Capillarity and wetting of non-Newtonian droplets915139/... · 2016. 3. 29. · non-Newtonian uids,(b) the Weissenberg e ect and (c) the Barus e ect. and inkjet printing. For emerging

Capillarity and wetting of non-Newtoniandroplets

by

Yuli Wang

April 2016Technical Report

Royal Institute of TechnologyDepartment of Mechanics

SE-100 44 Stockholm, Sweden

Akademisk avhandling som med tillstand av Kungliga Tekniska Hogskolan iStockholm framlagges till offentlig granskning for avlaggande av teknologiedoktorsexamen fredagen den 22 April 2016 kl 10.15 i Kollegiesalen, Brinellvagen8, Kungliga Tekniska Hogskolan, Stockholm.

c©Yuli Wang 2016

Tryckt av E-Print AB, Stockholm 2016

Capillarity and wetting of non-Newtonian droplets

Yuli WangLinne FLOW Centre, KTH Mechanics, The Royal Institute of TechnologySE-100 44 Stockholm, Sweden

Abstract

Dynamic wetting of non-Newtonian fluids plays important roles in manynatural and industrial processes, examples cover from a daily phenomenon assplashing of a cup of yogurt to advanced technologies such as additive manu-facturing. The applicable non-Newtonian fluids are usually viscoelastic com-pounds of polymers and solvents. Previous experiments observed diverse in-teresting behaviors of a polymeric droplet on a wetted substrate. However,our understanding of how viscoelasticity affects wetting and spreading remainsvery limited. A polymeric droplet meeting a bifurcation corner in a capillarytube is a common scenario in microfluidic devices for cell sorting, emulsion anddroplet formation under various biomedical circumstances. The behaviors of aviscoelastic droplet at a junction tip has not been studied. This work intends toshed light on viscoelastic effect on these small scale processes, i.e., the motionof a wetting contact line and droplet splitting at a bifurcation tip.

Numerical simulation is employed to reveal detailed information such aselastic stresses and interfacial flow field. A numerical model is built, combiningthe phase field method, computational rheology techniques and computationalfluid dynamics. The system is capable for calculation of realistic circumstancessuch as a droplet made of aqueous solution of polymers with moderate relax-ation time, impacting a partially wetting surface in ambient air.

The work is divided into three flow cases. For the flow case of bifurcationtube, the evolution of the interface and droplet dynamics are compared be-tween viscoelastic fluids and Newtonian fluids. The splitting or non-splittingbehavior influenced by elastic stress is analyzed for different droplet sizes andcapillary numbers. For the flow case of dynamic wetting, the flow field andrheological details such as effective viscosity and normal stress difference neara moving contact line are presented. The effects of shear-thinning and elasticityon droplet spreading and receding are analyzed, under inertial and inertialesscircumstances. The discussion relates the simulation results to experimentalobservations. In the last part, droplet impact of both Newtonian and viscoelas-tic fluids are demonstrated. For Newtonian droplets, a phase diagram coveringa wide range of Ohnesorge number and Reynolds number is drawn to visual-ize different impact regions for spreading, splashing and gas entrapment. Forviscoelastic droplets, the viscoelastic effects on droplet deformation, spreadingradius and contact line motion are revealed and discussed.

The major findings of this work can be summarized as follows. The criticalcapillary number for a droplet to split at a y-shaped junction is increased

iii

by the degree of elasticity. Elasticity also enhances the instability of a non-splitting droplet hanging on the junction. Shear-thinning generally acceleratesthe contact line while droplet spreading. Elastic effect is usually covered byshear-thinning when a droplet spreads fast or contracts on the wetted substrate.Its presence depends on local flow geometry in the contact region. The elasticityalters the Tanner-Cox-Voinov law when a droplet is spreading slowly. Howeverthis deviation is not observed for inertialess spreading, in which the spreadingis enhanced by both elasticity and shear-thinning.

Descriptors: dynamic wetting,contact line,diffusive interface, viscoelasticity,non-Newtonian,microfluidics, droplet impact, droplet spreading

iv

Preface

This thesis contains numerical studies of viscoelastic droplet dynamics in-cluding droplet based microfluidics, droplet spreading, wetting and impact ona solid surface. The thesis is divided in two parts, where the first one presentsbackground that motivates the work, basic concepts of the physics involvedin the simulation and a summary of the results. The second part consists offour journal articles. The layout of these papers has been adjusted to fit theformat of this thesis, but their content has not been changed with respect tothe original versions. The work was performed at KTH Mechanics during theperiod between September 2012 and March 2016.

Paper 1.Y. Wang, M. Do-Quang & G. Amberg, 2016Viscoelastic droplet dynamics in a y-shaped capillary channel. Phys. Fluids28, 033103

Paper 2.Y. Wang, M. Do-Quang & G. Amberg, 2015Dynamic wetting of viscoelastic droplets. Phys. Rev. E 92, 043002

Paper 3.Y. Wang, A. Gratadeix, M. Do-Quang & G. Amberg, 2016Events and conditions in droplet impact: a phase field prediction. under reviewfor publication on Int. J. Multiphase Flow

Paper 4.Y. Wang, M. Do-Quang & G. Amberg, 2016Impact of viscoelastic droplets. submitted to Soft Matter

v

Blow is a list of outcome from the work performed by the author con-ducting the research project funded by National Nature Science Foundation ofChina(Grant No.51176065) during the same period of time, but not includedin this thesis.

International archival journalsC. Gong, M.G. Yang, Y. Wang & C. Kang, 2015Wavelength measurement of a liquid jet based on spectral analysis of image in-tensity. Advances in Mechanical Engineering. 7(8),1-10

C. Gong, M.G. Yang, C.Kang & Y. Wang, 2015The acquisition and measurement of surface waves of high-speed liquid jets.Journal of Visualization.,1-14

C. Gong, M.G. Yang, C.Kang & Y. Wang, 2015Experimental study of jet surface structures and the influence of nozzle config-uration. submitted to Fluid Dynamic Research

vi

Division of work between authorsThe main advisor of the project is Prof. Gustav Amberg (GA) and the co-advisor is Dr. Minh Do-Quang (MDQ).

Paper 1The solver was developed and implemented by Yuli Wang (YW), with base codefrom Minh Do-Quang (MDQ). Simulations and analysis were carried out byYW. The paper was written by YW with feedback from Gustav Amberg(GA)and MDQ

Paper 2YW setup the simulation with the code implemented by YW and MDQ. YWperformed the computations and analyzed the data. The paper was written byYW with feedback from GA and MDQ.

Paper 3Modifications on code and case setup were done by YW . Simulations werecarried out by Anthony Gratadeix(AG). AG and YW analyzed the data. Thepaper was written by YW, with input from AG and feedback from GA andMDQ.

Paper 4Modifications on the code were carried out by YW. The computations andanalysis were performed by YW. The paper was written by YW with feedbackfrom GA and MDQ.

vii

Part of the work has been presented at the following international conferences:

Y. Wang, M. Do-Quang & G. AmbergSpreading of viscoelastic droplets. 68th Annual Meeting of the APS Division ofFluid Dynamics – Boston, USA, 2015

viii

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— Reid Hoffman

Building something nobody wants is the ultimate form ofwaste.

— Eric Rices

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— Elon Musk

x

Contents

Abstract iii

Preface v

Part I - Overview & Summary 1

Chapter 1. Introduction 3

Chapter 2. Capillarity and wetting 7

2.1. Capillarity 7

2.2. Wetting 8

2.3. Wetting of non-Newtonian fluids 11

Chapter 3. Viscoelasticity 13

3.1. Modeling viscoelasticity 13

3.2. Constitutive models 14

Chapter 4. The diffuse interface method 18

4.1. Phase field theory 18

4.2. Surface tension 19

4.3. Wettability, contact angle 20

4.4. Diffuse interface 21

Chapter 5. Numerical model 22

5.1. Non-dimensionalization 22

5.2. Cahn-Hilliard equations 23

5.3. Navier Stokes equations 24

5.4. Constitutive equations 25

5.5. Solving routine 27

Chapter 6. Summary of results 29

6.1. Validations 29

6.2. Viscoelastic droplet dynamics in microfluidic device 30

xi

6.3. Dynamic wetting of viscoelastic droplets 32

Chapter 7. Conclusions 41

Bibliography 43

Acknowledgements 51

Part II - Papers 53

Paper 1. Viscoelastic droplet dynamics in a y-shaped capillarychannel. 55

Paper 2. Dynamic wetting of viscoelastic droplets 81

Paper 3. Events and conditions in droplet impact: a phase fieldprediction 99

Paper 4. Impact of viscoelastic droplets 121

xii

Part I

Overview & Summary

CHAPTER 1

Introduction

Non-Newtonian fluids are common in our daily life. Toothpaste, shampoo,hand soap that we use to clean ourselves every morning, butter, cheese, fruitjam that are common residence of kitchen, cosmetics for make up, to namejust a few among many examples. Some of non-Newtonian fluids are importantcomponents of the human body such as blood, saliva and semen.

