounced minimum in δ indicatesthat there is a maximum in the elastic response of the suspensions at about 1 Hz.The position of this minimum is related to the bubble size, hence the broadening and



0

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6

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6

0

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5

6

0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5gas volume fraction, φ gas volume fraction, φ

rela

tive

visc

osity

, rη

rela

tive

visc

osity

, rη

rela

tive

visc

osity

, rη (a) (b)

(c) (d )

(e) ( f )

Figure 11. Equations for ηr(φ) for steady state flow presented by previous workers plotted againstour experimental data. Solid lines represent ‘limited’ fits, where parameters are restricted tolimits or values imposed by the originator of the model, dashed lines represent ‘free’ fits whereparameters are allowed to assume any positive value. Where only one line is shown, limitedand free fits are identical. The data of Stein & Spera (1992) are plotted as hollow squares witherror bars. (a) Barthes-Biesel & Chhim (1981), (b) Eilers (1943), (c) Mooney (1951), (d) Roscoe(1952), (e) Sibree (1934), (f) Stein & Spera (1992).

shallowing of the trough when a polydisperse model is used. The elastic maximumoccurs at the frequency where ω (= 2πf) is approximately equal to the inverse ofthe relaxation time (λ−1).

The φ dependence of δ is straightforward. The composite plot of all δ data infigure 13 shows that as φ increases, the suspension becomes more elastic. Whenφ = 0 the suspension is purely viscous (δ ≈ 90) and when φ = 0.461 its characteris more elastic than viscous (δ < 45) at its elastic maximum.

Figure 14 shows the dependence of viscosity |η∗| on φ and f . As φ increases, |η∗|becomes increasingly dependent on f : when φ = 0, |η∗| is essentially constant at|η∗| = η0 for all f , but when φ = 0.461, |η∗| decreases by a factor of 17 as f increasesfrom 0.01 to 10 Hz.

The most striking feature of the data is that at low f , |η∗| increases with increasing



Table 3. WSSQ for fits of various models to our data for ηr against φ

(The column headings ‘free’ and ‘limited’ refer to whether the parameters in the model wereallowed to vary freely or within the range set by the models originators. The top half of the tablerefers to models where ηr increases with increasing φ, and the bottom half to models where ηr

decreases with increasing φ.)

WSSQ︷︸︸︷

model free limited

Barthes-Biesel & Chhim (1981) 1.77 × 101 1.96 × 101

Eilers (1943) 5.22 × 100 5.45 × 100

Mooney (1951) 5.08 × 100 3.01 × 101

Roscoe (1952) 4.79 × 100 4.79 × 100

Sibree (1934) 2.05 × 100 2.05 × 100

Stein & Spera (1992) 1.12 × 100 8.26 × 100

Bagdassarov & Dingwell (1992) 3.85 × 10−2 1.52 × 100

Ducamp & Raj (1989) 3.49 × 10−2 6.81 × 10−2

Mackenzie (1950) 4.15 × 10−2 4.22 × 10−2

Rahaman et al . (1987) 2.23 × 10−2 1.48 × 100

Scherer (1979) — 2.48 × 10−2

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60

70

80

90

1 1010−2 10−1

frequency (Hz)

phas

e sh

ift,

(

deg)

δ

Figure 12. δ as a function of frequency of oscillation for φ = 0.235. Solid line is monodispersemodel fit. Dashed line is polydisperse model fit based on bubble radius distribution.

φ, whereas at high f , |η∗| decreases with increasing φ. Thus total resistance to defor-mation can either increase or decrease with increasing φ, depending on the frequencyof oscillation. This reversal in the effect of φ on viscosity immediately brings to mindthe apparently contradictory results from different groups of models discussed in§ 2 and from different experimental studies (see, for example, Stein & Spera 1992;



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100

0.000 0.0360.0570.0630.085

0.103

0.140

0.235

0.302

0.461

1 1010−2 10−1

frequency (Hz)

phas

e sh

ift,

(

deg)

δ

Figure 13. Composite plot of δ against frequency of oscillation for all gas volume fractions. Thelines are not curve fits but simple interpolations between neighbouring datapoints. The randomerror on each datapoint is estimated to be ±10%. δ decreases steadily as φ increases.