One feature that classifies non-Newtonian fluids is the dependence of vis-cosity on the shear rate, see Fig.1.1(a). The kitchen can serve as a good demofor classifying non-Newtonian fluids. A Newtonian fluid, the tap water, has alinear dependence of shear stress on shear rate. A shear-thickening fluid has ashear-hardening behavior, the example is picked as the solution made of corn-starch dissolved in water. The viscosity of ketchup decreases with shear rate soit belongs to the shear-thinning class. Some fluids need a yield stress to startflowing, like chocolate and mustard, they are named as Bingham plastic. Thereare also fluids that have time-dependent viscosity. For instances, the suspen-sion of gypsum with an apparent viscosity increasing with the time of stressapplied, and xanthan gum of which the viscosity decreases with the durationof stress are referred to as the thixotropic fluids.

One class of non-Newtonian fluids are labeled as viscoelasic fluids which, asstated by their title, present both viscous and elastic characteristics. Examplesof viscoelastic materials include toothpaste, gelatine, the blood clots, printingink and various types of polymeric compounds in nature and industry. Oneimportant feature of viscoelastic fluids is the normal stress difference. Twofamous demos of its effect are the Weissenberg experiment and the Barus ex-periment. In the Weissenberg experiment, the mixer stirs a polymer solution,the stretching of the polymers generates a positive normal force which allowsthe fluid to rise up along the rod, see Fig.1.1(b), while the Newtonian fluidwill be swaped away from the rod due to centrifugal force. When one tries tosqueeze shampoo out of a fine nozzle, we can observe that the diameter of theliquid stream is larger than the diameter of the nozzle, see Fig.1.1(c). Thisis because the deformed polymers tend to regain their spherical conformation.For a Newtonian fluid, the stream diameter will be smaller than the nozzlediameter, due to the acceleration of the fluid downstream.

Viscoelastic rheology attracts increasing interest due to the extensive ap-plication of polymeric materials in the industrial and medical scenarios such asthe production of plastics and fibers, additive manufacturing, module injection

3

4 1. INTRODUCTION

(a) (b) (c)

Figure 1.1. Demonstrations of (a) shear-dependent viscosity ofnon-Newtonian fluids,(b) the Weissenberg effect and (c) the Barus

effect.

and inkjet printing. For emerging techniques such as microfluidics and nanoflu-idics, many applications use polymer melts, polymer suspension and biofluids,most of which are viscoelastic (Kim et al. 2014; Pedron et al. 2011; Sun, Lin,and Barron 2011). Droplets are a common presence for the polymeric materi-als in industrial processes. For example, a DNA solution traveling inside a cellsorting device (Johnson et al. 1989) and a liquid slug of polymer melt insidea T-junction in an emulsion device (Garstecki et al. 2006). Droplet dynamicstherefore directly affect the outcome of these applications.

The rheological complexity brings many interesting observations on dropletdynamics. The viscoelastic droplet becomes less deformable in both shear andextensional flow while a viscoelastic medium can enhance the deformation ofa Newtonian droplet in it (Yue et al. 2005; Hooper et al. 2001). A dropletsubject to shear is more likely to breakup in a viscoelastic medium (Cardi-naels & Moldenaers 2011). Research inspires innovation of new techniques.The unstable nature of by a viscoelastic droplet in a junction can be used todevelop rheometer to measure the extensional viscosity (Seemann et al. 2012).The vertical migration of viscoelastic droplets towards the channel wall canbe used for cell sorting, see Fig.1.2(a) (Hatch et al. 2013). Junctions are ele-mentary components in microfluidics, also see Fig.1.2(a) (Hatch et al. 2013).Droplet behaviors at junctions have been extensively studied for Newtonianfluids (Jullien et al. 2009; Leshansky and Pismen 2009; Carlson, Do-Quang,and Amberg 2010), while the story for non-Newtonian fluids is far from com-plete. For the particular behavior of droplet splitting, relevant studies is rarelyreported (Christopher and Anna 2009), leaving one niche to be filled.

Spreading and wetting of droplets is another important aspect of dropletdynamics. The wetting of non-Newtonian fluids is not well understood, despitethe fact that viscoelastic fluids occupy a large portion among the fluids that areused for spray coating, agrochemicals deposition, ink-jet printing and additive

1. INTRODUCTION 5

(a)

(b) (c)

Figure 1.2. (a)1Passive cell sorting by lateral migration of vis-coelastic droplets, TMG is short for anhydrous trimethylglycine(Hatch et al. 2013). (b)2Spreading of a cell aggregate on a solidsubstrate (Douezan et al. 2011). (c)3Comparison between a waterdroplet(upper chart) and a polyethylene oxide(PEO) droplet(lowerchart) behavior after impacting at a flat substrate (Bartolo et al.

2007).

manufacturing, all of which involves droplet spreading. Besides, spreading ofcomplex fluids is fundamental in many biomedical circumstances such as woundhealing and cancer propagation, see Fig.1.2(b). One of the core problems ishow viscoelastic rheology affects the contact line motion. A natural expecta-tion is that a viscoelastic droplet shall spread slower as the extensional viscos-ity increases in the droplet. This expectation however is not true accordingto several previous studies on droplet impact. The polymeric droplet spreadssimilar or to Newtonian droplet (Wei et al. 2007; Han & Kim 2013). However,

1Hatch, A. C., Patel, A., Beer, N. R. & Lee, A. P. (2013) Passive droplet sorting using

viscoelastic flow focusing. Lab on a Chip 13, 1308-1315, part of fig.5(b). c©2013 RoyalSociety of Chemistry2Bartolo, D., Boudaoud, A., Narcy, G. & Bonn, D. (2007) Dynamics of non-newtoniandroplets. Physical review letters 99, 174502, part of fig.1. c©2007 American Physical Society.3Douezan, S., Guevorkian, K., Naouar, R., Dufour, S., Cuvelier, D. & Brochard-Wyart, F.(2011) Spreading dynamics and wetting transition of cellular aggregates. Proceedings of the

National Academy of Sciences 108, 7315-7320, fig.1F. c©2011 National Academy of SciencesU.S.A.

6 1. INTRODUCTION

the contraction speed is indeed slower than the spreading speed and eventu-ally the rebound is suppressed, see Fig.1.2(c) (Bergeron et al. 2000; Smith &Bertola 2010). However, a controversial argument is reported that the max-imum spreading diameter is reduced by viscoelasticity (Guemas et al. 2012),leaving this issue for further investigation. Another missing part in the previ-ous research is that the contact line rheology is by far not fully resolved as theviscoelastic stress is not directly measurable in an experimental setup, while isreadily available in a simulation.

The presented work intends to explore the viscoelastic effects on smallscale dynamics such as droplet splitting in microfluidics, droplet spreading anddroplet impact, using numerical simulation. Part I provides background knowl-edge that helps readers understand the theories and methods that are employedto gain the results presented in Part II. Chapter2 is the concepts of capillarity,wetting and spreading. Chapter3 explains viscoelasticity and the models formodeling viscoelastic flow. Chapter4 revisits the diffuse interface method formodeling multiphasic flow. Chapter5 covers details of the formulations and nu-merical schemes that are used in this work. Chapter6 contains a brief summaryof the results in Part II. Chapter7 includes conclusions and suggests potentialextension from this work. Part II is the collection of results, presented by fourjournal articles, each of them addresses one particular problem.

CHAPTER 2

Capillarity and wetting

This chapter covers basic concepts of dynamic wetting. The emphasis is put ona moving contact line that involves in natural wetting, while the comprehensivereviews on wetting and spreading can be found in (Bonn et al. 2009) and(Snoeijer & Andreotti 2013).

2.1. Capillarity

Capillarity relates to the surface tension force existing at the interface of twophases. Taking a water-air interface as an example, the surface tension comesfrom the fact that the water molecules in the interface are subject to stronger at-traction force from other water molecules than the force from the air molecules,creating an inward net force on the interface. The main outcome of this forceis that the interface always evolves towards a state that minimizes the surfacearea/energy, which is the reason why a droplet favors a spherical shape. TheYoung-Laplace law well demonstrates the power of surface tension (de Genneset al. 2013).

4P = 2σ/K (2.1)

where 4P is the Laplacian pressure that indicates the pressure jump crossthe interface. It is related to the surface tension coefficient σ and K the localcurvature. Strong capillarity allows water striders to stride on the river surface.The Young-Laplace law explains why capillarity is an important factor formotions in small scale. As K → 0, the strength of surface tension becomessignificant. Typically a capillary length lc =

√σ/ρg can be defined as the

length scale below which capillarity becomes important in relation to gravity,where g is gravitational acceleration, ρ is the fluid density.

The non-dimensional number that reflects the relative importance of sur-face tension is the Capillary number:

Ca = Uµ/σ (2.2)

where U and µ are the characteristic speed and viscosity of a bubble/drop,respectively. Ca = 10−1 ∼ 10−3 denotes strong capillary flow and such flowcommonly exist in microfluidic devices (Stone et al. 2004; Carlson, Do-Quang,and Amberg 2010). The capillary force is important for droplet dynamicsin microfluidics as it resists deformation of the interface, thus affecting thesplitting/non-splitting of a droplet, an important phenomenon in cell sorting,drag delivery and droplet formation. Moreover Ca is used to form critical

7

8 2. CAPILLARITY AND WETTING

(a) (b)

Figure 2.1. (a)Schematic of a static drop partially wetting on asolid substrate, θe is the equilibrium contact angle. (b) Schematicof a moving droplet on a solid substrate, θa is the advancing contactangle, θr is the receding contact angle, the arrow shows the directionof the droplet motion

conditions for droplet splitting/non-splitting (Christopher et al. 2009; Linket al. 2004; Carlson, Do-Quang, and Amberg 2010).

2.2. Wetting

Wetting is the behavior of a liquid in contact with a solid surface, characterizedby the contact line where the gas-liquid-solid phases join, see Fig.2.1(a). Astatic contact line follows the Youngs equation as

σcosθe = σsl − σsg (2.3)

balancing the surface energies between gas-liquid(σ), gas-solid(σsg) and liquid-solid(σsl). θe is the equbrilium or static contact angle, decided by the surfacewettability. θe > 90 gives a hydrophobic surface such as the surface of a lotusleaf, while θe < 90 refers to a hydrophilic surface such as a glass surface. Onespecial state in the hydrophilic regime is the complete wetting, i.e., θe ≈ 0. Inthis state the liquid spreads into a film with a nanoscopic thickness, coveringthe substrate. On the other hand, θe > 120 means a superhydrophobic surfacewhich is difficult to wet. A water droplet hitting on such surfaces can reboundlike a ball. The Lotus leaf is a natural example of this kind of surface.

In practical, the actual contact angle is not equal to the equilibrium angle,due to the fact that the surface is not perfectly homogeneous. The differencefor a moving contact angle away from the equilibrium value is called contactangle hysteresis, θr < θe < θa, where θr is the receding contact angle formed bythe contact line that moves towards the wetting liquid and θa is the advancingcontact angle formed by the contact line that moves towards the ambient gas,as shown in Fig.2.1(b).

The physics of a moving contact line in dynamic wetting is a multi-scaleproblem. The macroscopic dynamics present in the length scale above thecapillarity length, see the top layer in Fig.2.2(b) where the apparent/dynamiccontact angle θ can be measured. θ deviates from θe and varies when the contactline moves. Flow in the outer region can be described by the hydrodynamic

2.2. WETTING 9

(a) (b) (c)

Figure 2.2. Schematics of (a) the molecular process in the innerregion of a contact line,(b)the contact line regions and the stream-lines of corner the flow in it, θ is the apparent contact angle, θmis the molecular contact angle, θe is the equilibrium contact angleand (c) a droplet wetting on a solid substrate.

theory, i.e., the incompressible Navier Stokes equations:

∇ · u = 0 (2.4)

ρDu

Dt= −∇P +∇ · (η(∇u + (∇u)T)) + f (2.5)

where ξ = ∇u + (∇u)T is the strain rate tensor. Eq.2.4 guarantees the massconservation while Eq.2.5 ensures the momentum conservation. Du/Dt is thematerial derivative of u. f is the source term from gravity, surface tension forceand other body forces.

The mesoscopic aspect comes from the near vicinity of the contact, with alength scale spanning 10 nm ∼ 1 mm, see the second layer in in Fig.2.2(b. Themicroscopic contact angle θ is approximately equal to θe appears in this region.Flow in this region can be described by the hydrodynamics. It is featuredby a wedge-like flow geometry, (Huh & Scriven 1971), presented by the bluestreamlines in the Fig.2.2(b). Using Navier Stokes equations, assuming Ca 1and the interface profile has a shape like a wedge, a scaling law is obtainedthat relates contact line speed Uc and θ, as θ ∼ Ca1/3 = (Ucη/σ)1/3 (Hoffman1975), also the spreading radius r is found to be a function of spreading time asr ∼ t1/10 (Tanner 1979). As inertia becomes negligible when approaching smallscale, Eq.2.5 is reduced to the Stokes equation, i.e., the left hand side vanishes.The contact line motion is decided by the force balance between viscous stressesand surface tension. Given assumptions as θ < 1 and Ca ≤ 0.1, using thelubrication approximation, the Stokes equation can be further reduced to:

3kCa

h2= hxxx (2.6)

where the subscript x denotes the derivative to the horizontal coordinate alongthe wetted surface, h(x) is the profile of the liquid-gas interface, k = 1 for


receding contact line while k = −1 for advancing contact line. Solution of thisthird equation reads:

h3x = θ3

e + 9Ca ln(x/Lm) (2.7)

Starting from Eq.2.7, a matching law bridging microscopic equilibrium andmesoscopic hydrodynamics can be derived (Voinov 2000; Cox 1986):

θ3 − θ3e = Ca ln(Λ

L

Lm) (2.8)

where Uc is the contact line speed, L is the length scale of the outer region,for instance the diameter of a droplet. The factor Λ depends on the boundaryconditions that are proposed (Eggers & Stone 2004). Lm is the characteristiclength of the inner region in which molecular kinetics can be important. Eq.2.8is valid when θ ≈ θe and L/Lm is sufficiently large. Considering a water dropletspreading on the wall, substituting θ ≈ 4V/r3 and Uc ≈ dr/dt into Eq.2.8,where V is the volume of the droplet, the Tanner’s law for viscous spreading isobtained:

r = L(σL

ηt)

110 (2.9)

The inner region of a contact line has an approximate thickness of 10nm, shown by the bottom layer in Fig.2.2(b). The continuum breaks downwhile molecular process has to be accounted. The contact line motion is thecollective result from the jumping of molecules near the three-phase interface,see Fig.2.2(a). Defining a jumping length is lj, the contact line speed can beexpressed as Uc = lj(k

+ − k−), where k+, k− are the frequencies of moleculesjumping out of and jumping into the interface, respectively. For a contact lineaway from equilibrium, the capillary force will feed Uc :(Blake & Haynes 1969;Blake 2006)

Uc = 2k0ljsinh(σ(cosθe − cosθm)

2nkBT) (2.10)

where lj is the hooping length, k0 = k+ − k− is the net jumping frequency, kBis the Boltzmann constant, n is the number of jumping molecules and T is thetemperature, θm is the contact angle in the molecular scale.

The molecular kinetic description of the inner region does not fit into thehydrodynamic framework. while the hydrodynamic description fails in theinner region because of several drawbacks, When R→ 0 the shear stress scalingwith µU/R grows to infinite, the velocity at r = 0 has multiple directions, due tothe corner flow geometry. Further modeling is required to close hydrodynamicdescription in the inner region. One way is to introduce the slip length ls(Eggers 2004), allowing the fluid at the wall to slip at a speed decided by:

U(y=0) = ls∂u

∂y(2.11)

where y denotes the vertical distance from the wall. Eq.2.11 is well establishedto described the first few molecular layers above the surface and the lengthls can be an approximation of Lm. Another approach is to utilize the diffuse

2.3. WETTING OF NON-NEWTONIAN FLUIDS 11

interface, which will be explained in Chapter5. Considering that the liquid-gasinterface has a finite thickness ε, which is typically in order of 10 nm, inside theinterface the material properties such as viscosity and density transit smoothlyfrom the dense liquid phase to the dilute gas phase. The contact line motioncan be realized by an diffusive process of phase, even if a non-slip condition isapplied. The phase transition is then characterized by a diffuse length ld whichis a natural slip length for the diffuse interface method (Yue & Feng 2012).There is also other remedies such as precursor film and disjoining pressure(Eggers 2005; Hervet & Degennes 1984), which are not included in this chapter.

Back to the hydrodynamics description, Tanner’s law is not universal as it isnot valid when Ca > 0.1 (Chen et al. 1995). Different values of the exponent inEq.2.9 are reported for larger Ca (Biance et al. 2004; De Coninck et al. 2001).Beside surface tension and viscosity, there are other factors, such as inertia,that can affect the contact line motion. In the initial stage of spreading of a

water droplet r is found to scale with the inertial time√ρR3/σ and r ∼ t1/2

(Biance et al. 2004; Bird et al. 2008; Winkels et al. 2012). Studies on rapidspreading show that there can be another energy dissipation mechanism for thecontact line motion. It is interpreted as contact line friction in the simulationsand experiments by (Carlson et al. 2012, 2011), where data of r for variouswettability and viscosities collapses into a scaling law based on a contact linefriction factor.