0

100

200

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400

0.0000.0360.0570.0630.085

0.103

0.140

0.235

0.302

0.461

0.000

0.461

1 1010−2 10−1

frequency (Hz)

| ∗ |

(Pa

s)

η

Figure 14. Composite plot of |η∗| against frequency of oscillation for all gas volume fractions.Polydisperse model fits to data are plotted. At low frequency, |η∗| increases steadily as φincreases, at high frequency, the opposite is true.

Bagdassarov & Dingwell 1992) and deserves careful consideration. All experimentsare carried out at small Ca and bubbles remain essentially spherical throughout.Therefore, the response cannot be explained on the basis of the steady-state shear-thinning rheology calculated by Manga & Loewenberg (2001), which depends uponbubbles undergoing considerable elongation. In the following sections, we show thatthe behaviour is a viscoelastic response that is independent of Ca. It is a result of



unsteady bubble deformation with frequency and can be described by a dimensionlessparameter Cd, which we call the ‘dynamic capillary number’.

(b) Dynamic capillary number

In this investigation, the bubble suspensions are subjected to a varying strain rateand so the deforming (viscous) and restoring (surface tension) forces acting on abubble may not be in equilibrium. In such an unsteady flow, the capillary numbergiven in equation (2.2) has little meaning and it is more instructive to consider theflow in terms of the rate of change of bubble deformation. When the strain rate isconstant (and has been for a time t λ) bubbles are in equilibrium deformationand are not actively changing shape. When the strain rate is changing, the bubbleschange shape and, in general, the rate at which a bubble deforms depends on therate of change of the imposed viscous forces, i.e. on γ, and on the relaxation time λ.The bubble deformation rate is given by λγ. A suitable dimensionless number withwhich to describe the flow is the ratio of this bubble deformation rate to the bulkstrain rate γ. We call this the dynamic capillary number Cd,

Cd = λγ

γ. (6.2)

Cd is a measure of how strain is accommodated in a bubble suspension subject toa time-dependent shear. It is also a measure of the steadiness of the flow. Cd 1indicates that the rate of bubble deformation is small compared with the bulk strainrate and, therefore, that most of the strain must be taken up by viscous flow of theliquid phase around the bubbles. As φ increases, flow lines are increasingly distorted,leading to an increase in viscosity with φ. By contrast, Cd 1 indicates that the rateof bubble deformation is large compared with the bulk strain rate and, therefore, thatmost of the strain can be accommodated by deformation of the gaseous phase. Sincethe gaseous phase has a negligible viscosity, this leads to a decrease in viscosity as φincreases. The flow approaches steady flow conditions as Cd → 0 and is increasinglyunsteady with increasing Cd. It is apparent that calculating Cd for a bubbly flowoffers a convenient way of establishing how the system will respond to changes in φ.

For samples subject to forced sinusoidal oscillations (see equations (4.1)–(4.3))γ/γ is a tangent function, so Cd varies from ±∞. An average value for Cd can befound by using the ratio of the amplitudes γ0/γ0, which is equivalent to the angularfrequency ω hence equation (6.2) becomes

Cd = λω. (6.3)

The elastic maximum (see § 6 a) occurs when ω ≈ 1/λ, i.e. when Cd ≈ 1. This is alsothe value which separates the regimes of increasing and decreasing viscosity with φ(see figure 14).

It is worth considering the significance of Ca and Cd. Ca defines the steady com-ponent of deformation and Cd the perturbation of this component. It is then clearthat, whereas Cd can be defined for all flows (i.e. for all Ca), Ca is only physicallymeaningful at low values of Cd, i.e. where a steady component can be clearly identi-fied. Thus, at high Ca (and low Cd), Cd is a measure of the rate of change of shapeof the elongated bubbles around their equilibrium deformation.



0

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0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5

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0 0.1 0.2 0.3 0.4 0.5

gas volume fraction, φ gas volume fraction, φ

rela

tive

visc

osity

, rη

rela

tive

visc

osity

, rη

rela

tive

visc

osity

, rη (a) (b)

(c) (d )

(e)

gas volume fraction, φ

Figure 15. Equations for ηr(φ) for unsteady flow presented by previous workers plotted againstour data for the high Cd limit of our experiments (i.e. the data points give the value ofηr = |η∗|/η0 at maximum frequency (10 Hz → Cd > 5) for each of our samples). Solid linesrepresent ‘limited’ fits, where parameters are restricted to limits or values imposed by the origi-nator of the model, dashed lines represent ‘free’ fits where parameters are allowed to assume anypositive value. Where only one line is shown, the model has no adjustable parameters. (a) Bag-dassarov & Dingwell (1992), (b) Ducamp & Raj (1989), (c) Mackenzie (1950), (d) Rahaman etal . (1987), (e) Scherer (1979).