2.3. Wetting of non-Newtonian fluids

Wetting of non-Newtonian fluids has not been extensively studied. The rhe-ological properties that are mostly concerned near a contact line are shear-thinning and elasticity which have been introduced in Chapter1. Some non-Newtonian effects have been well understand. The interface is less curved byshear-thinning(Seevaratnam et al. 2007), because lower capillarity is requiredto drive a contact line as viscosity is reduced. The interface curvature is en-hanced by elasticity, due to the normal stress that requires extra capillarityto balance (Wei et al. 2007; Yue & Feng 2012). More interests lie on de-tecting the rheological effects on the contact speed, e.g., θ vs Ca. Attemptsthat explores this problem have not converged to a general conclusion. Someelastic fluids, e.g., the Boger fluids, obey Tanner-Cox-Voinov law in Eq.2.8.However the dependence of the slope ln(ΛL/Lm) on the relaxation time λ hasnot been determined. With an increase of λ, the slope can be either (Han& Kim 2013; Wang et al. 2015) reduced or increased(Wei et al. 2007) . Onehypothesis can be made even without considering the rheological aspect. If weadmit Lm ∼ ls, and the polymeric solution has a larger ls than the Newtonianfluid(Snoeijer & Andreotti 2013), one can expect the slope will be smaller forthe polymeric fluid. Deviating from Eq.(2.8), some elastic fluids such as thePAM solution are found to follow a quadratic dependence as θ ∼ Ca2(Kim& Rothstein 2015) , provided with sufficiently large λ. A power law scalingas θ1+2/n ∼ Ca[ηl/(γcσ)]1−1/n is found for the shear-thinning fluids such as


Xanthan gum, where γc is the critical shear rate for shear-thinning, n is theshear-thinning exponent(Seevaratnam et al. 2007).

Some macroscopic outcomes of the contact line motion of non-Newtonianfluids are also reported but the underlying physics of these results are notcompletely clear. The Tanner’s law r ∼ t1/10 is also valid for some shear-thinning fluids and elastic dominated fluids(Rafaı et al. 2004). Interestinglyshear-thinning fluids can spread slower than its Newtonian counterpart, withthe exponent of Tanner’s law slightly less than 0.1 (Bonn et al. 2009), on con-trast to the acceleration of the contact line. The droplet impact experimentin Fig.1.1(c) observes that the maximum spreading ratio has no differnce be-tween a shear-thinning elastic droplet and a water droplet, similar observationis made by (Smith & Bertola 2010). This result is against the naive expecta-tion that the fast spreading involved in droplet impact should have sufficientlylarge shear rate to activate the effects of shear-thinning and normal stress. Onthe contrary, the maximum spreading ratio is found to be significantly sup-pressed due to the presence of viscoelasticity in a Carbopol droplet (Guemaset al. 2012), comparing to the Newtonian scaling as We1/2, where We is theimpact Weber number. This indicates an energy dissipation due to viscoelasticstresses.

The divergence of experimental observations implies that the role of non-Newtonian rheology can be case dependent, upon the material properties of thetest fluids and details of the wetted surface. Overall the puzzle still remains:whether and how shear-thinning and elasticity affects dynamic wetting?

CHAPTER 3

Viscoelasticity

This chapter provides background on the origin and derivation of basic non-Newtonian constitutive models employed by the simulation in this work. Read-ers can refer to (Bird & Wiest 1995),(Bird et al. 1977) and (Spagnolie 2015)for comprehensive knowledge on viscoelastic rheology.

3.1. Modeling viscoelasticity

Viscoelasticity is often observed in polymeric fluids. It states that a fluid holdsboth viscosity and elasticity. The viscoelastic response to deformation can besimulated by a serial connection of an elastic spring and a viscous dashpot,called the Maxwell model, see Fig.3.1(a). The elastic stress on the the springτs and the viscous stress on the dashpot τd follow the Hooke’s law and theNewton’s law, respectively:

τs = Gξs (3.1)

τd = ηξd (3.2)

where G is the elastic modulus, η is the viscosity, ξs is the strain of thespring, ξd is the strain of the dashpot. For a serial connection the stress shallbe equal between elements while the total strain shall be the sum of the strainon each element, i.e., τ = τd = τs, ξ = ξs + ξd. Taking the time derivative of ξand multiplying by η, one obtains:

ηξ =η

Gτ + τ (3.3)

where ξ denotes the strain rate, η/G = λ defines a relaxation time. Eq.3.3is constitutive and serves as the ground basis of the linear viscoelastic models.Its solution reads:

τ(t) =1

λ

∫ t

−∞e−

t−t′λ ηξ(t′)dt′ (3.4)

Imposing a simple deformation as ξ(t) = ξ0sinωt, where ω is the frequency,Eq.3.4 becomes:

τ(t) = ξ0ηωcosωt+ λωsinωt

1 + (λω)2=

η

1 + (λω)2ξ(t) +G

(λω)2

1 + (λω)2ξ(t) (3.5)

13

14 3. VISCOELASTICITY

(a) (b)

Figure 3.1. Schematics of (a) a Hookean spring connected with adashpot and (b)a Hookean dumbbell.

Eq.3.5 describes the nature of linear viscoelasticity and helps to understandthe behavior of viscoelastic fluid in a general sense. For a short period of time,1/w λ, τ(t) ≈ Gξ(t), the material reacts as a solid; while for a long term

deformation, 1/w λ, τ(t) ≈ η ˙ξ(t), the material turns to be a viscous fluid.The transition occurs when 1/w is comparable with λ. For a moderate period,the material reacts a solid like fluid, its elastic part maintains the strain andstores energy, meanwhile its viscous part relaxes the strain and dumps energy.

3.2. Constitutive models

UCM model

To model an arbitrary flow, Eq.3.4 needs to be rewritten in the form of thestrain tensor ξ in terms of the velocity gradient ∇u, and the stress tensor τ .The direct replacement of τ and ξ will leave the resulting equation not frame-invariant. Therefore, a few steps are necessary to derive the frame-invarianttime derivative of a second rank tensor as τ (Spagnolie 2015). With this beingdone, the constitutive equation reads:

τ + λ∇τ = ηξ (3.6)

where∇τ is called the upper convective derivative of τ :

∇τ =

∂τ

∂t+ u · ∇τ +∇u · τ + τ · (∇u)T (3.7)

Eq.3.6 is called the upper-convective-Maxwell(UCM) model which is the sim-plest constitutive law for polymeric fluids (Olsson & Ystrom 1993; Bird et al.1977). It is also the base for many more elaborate models.

Oldroyd-B model

If considering a complex fluid as a dilute suspension of polymer molecules,the constitutive law can also be described from a kinetic theory. The simplestmodel for a polymer molecule is the linear Hookean dumbbell, i.e., two beadsconnected by a Hookean spring, shown in Fig.3.1(b). The force on the springF = GQ, then polymeric stress is computed by

τ = G(< QQ > −I) = G(C− I) =ηpλ

(C− I) (3.8)

3.2. CONSTITUTIVE MODELS 15

where ηp is the viscosity of polymers, Q is the end-to-end vector between thebeads, reflecting the entropic effect of the spring caused by the rotation andstretching of the dumbbell. The symbol <> denotes the ensemble averagedyadic product, and the tensor C =< QQ > is called the conformation tensorthat records the deviation away from the equilibrium. C = I, i.e., no defor-mation, for equilibrium thus the evolution of C is governed by (Hulsen et al.2005):

∇C = − 1

λ(C− I) (3.9)

where∇C is the upper convective derivative of C. The kinetic theory does not

give a new constitutive model, instead it relates the model parameters with themolecular properties: λ = ζ/4G, ζ = 6πηsa and ηp = λnkBT , where ζ is theStokes drag coefficient of the spherical bead, a is the diameter of the bead, ηsis the viscosity of the Newtonian solvent, n is the number density of beads .Given these properties τ follows Eq.3.6, but with the viscosity η replaced byηp:

τ + λ∇τ = ηpξ (3.10)

where ηs+ηp = η. This is called the Oldroyd-B model, (Bird & Wiest 1995; Ol-droyd 1950) therefore the UCM fluid is the upper limit of the Oldroyd-B modelwith ηs = 0. Physically the Oldroyd-B model suffers from several drawbacksas it does not present shear-thinning, also it produce unbounded extensionalviscosity as there is no limits for the extensibility of the dumbbells. However,the model still predicts the Boger fluids in the shear and some extensional flowsand because of its simplicity it is popular in computational rheology. (Ardekaniet al. 2010; Bhat et al. 2010; Chinyoka et al. 2005; Hooper et al. 2001; Yue et al.2006)

With the viscosity being split, the total stress on a fluid element is split intothe Newtonian stress ηsξ and the τ contributed by the dumbbells. Consideringthe momentum equation in Eq.2.5, τ needs to be added in the right hand sideto include the contribution from the polymer molecules. To differentiate withthe normal stress, we starts here to write τ p instead of τ for the polymericstress.

Giesekus Model

Based on the Hookean dumbbell, Giesekus proposed that the drag force onthe bead shall be anisotropic (Giesekus 1982), i.e., the drag will be less in thedirection in which the flow is stressed and the molecules are aligned to. Basedon his assumption, the Giesekus model is developed as:

τ + λ∇τ +

αλ

ηpτ 2 = ηpξ (3.11)

where α is a mobility factor that limits the extensional viscosity, ranging from0 to 1. α = 0 represents the Oldroyd-B model. This constitutive law in terms

16 3. VISCOELASTICITY

of C reads:

C + α(C− I)2 + λ∇C = I (3.12)

The Giesekus model predicts shear-thinning properties despite its simpleform, while it has problem of tension-thinning at high shear rates (Luo & Tan-ner 1987; Oldroyd 1950).

FENE Model

The FENE constitutive laws including FENE-P (Peterlin 1955) and FENE-CR (Chilcott & Rallison 1988) models originates from non-linear viscoelastic-ity. Back to the Hookean dumbbells, the FENE models introduce a maximumextensibility of the spring:

F =G

1− Q2

b2

Q (3.13)

where b is the maximum extensibility factor of Q. However, due to the non-linearity it is impossible to have the evolution equation for < QQ >. As-sumptions are needed, applying the closure treatment by (Peterlin 1955) theFENE-P model in terms of the conformation tensor reads:

C +λ

f(Q)

∇C =

λ

f(Q)I (3.14)

τ = G(f(Q)C− I) (3.15)

where f(Q) = 1/(1 − tr(C)/b2), tr(C) is the trace of C. These models aremore realistic than the UCM type of models and are widely employed in an-alyzing polymeric turbulent flows (Dimitropoulos et al. 1998; Vaithianathanet al. 2006; Zhou & Akhavan 2003).

Selection of model

Given different assumptions on structure and forcing from the molecular scale,with the help of MKT and statistical mechanics, various constitutive lawscan be derived. There are other famous models such as the Phan-Thien-Tanner(PTT) model (Thien & Tanner 1977) that describes polymer moleculesas network of springs with junctions, rather than Hookean dumbbells. In thiswork, the Oldroyd-B and Giesekus model are employed to present viscoelas-ticity. The reasons are: The Oldroyd-B model, despite its drawbacks, is con-venient to implement and validate against literature. It is even preferable inthe case of studying the effect of elastic stress, giving a constant viscosity likethe Boger fluid. It is straightforward to switch to the realistic Giesekus modelfrom the Oldroyd-B model, by just tuning one parameter α. However the priceto pay is the numerical difficulty to resolve the unbounded growth of exten-sional stresses, see Fig.3.2(b), discussed in Chapter5. The Giesekus model isa qualified candidate to be the representative of polymeric fluids in reality, it

3.2. CONSTITUTIVE MODELS 17

100

105

10−4

10−2

100

η/η

0

γλ

Giesekus,α = 0.01Giesekus,α = 0.1Giesekus,α = 0.5Oldroyd−B

(a)

100

105

10−5

100

105

1010

N1λ/(2η0)

γλ

Giesekus,α = 0.01Giesekus,α = 0.1Giesekus,α = 0.5Oldroyd−B

(b)

Figure 3.2. Shear viscosity η (a) and normal stress difference N1

(b) as functions as shear rate γ of Giesekus fluids and Oldroyd-Bfluids under steady shear. η0 denotes the viscosity at γ = 0.

presents both shear-thinning and elasticity which are common features of dilutesuspensions of polymers, see Fig.3.2.

CHAPTER 4

The diffuse interface method

This chapter briefly introduces the diffuse interface method and common issueson its application on simulating multiphase flows. This include the representa-tion of surface tension and wetting conditions and effects of phase field param-eters on the numerical result. A comprehensive review on this method appliedin fluid mechanics can be found in (Anderson, McFadden, and Wheeler 1998).

4.1. Phase field theory

The phase field theory proposes that the interface between two immiscible fluidshas a finite thickness. Within this thickness, the two fluids mix and store freeenergy. A phase variable is introduced to represent the concentration of eachcomponent, φ = 1 denotes pure fluid one and φ = −1 denotes pure the secondfluid, −1 < φ < 1 denotes the mixture of two fluids. Following Cahn andHilliard (Cahn & Hilliard 1958), the density per unit length of the free energyis written as a function of φ and its gradient:

f =1

2κ|∇φ|2 +

κ

ε2Φ(φ) (4.1)

where κ is the magnitude of the free energy density, and ε represents the in-terface thickness. Eq.10 shows that mixing energy is composed of two parts,κ|∇φ|2/2 is the gradient energy which favors mixing, the term containing Φ isthe bulk energy that prefers separation. The term Φ(φ) is a double-well func-tion that has two minima at C ± 1, in phase field theory it is widely chosenas:

Φ(φ) =1

4(φ− 1)2(φ+ 1)2 (4.2)

shown by Fig.4.1(a). The variation of F =∫fdΩ with respect to φ gives the

chemical potential ψ:

ψ =κ

ε2Φ′(φ)− κ∇2φ (4.3)

The derivation of ψ can be found in Sec.4.3. The equilibrium interface profileis the one that minimizes F and keeps ψ = 0 (Van der Waals 1893). Thus thevariation of ψ drives the evolution of φ and any disturbance on φ results in areduction of F . The rate of change of φ is driven by a diffusive interfacial fluxJ as

δφ/δt = −ζ∇ · J (4.4)

18

4.2. SURFACE TENSION 19

where ζ is a mobility factor. Cahn and Hilliard assumed that J is proportionalto the gradient of the chemical potential J = ∇ψ. Extending this to an interfacethat evolves in a flow field, we use the material derivative to describe the rateof change, this gives the Cahn-Hilliard equation

∂φ

∂t+∇ · (uφ) = −ζ∇2ψ (4.5)

Eq.4.5 forms the governing equation of the interface profile, with the phasefield parameter κ, ε and ζ.

4.2. Surface tension

Equation.4.5 can couple with the Navier-Stokes equations Eq.2.4, Eq.2.5, withthe velocity u to describe the multiphase flow system. However, surface tensionis not represented in this form. Thus, to complete the coupling and match thistheory to the appropriate physics, it is important to recover the surface tensionforce using the phase field parameters.

Starting with the statement that the local interfacial free energy is equalto the surface energy, the problem is simplified to a 1D interface profile, thelocal surface energy is now

σ =

∫ ∞−∞

(0.5κφ2x +

κ

ε2Φ(φ))dx (4.6)

where the subscript x denotes the direction normal to the interface, φx is thederivative of φ along x. Assume the interface is at equilibrium, i.e., ψ = 0, thisgives an ordinary differential equation as:

κ

ε2Φ′(φ)− κφxx = 0 (4.7)

multiplying this equation by φx and integrating, the following is obtained

φ2x =

κ

2ε2Φ(φ) (4.8)

giving the condition φ(x = 0) = 0, the solution of Eq.4.8 is

φ(x) = tanh(x√2ε

) (4.9)

This equation is used to implement the initial interface profile for a simula-tion, see Fig.4.1(b). Substituting it into Eq.4.6, the matching of phase fieldparameters with surface tension is gained

σ =

√8

9

κ

ε(4.10)

One can see that κ/ε converges to the surface tension when ε→ 0.

The other task is to derive the phase field surface tension force for theincompressible momentum equation Eq.2.5. This is achieved by equating thevariation of the free energy (surface energy) due to convection

∫Ωψ∇ · (uφ)dΩ

20 4. THE DIFFUSE INTERFACE METHOD

−2 −1 0 1 20

0.5

1

1.5

2

2.5

φ

Ψ(φ)

(a)

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

φ(x)

x

(b)

Figure 4.1. (a)The double-well function Φ(φ) and (b) the equilib-rium interface profile defined by Eq.4.9.

=∫

Ωψu∇φdΩ, with the rate of change of the kinetic energy due to surface

tension force∫

ΩFudΩ. This gives:(Jacqmin 2000)

F = ψ∇φ (4.11)

F is the potential form of the surface tension force. The phase field surfacetension also has a stress form (Yue et al. 2004) and a form for compressibleflow (Jacqmin 1999). In this thesis, the potential form is used and it is onlynon-zero inside the interface can produce the correct surface tension within thesharp interface limit(ε→ 0). The validation of this is made in Chapter6.

4.3. Wettability, contact angle

The total free energy of the two phase system contains the integral of f overthe volume Ω plus the contribution of from the surface Γ(Van der Waals 1893).

F =

∫Ω

(1

2κ|∇φ|2 +

κ

ε2Φ(φ))dΩ +

∫Γ

(σsg + (σsl − σsg)g(φ))dΓ (4.12)

The variation of F with respect to φ reads:

δF =

∫Ω

κ

ε2Φ′(φ)δφdΩ +

∫Ω

κ∇φ∇(δφ)dΩ +

∫Γ

(σsl − σsg)g′(φ)δφdΓ (4.13)

Integrating by parts the second term as∫

Ωκ∇φ∇(δφ) =

∫Γκ∇φδφ−

∫Ωκ∇2φδφ,and

substitute Eq.2.3 into the third term, we obtain

δF =

∫Ω

(κ

ε2Φ′(φ)− κ∇2φ)δφdΩ +

∫Γ

(κ∇φ · n− σcosθeg′(φ))δφdΩ (4.14)

where n is the unit vector normal to the wetted surface. The first term in theright hand side gives f and the second term is used to implement the surfacewettability. Assuming local equilibrium at the boundary, i.e., the microscopiccontact angle is equal to θe, we have

κ∇φ · n− σcosθeg′(φ) = 0 (4.15)

4.4. DIFFUSE INTERFACE 21

the polynomial g(φ) = 1/4(−φ3 + 3φ + 2) is not arbitrary but can be derivedfrom Eq.4.15: giving ∇φ · n = cosθeφx, subsisting Eq.4.8, Eq.4.10 into Eq.4.15and solve the resulting Eq.4.16 with the boundary conditions g(−1) = 0, g(1) =1 (Carlson 2012).

dg(φ)

dφ=

3

4(1 + φ)(1− φ) (4.16)

If the microscopic contact angle changes while the contact line is moving, thisleads to the non-equilibrium boundary condition. Postulating that the variationof φ at the boundary is always countered by a diffusive boundary flux Dφ/Dt =Dw, Eq.4.15 can be expanded to general cases as:

Dw(∂φ

∂t+ u∇φ) = κ∇φ · n− σcosθeg

′(φ) (4.17)

with a non-slip condition at the boundary, u∇φ can be dropped off fromEq.4.17. Dw is a diffusion rate with a dimension of mPa.s. It is re-definedas the contact line friction Dw = µf ε, where µf is a friction parameter thatmatches phase field model to surface patterns such as surface roughness (Carl-son et al. 2012; Do-Quang et al. 2015).