(c) Behaviour at high Cd

We are now in a position to reconsider the models and data for porous glassesthat show a decrease in viscosity with increasing gas volume fraction (Rahaman etal . 1987; Ducamp & Raj 1989; Mackenzie 1950; Scherer 1979; Sura & Panda 1990;Bagdassarov & Dingwell 1992, 1993; Lejeune et al . 1999).

We can calculate the approximate minimum relaxation time for a bubble in aporous glass using equation (2.4). When r = 1 µm (minimum bubble size presentin samples for which rheological data have been presented), η0 = 109 Pa s, Γ =



0.3 N m−1 (typical value for a volcanic glass from Murase & McBirney (1973)) andk = 1, then λ ≈ 3000 s. A reconsideration of reported data on porous glasses showsthat deformation time-scales in the experiments are typically much shorter thanone hour and therefore flow is highly unsteady. As this represents the minimumrelaxation time, we would expect all flows with deformation time-scales shorter thanabout 1 h to be highly unsteady. For example, the dynamic capillary number neverfalls below 100 in the experiments of Bagdassarov & Dingwell (1993) and in theexperiments of Lejeune et al . (1999) the longest experiments ran for only ‘tens ofminutes’, indicating that steady conditions could not have been reached. Similarly,in the calculations of Manga & Loewenberg (2001), steady-state conditions were notattained for Ca > 1 due to numerical limitations, as the authors freely admit. We cancompare our measurements with the predictions of the models presented in § 2 b byconsidering the high Cd (10 Hz → Cd > 5) end-member of our curves (see figure 15).Since the error in measured values of η is 2%, the error in the relative viscosity is4%.

All of the models provide a good fit to the data; WSSQ values for each fit areshown in the bottom half of table 3. Thus, our data demonstrate that the apparentcontradiction in the published rheological data on suspensions and porous glassescan be resolved by considering the steadiness of the flow as described by the Cdregime.

We described in § 6 how, in the steady limit as Cd → 0, equation (6.1) becomes

τij = η0(1 + 9φ)γij . (6.4)

Effectively, the τ and γ terms can be ignored. This is equivalent to the Stein & Speramodified Taylor equation (2.7) and is valid for small Ca. Similarly, in the unsteadylimit, as Cd → ∞, the τ and γ terms can be ignored and equation (6.1) becomes

τij = η0(1 − 53φ)γij . (6.5)

Since τ /γ has the units of viscosity, this can be rewritten as

ηr = 1 − 53φ, (6.6)

which is the same as Mackenzie’s model, and agrees well with our high Cd data.Thus, when Cd 1, the rheology of a bubbly liquid undergoing small deformationsis described by equation (6.4) and when Cd 1, it is described by equation (6.5).Equation (6.1) is only required in its entirety when the dynamic capillary number isclose to unity.

7. Summary of main conclusions

We have developed a constitutive law for the time-dependent rheological behaviourof bubbly flows valid for φ 0.5 and small bubble deformations (i.e. Ca 1), whichallows the viscous and elastic moduli of a bubbly flow to be calculated provided φ,η0, λ and the bubble size distribution are known. This model is based on a Jeffreysmodel and the theoretical analysis of Frankel & Acrivos (1970). It is underpinned bydata collected from oscillatory rheometric measurements of bubbly golden syrup andcontains a semiempirically determined parameter. The model is consistent with allpreviously published data from experiments on bubble-bearing Newtonian materials



(Stein & Spera 1992; Bagdassarov & Dingwell 1992, 1993; Sura & Panda 1990).The model and experiments show that the viscosity of a bubbly fluid can vary overmore than an order of magnitude depending on φ and the exact conditions of shearit experiences. We have shown that the presence of a range of bubble sizes in thesuspension can be modelled by weighting the contribution of several monodispersefractions according to the distribution of bubble radii. We have also shown that therelaxation time for a non-dilute bubble suspension is significantly higher than for anequivalent emulsion with a viscosity ratio of 1.