4.4. Diffuse interface

We consider the interface to be diffuse over a length scale ld. However this in-terfacial width should be small enough to match the model to a sharp interface.Elder (Elder et al. 2001) proposed ld/K 1 and ldU/D 1, where 1/K is theinterface curvature, U and D are the characteristic velocity and diffusion coef-ficient of the bulk flow, respectively. ld is determined by both the parameter εand the mobility factor ζ. In principle, ε shall be chosen as small as possible toapproach the sharp interface limit and the correct interfacial force. Once ε isfixed, κ is decided by Eq.4.10. ζ needs to be chosen so that it is large enoughthat the interface is stable and resolved in the numerics and small enough toavoid over dissipation of the surface energy. A typical way to decide ζ is theconvergence study such that the numerical results converges while reducing ζ,such validation is made in Chapter5. Therefore, the right combination of ε andζ is important, guidelines for choosing their proper values can be found in (Yueet al. 2010)

CHAPTER 5

Numerical model

From the previous chapters, all elements to model a non-Newtonian two phaseflow are available: the Navier-Stokes equations to describe the flow field, theconstitutive equations to calculate non-Newtonian stresses and the Cahn-Hilliardequation to evolve the interface and calculate the surface tension force. Byadding ∇ · τ p and G∇φ to the right hand side of Eq.2.5, the coupling of thethree models for single phase flow is complete:

ρDu

Dt= −∇p+∇ · (ηs(∇u + (∇u)T)) +∇ · τ p +G∇φ (5.1)

where ηs is the Newtonian part of the zero shear viscosity η = ηs + ηp. Eq.2.4,Eq.3.8, Eq.3.12, Eq.4.5 and Eq.11 compose the governing equations of all simu-lations presented in this thesis. This chapter contains numerical methodologiesto solve this set of equations, with emphasis on schemes for computationalrheology.

5.1. Non-dimensionalization

The normalization uses reference variables such as η∗ = ηl, ρ∗ = ρl, U

∗ =U,L∗ = Rd, t

∗ = Rd/U, p∗ = τ∗ = URd/ηl, G

∗ = 3σ/2√

2ε, where Rd isthe characteristic length of the flow, e.g., the radius of droplet, the subscriptl denotes the property of the liquid phase. The following non-dimensionalnumbers appear in the equation set:

Cn =ε

Rd(Cahn number)

Pe =2√

2UεRd3σζ

(Peclet number)

Wi =λU

Rd(Weissenberg number)

Ca =ηlU

σ(Capillary number)

Re =ρlURdηl

(Reynolds number)

Pe expresses the ratio of interfacial convection and diffusion, Cn representsthe normalized interface thickness, Ca is the relative importance of viscosityover surface tension, Re states how inertia weighs over viscosity, Wi = λU/Rd

22

5.2. CAHN-HILLIARD EQUATIONS 23

expresses the degree of elasticity. In the following text we will use the non-dimensional form of equations.

The coupled system is solved in a segregated routine. First the system ofEq.4.3 and Eq.4.5 is solved to update ψ and φ; the system of Eq.2.4 and Eq.11is solved to update u, then the set of equations(3 in 2D, 4 for axisymmetric, 6for 3D) defined by Eq.3.12 is solved to update the conformation tensor, FinallyEq.3.8 is computed to update τ p. In the time dependent formulation, we usethe superscript i−1 to denote the solution of previous time step, the superscripti to denote the variable to be solved.

5.2. Cahn-Hilliard equations

The original Cahn-Hilliard equation in Eq.4.5 is difficult to solve because itcontains a forth order derivative term and non-linearity caused by the double-well function Φ(φ). The standard approach is to separate the equation into twosecond order equations: one equation for φ as Eq.4.5 with the chemical potentialψ as a variable and another equation for ψ as Eq.4.3. The two equations arediscretized using the Galerkin method, in the same finite element space P 1(Ω)that contains all piecewise linear polynomials. The second order derivativesare all reduced to the first order using integral by parts. The time derivativeterm is discritized by a semi-implicit scheme. The weak formulation of theCahn-Hilliard equation reads now finding φ ∈ P 1(Ω), ψ ∈ P 1(Ω) that for allν ∈ P 1(Ω)

(φi − φi−1

4t, ν)+(

1

Pe∇ψi,∇ν)+(−φiui−1,∇ν)+ui−1φin, ν+− 1

Pe∇ψin, ν = 0

(5.2)and

(ψi, ν) + (φi − (φi)3, ν) + (Cn2∇φi,∇ν)− Cn2∇φin, ν = 0 (5.3)

are satisfied, where (.) denotes integral over the domain, . denotes integralalong the boundary. The two boundary integral terms in Eq.5.2 are zero, giventhe non-slip condition u = 0 and ∇ψ ·n = 0, i.e., there is no flux of φ from thewall. The boundary integral term in Eq.5.3 enables the implementation of thecontact angle by substituting Eq.4.15 or Eq.4.17 into it.

In every time step, the Newton method is used to solve Eq.(5.2) andEq.(5.3) simultaneously. The solving procedure is as follows: the initial guessis prescribed by the initial condition φ0 and ψ0 at the beginning of the com-putation, or as φi−1 and ψi−1. The non-linear problem is solved by a numberof Newton iterations, in each iteration the Jacobian matrix is formed auto-matically by FemLego and the resulting linear problem is solved by the directsparse matrix solver MUMPS, obtaining the incremental solution. In the endof each iteration the solution is updated by the incremental solution and servesas known for computing the Jacobian matrix for the next iteration. The iter-ation ends at a prescribed residual. Similar procedures with more details arereported in (Do-Quang & Amberg 2010; Boyanova, Do-Quang, and Neytcheva

24 5. NUMERICAL MODEL

2012). Besides, an linear iterative solver based pre-conditioning techniques forsuch Cahn-Hilliard system is also available (Villanueva 2007) and is only usedin the simulation in Part II, paper II.

5.3. Navier Stokes equations

Eq.11 needs to be generalized to calculate cases of phases with unequal densityand viscosity. This is done by interpolation of physical properties. For instance,for the gas-liquid two phase cases presented in Part II, we have the liquid phase(ηl, ρl, λl) as a dilute polymer solution (ηl = ηls + ηlp) while the gas phase asanother viscoelastic fluid(ηg = ηgs + ηgp, ρg, λg). The subscripts l denotesliquid property, g the gas property, s the Newtonian property, p the polymericproperty. The normalized zero shear viscosity η(φ) and the normalized mixturedensity ρ(φ) of the two phase system are interpolated as

η(φi) = 0.5(1 + φi) + 0.5(1− φi)ηg/ηl (5.4)

ρ(φi) = 0.5(1 + φi) + 0.5(1− φi)ρg/ρl (5.5)

while the mixture’s Newtonian vs polymeric viscosity ratio β(φ) is also inter-polated:

β(φi) = 0.5(1 + φi)βl + 0.5(1− φi)βg (5.6)

where βg = ηgs/ηg, βl = ηls/ηl. We seek for the solution of velocity in a finiteelement space P 2(Ω) that contains all piecewise quadratic polynomials whilestill looking for the solution of pressure in P 1(Ω). The weak formulation forthe Navier Stokes equations reads: finding u ∈ P 2(Ω), p ∈ P 1(Ω), G ∈ P 1(Ω)for all ν ∈ P 1(Ω), ν2 ∈ P 2(Ω)

Reρ(φi)(ui − ui−1

4t, ν2) +Reρ(φi)(ui−1∇ui, ν2) = (piI,∇ν2) (5.7)

− (η(φi)(∇ui + (∇u)iT),∇ν2) + (η(φi)(1− β(φi))(Gi−1 + (Gi−1)T),∇ν2)

− (τ ip,∇ν2) +1

CnCa(ψi∇φi, ν2)

(∇ · ui, ν) = 0 (5.8)

(Gi, ν)− (∇ui, ν) = 0 (5.9)

where u,G represent components of u and G, respectively. Here all boundaryintegral terms coming from the step of integral by parts are neglected, this

implies a stress free boundary condition, i.e., pi + τ ip + η(φi)β(φi)ξi = 0.

Eq.5.7 and Eq.5.8 are solved simultaneously by the sparse direct solverMUMPS, in which direct LU decomposition of the operator matrix is performed(Amestoy et al. 2000). A projection scheme developed by (Guermond & Quar-tapelle 2000) is also implemented to solve the coupled problem with unequalproperties, following the implementation in (Carlson 2012). This projectionscheme allows the usage of piecewise linear polynomial for u, which causes sad-dle point (Girault & Raviart 2012) problem while looking for both pressure andvelocity in linear space. Therefore the computational cost reduces significantly.

5.4. CONSTITUTIVE EQUATIONS 25

Eq.5.9 is solved component by component using the conjugate gradient(CG)solver.

In cases of Re = 0, the equation reduces to the augmented Stokes equa-tion. One approach is to solve it using MUMPS, another way is to applypre-conditioning techniques and solve it iteratively, however the mixed finiteelement formulation of a Stokes system has zero pivot block because there is nopressure term in the continuity equation, thus typical pre-conditioners offeredby krylov solvers can not be used. A solution to this issue is discussed in (Sil-vester & Wathen 1994; Wathen & Silvester 1993). Here the implementationof the iterative Stokes solver largely follows the code published in DEAL.IItutorial Step-22. The iterative Stokes solver is implemented for the simulationpresented in paper II, Part II. In the remaining part of this thesis, MUMPS isemployed for both Navier Stokes and Stokes equations.

5.4. Constitutive equations

The computation of constitutive equations often encounters numerical diffi-culties, due to the advective nature of the equations. The high Weissenbergnumber problem (HWNP) (Keunings 1986) is a classic numerical issue in com-putational rheology. It indicates that the computation breaks down when Wiis large and numerical methods fail to resolve sharp gradients of τ p when theflow encounters a geometric singularity or a stagnation point. For instancethe upper limit of computable Wi for the case of single phase viscoelastic flowpassing a cylinder is 5 in early studies. During the past two decades continuousefforts have been made to overcome HWNP, resulting in numerical techniqueswhich enables simulation of highly elastic fluids, a review of contemporarystandard methods can be found in (Baaijens 1998). In this thesis we adopt theDEVSS-G/SUPG scheme. DEVSS abbreviates Discrete Elastic-Viscous StressSplitting, initially developed by (Guenette & Fortin 1995). In case ηs is small,or more extremely for the Maxwell model, i.e., ηs = 0, Eq.11 is without anydiffusion term and consequently becomes fragile to numerical disturbance. Toenhance the ellipticity of this equation, the DEVSS formulation introduces apolymeric viscous stress E = ηpξ and the momentum is rewritten as:

ρDu

Dt= −∇P +∇ · (ηξ)−∇ ·E +∇ · τ p + ψ∇φ (5.10)

where E is an additional variable computed from the new velocity. G denotesthe gradient of velocity, rather than solving for E, one can solve for

G = ∇u (5.11)

and replace E with ηp(G + GT) in Eq.12, and also replace ∇u with G inthe constitutive equation. Therefore the coupling of Navier Stokes equationsand viscoelastic constitutive equations becomes stronger. The block matrixresulting from the mixed finite element discritization of the equation systemcan be solved implicitly by developed iterative schemes (Baaijens et al. 2004;Kim et al. 2004). Though DEVSS-G is designed to improve the block matrix of


the coupled system composed by the constitutive equations and Navier Stokesequations, numerical experiment has shown that this scheme also stabilizes asegregated solver. SUPG is short for the Streamline Upwind Petrov Galerkinmethod developed by (Brooks & Hughes 1982; Johnson et al. 1984), whichintroduces numerical diffusion to discritized equations.

A recent development called the logarithmic conformation scheme is intro-duced by (Fattal & Kupferman 2005; Hulsen et al. 2005) to further enhance thenumerical stability of computational rheology. Instead of solving for C directlyfrom Eq.(3.12), this scheme solves for the matrix-logarithm H of the symmetricpositive definite tensor C. This requires a transformation of Eq.(3.12) into anevolution equation for H. This scheme benefits numerics such that the growthof the unbounded extensional viscosity becomes linear instead of exponential.The core task is to transform the material derivative C = ∂C/∂t+u ·∇C into:

∂H

∂t+ u · ∇H = H (5.12)

according to (Hulsen et al. 2005) the transformation obeys :

nk · H · nj =

dim∑k=1

1

cknk · C · nk +

dim∑k=1

dim∑j=1

hk − hjck − cj

nk · C · nj(k 6= j) (5.13)

where hk, hj are eigenvalues of H. ck, cj are eigenvalues of C, with hk = log(ck).The derivation of Eq.(5.13) can be found in (Jafari). nk,nj are eigenvectors ofH. We can use eigenvectors either from H or C to conduct this transformationbecause H and C are coaxial.

Subsisting Eq.(5.13) into Eq.(5.12), the weak formulation for H as: findingH ∈ P 1(Ω) for all ν ∈ P 1(Ω)

(Hi −Hi−1

4t+ ui−1 · ∇Hi −Mi−1(nk,nj)C

i−1, ν + χui−1 · ∇ν) = 0 (5.14)

is satisfied, where the factor χ controls the weight of numerical diffusion in-troduced by the SUPG scheme. M is a matrix composed by eigenvalues of Cand

Ci−1 = Ci−1 ·Gi−1 + (Gi−1)T ·Ci−1 − (Ci−1 − I + α(Ci−1 − I)2)/Wi(φi−1)(5.15)

is treated explicitly during time stepping. Giving Hi−1, an eigenvalue solverfor nk,nj is required to compose Mi−1. With the solution of Hi, the tensor Ci

can be constructed by:

Ci = QieH′i

(Qi)T (5.16)

where Qi is the matrix formed by all eigenvectors of Hi as columns. eH′i

is the matrix exponential of H′ which is a diagonal matrix composed by alleigenvalues of Hi. Finally we look for τp ∈ P 1(Ω) that for all ν ∈ P 1(Ω)

(τ ip, ν)− (1− β(φi)

Wi(φi)(Ci − I), ν) = 0 (5.17)

5.5. SOLVING ROUTINE 27

is satisfied, where Wi(φi) is computed in a same manner as Eq.5.4. Besides im-provement of numerical instability, another benefit from the presented schemeis decoupling the constitutive equations into a set of first order advection equa-tion that can be solved one by one to reduce computational cost, especially inthe case of solving the nonlinear Giesekus model because the model can alsobe linearized by this scheme. An application using logarithmic conformationtensor in 2D is presented in Paper 2, Part II, where the calculation of nk,nj iscalculated analytically.

5.5. Solving routine

The numerical procedure is summarized as follows:

1. Conduct mesh refinement or coarsening based on values of the phasevariable φi−1. The criteria are |φi−1| < 0.975 and hcell > hmin forrefinement. |φi−1| > 0.975 and max(|Gi−1|) < 3 for coalescence. hmin

is a prescribed value for the minimum mesh size. |.| means the absolutevalue.

2. Calculate φ and ψ: Eq.(5.2) and Eq.(5.3) are coupled together. Theresulting non-linear vector-valued system is solved by Newtons method,in each Newton iteration a preconditioner proposed by (Boyanova, Do-Quang, and Neytcheva 2012) is applied to the Jacobian matrix for-med by the coupled Cahn-Hilliard system. Another way is to employMUMPS and solve directly.

3. Update the conformation tensor: solving Eq.(5.14) to obtain Hi, withinput of Ci−1,ui−1,Hi−1,Gi−1. The solver can be MUMPS or GMRESwith prescribed small residual, then construct new conformation tensorCi by Eq.(5.16).

4. Calculate the polymeric stress τ ip by Eq.(5.17) with input of Ci, usinga CG solver.

5. Compute the new velocity field: solving the coupled Eq.(5.7) and Eq.(5.8)system, with the input of φi,ψi,Gi−1 and τ ip, using either the pre-conditioning or direct method. This step gives a new temporal velocityu∗ and pressure p∗ .

6. Solve Eq.(5.9) to obtain a new velocity gradient G∗. If the followingcondition is satisfied, update u∗ = ui, p∗ = pi,G∗ = Gi and moveforward to step 7, otherwise substitute G∗ as Gi−1 back into Eq.(5.7)and use u∗ to replace ui−1, p∗ to replace pi−1, then repeat from step5. ||.|| denotes the modulus of a vector. n is an iteration index, forinstance, pn−1 is the value for p∗ in the previous iteration.

||[p∗,u∗,G∗]|| − ||[pn−1,un−1,Gn−1]||||[pn−1,un−1,Gn−1]||

< 10−6

7. Conduct mesh refinement and coalescence based on Gi : max(|Gi|) > 10and hcell > 0.225hmin for refinement; max(|Gi|) < 3 and hcell < 0.3hmin

for coalescence.


8. Adjust time step size ∆t according to the following criteria, restart fromstep1 with the obtained solution as previous solution used to next timestep.