We distinguish between ‘steady’ flows in which bubbles have reached equilibriumdeformation and ‘unsteady’ flows in which active bubble deformation accounts for aproportion of the total strain accommodation. A dimensionless number, the dynamiccapillary number Cd = λγ/γ, gives a measure of the unsteadiness of a flow. Cd 1corresponds to steady flow in which viscosity increases with increasing φ and Cd 1corresponds to an unsteady flow in which viscosity decreases with increasing φ. Amaximum in the elastic behaviour of the material occurs at Cd ≈ 1. The full model,equation (6.1), is only required when Cd = O(1): in the steady (low Cd) regime,and when Ca is small, equation (6.4) is sufficient; in the unsteady (high Cd) regimeequation (6.5) is sufficient.

Carl Stuart Ltd has provided invaluable support; we thank Steve Robinson, Kevin Barber andTim Green for assistance in resolving instrumental problems at the start of the project andPaul Heath for the measurement of the surface tension of golden syrup. We are grateful to BillBardsley for his patient and generous help with Simfit and with the preparation of graphics.Michael Manga provided a very constructive review, which has lead to significant improvements.We are indebted to Harry Pinkerton for contributions to the grant proposal. This work wasfunded by the Natural Environment Research Council UK via a research grant GR3/10500 anda studentship for E.W.L. GT16/97/ES/14.

Appendix A.

The constitutive equation for a dilute emulsion, or suspension of deformable fluiddroplets, has been derived by Frankel & Acrivos (1970, eqns 3.4 and 3.5). The devi-atoric stress tensor τij is, given in the notation of the present paper,

τij = 2η0eij + φη0(103 eij + 8Fij + 40

7 λ[Fipepj ] − 1927 λ[FipFpj ]) (A 1)

and the auxiliary tensor Fij is given by

Fij + 65λ

D

DtFij = 2

3eij − 127 λ[Fipepj ] + 186

35 λ[FipFpj ], (A 2)

where eij is the rate of strain tensor and λ is given by equation (2.4) with k = 1(case for dilute limit).

Their equation has been specialized to the present case of gas bubbles (regardedas inviscid) by setting their viscosity ratio parameter equal to zero. The symbol denotes the symmetric, traceless part:

Aij = 12(Aij + Aji) − 1

3Allδij . (A 3)

The operator D/Dt is the Jaumann derivative:

D

DtAij =

∂Aij

∂t+ uk

∂Aij

∂xk+ ikAkj − kjAik, (A 4)



where uk is the velocity vector and ik is the vorticity tensor. (Frankel & Acrivosgive a slightly different definition which appears to contain a misprint: Fki should beFik.)

This model is an exact asymptotic result derived from a microscopic theory validin various limits as explained in the paper. These are (1) the dilute limit, φ 1;(2) large surface tension, resulting in small bubble deformations. The bulk strainingmotion, however, is unrestricted.

The Jaumann derivative simplifies when the theory is applied to the present exper-imental arrangement. We have (approximately) oscillatory rectilinear shear flow, sothat the convective term uk∂/∂xk vanishes. Suppose that the displacement of themoving boundary is given by x = l sin ωt. The plate separation is h, and we havel/h 1. The components of ik and eik are of order ωl/h and it is easy to seethat the nonlinear terms are of order l/h relative to the time derivative terms ∂/∂t.Thus the operator D/Dt reduces to simply ∂/∂t. The approximate (dimensional)equations corresponding to (A 1) and (A 2) are then

τij = 2η0eij + φη0(−103 eij + 8Fij), (A 5)

Fij + 65λ

∂Fij

∂t= 2

3eij . (A 6)

Now operating on (A 5) with 1 + 65λ∂/∂t allows Fij to be eliminated reducing the

model to the form of a Jeffreys model. This gives (2.19) of the main text, whereγ = 2e.

When the Jeffreys model is used to describe a system undergoing forced oscillations(equations (4.1), (4.2) and (4.3)), as in the present study, equation (2.18) can bewritten:

τ0(sin ωt cos δ + cos ωt sin δ) + τ0α1ω(cos ωt cos δ − sin ωt sin δ)= γ0β1 cos ωt − γ0β2ω sin ωt. (A 7)

Comparing terms allows us to write the Jeffreys model in terms of its real andimaginary components η′ and η′′ (equations (4.5) and (4.6)), giving us equations(4.8) and (4.9) in the main text.

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The rheology of a bubbly liquid - · The rheology and ﬂow behaviour of suspensions of particles in liquids is of interest because of the enormous range and complexity of the phenomena