0.1 ≤ maxall domain cells

(||ucell||∆thcell

)≤ 0.25

The presented numerical methodology is implemented by FemLego, a sym-bolic FEM simulation toolbox (Do-Quang et al. 2007; Amberg et al. 1999).The recent development of FemLego includes linear and quadratic polynomi-als, mesh size adaptivity and allows parallel computing for large scale problem.The users can define the weak formation, numerical parameters and solvingprocedure in a Maple sheet. Compiling the Maple sheet will automaticallygenerates C++ source code. Beside auto generation, FemLego also offers inter-faces with several numerical packages such as MUMPS, PETSC (Balay et al.2015) and TRILINOS(Heroux et al. 2003), the users can access their sourcecode and customize the solving strategy.

CHAPTER 6

Summary of results

6.1. Validations

Several test cases are set up to validate the presented numerical model inChapter5. The simplest test case is built to verify that the implementation ofCahn-Hilliard equation predicts the correct Laplacian pressure, i.e., the surfacetension coefficient. For a 2D steady Newtonian droplet with ηl/ηg = 1, ρl/ρg =1 and radius R in the center of a square domain of 5R × 5R, the analyticalvalue of the dimensionless Laplacian pressure is

√8/9/Ca. The computed

4P is obtained, using the result of pressure field obtained by solving Eq.(5.2),Eq.(5.3), Eq.(5.7) and Eq.(5.8). Fixing Cn = 0.005, Re = 1 we varied Pe from500 to 3000, the deviation of computed value from the analytical value is within0.5%.

The test cases on droplet dynamics are reported in Part II. The transientdeformation of VE and Newtonian droplets subject to an extensional flow isreported in Fig.2, Tab.1 in Paper 1, Part II. The results are validated againstthe simulation by (Yue et al. 2006). For this test case the convergence of resultson phase field parameter Pe,Cn and the volume conservation are examined,see Fig.6.1. The spontaneous volume shrinkage is an inherent issue of reportedphase field method (Yue et al. 2007), while it is effectively suppressed by thepresented numerical scheme. The spreading diameter and the profile of New-tonian droplets hitting a homogenous flat substrate are validated against theexperiments by (Mongruel et al. 2009; Palacios et al. 2013; Rioboo, Marengo,and Tropea 2002), reported in Fig.2 and Fig.3 of Paper 3, Part II. The splash-ing and non-splashing phase diagram drawn by simulation is compared withthe experiments by (Palacios et al. 2013; Vander Wal et al. 2006), reported inFig.5 of Paper 3, Part II.

The results contains four articles studying viscoelastic(VE) droplets, col-lected in Part II. One article is on VE droplet migration in a microfluidic devicewith a bifurcation junction. Three articles are on dynamics wetting of droplets.This chapter highlights the most important findings of every article in Part II.Readers are encouraged to refer to Part II for comprehensive discussion ontopics.

29

30 6. SUMMARY OF RESULTS

0 1 2 3 4 5Time

1

1.1

1.2

1.3

1.4

1.5

R/R

0Cn=0.005,Pe=100Cn=0.002,Pe=300Cn=0.01,Pe=100

(a)

0 1 2 3 4 5Time

2

2.05

2.1

2.15

2.2

Dro

plet

Vol

ume

Cn=0.005,Pe=100Cn=0.002,Pe=300Cn=0.01,Pe=100

(b)

Figure 6.1. Effects of phase field parameters Cn and Pe on (a)the computed deformation ratio and (b) the total volume of a New-tonian droplet under uniaxial extension, the case is explained ac-cording to Fig.2 of Paper 1, Part II.

6.2. Viscoelastic droplet dynamics in microfluidic device

Many droplet-based microfluidic devices have junctions which are commonlyin a T-shape or a Y-shape(Teh et al. 2008). The splitting or non-splitting ofa liquid slug at junctions is utilized to enable emulsion, droplet formation andcell sorting(Guo et al. 2012; Mazutis et al. 2013; Vladisavljevic et al. 2012;Wehking et al. 2014). The breakup of Newtonian droplets at a T-junctionhave been extensively studied, examples among many others are (Christopher& Anna 2007; Link et al. 2004; Jullien et al. 2009). The splitting of viscoelasticdroplets at a T junction is investigated by (Christopher and Anna 2009). Thebehaviors of a Newtonian droplet at a Y-junction are reported by (Calderon,Fowlkes, and Bull 2005; Calderon et al. 2006; Carlson, Do-Quang, and Amberg2010). In what follow we will extend this to study viscoelastic droplets at Yjunction. This section gives a summary of Paper 1.

The challenge is computing polymeric stress at a bifurcating corner wherethe flow field diverges and the interface is subject to large morphological change.The DEVSS-G/SUPG logarithmic conformation scheme combined with meshadaptivity is implemented to run with larger Wi. Fortunately the Wi wecan reach is sufficient for observable viscoelastic effects, although we are stillrestricted to Wi ≤ 5.

A set of simulations with different Wi, Ca and droplet size L associ-ated with microfluidic environment was conducted. The motion of Newtoniandroplets are compared with viscoelastic droplets with equal properties andinitial position. The polymeric stress τ p is plotted to reveal when and howelasticity affects the droplet.

The major mechanism is that when a droplet encounters the junction, thestretching of polymers in the bulk creates tensile stresses to play against further

6.2. VISCOELASTIC DROPLET DYNAMICS IN MICROFLUIDIC DEVICE 31

(a) (b)

Figure 6.2. Time sequences of droplets (Ca = 0.1, L = 0.28) pro-files and polymeric tensile stress. The read line denotes the Newto-nian droplet, the white line denotes the VE droplet of Wi = 1. Thecontours plot the magnitude of τpyy − τpxx. The small chart is theenlarged representation of the bifurcation corner. (a)T = 2.8 whendroplets are deforming (b)T = 3.6 when the Newtonian dropletsplits, figures are taken from Fig.4 in Paper 1, Part II.

(a) (b)

Figure 6.3. Time sequences of droplets (Ca = 0.05, L =0.27)profiles and polymeric tensile stress. The read line denotesthe Newtonian droplet, the white line denotes the VE droplet ofWi = 1. The contours plot the magnitude of τpyy−τpxx. (a)T = 6.5when the VE droplet overtakes the Newtonian droplet (b)T = 8.5when the VE droplet travels in advance, figures are taken from Fig.6in Paper 1, Part II.

deformation. However it has two different outcomes, depending on Ca and L.When a droplet comes close to splitting at the junction, the stress concentratesin the thread which is thinning to breakup, the stress provides extra drag onthe beads so the breakup is delayed in time, see Fig.6.2. This effect is enhancedby using larger Wi but the trend is not monotonic. In the case of a dropletthat does not split and entirely slips into one daughter branch, the stressesconcentrate along the rear of the droplet, where high shear rate across dropletsurface is observed. The droplet becomes more sensitive to the asymmetry offlow condition at the junction. The asymmetric tensile stress pulls the dropletaway from the junction. This makes a viscoelastic droplet slip faster than itsNewtonian counterpart, as shown in Fig.6.3.


At some conditions of Ca and L, we found that the Newtonian dropletsplits while the viscoelastic droplet does not. A set of cases for various circum-stances of droplet size and Ca is conducted. In the end a splitting/non-splittingphase diagram of all viscoelastic and Newtonian droplets is drawn. In the phasediagram the splitting/non-splitting threshold of Newtonian droplets and vis-coelastic droplets are compared.

6.3. Dynamic wetting of viscoelastic droplets

Wetting of polymeric fluids on a flat substrate is a fundamental process inbiomedical, printing, coating and agricultural applications. The presence ofviscoelastic rheology has yet brought rich complexity to the droplet behaviorsuch as spreading and receding on the substrate. However the understandingof viscoelastic effect on spreading is so far very limited. Conclusion on theintuitive question weather elasticity hinders or accelerates a droplet has notbeen reached, reviewing previous studies on this issue. There are also evi-dences showing that the elasticity does not affect the spreading. However, afew uncertainties arises which make the experimental results ambiguous. It isimpossible to avoid viscosity variation that is not caused by rheology, eitherdue to evaporation near the contact line (Berteloot et al. 2008) or due to shearinduced motion of polymer molecules away from the wetted surface (Fang et al.2005; Ma & Graham 2005). There are other factors such as inertia, moleculesgoing out of the drop(Smith & Bertola 2010) and contact angle hysteresis whichcan cover the effects of viscoelasticity. We set up several simulations on wet-ting of Newtonian droplets and viscoelastic droplets, in order to answer thequestion what is the alteration of contact line motion caused by viscoelasticitycoming from the droplet fluid. The results are presented by Paper 2, Paper 3and Paper 4 in Part II.

6.3.1. Spreading without inertia

This study focuses on droplet spreading in microscopic scale that results inRe 1. The complex flow is reduced so that the spreading is only governedby capillarity, viscosity and elasticity. Given identical viscosity and density,the spreading speed of Newtonian droplets is compared to that of viscoelasticdroplets. The simulation uses the Oldroy-B model to represent the Boger fluidand the Giesekus model to mimic a shear-thinning polymeric fluid, e.g., thePEO solution with certain molecular weight and concentration. The simulationis set up in an axisymmertic r-z plane. It solves Stokes equation instead ofNavier Stokes equation, thus inertia completely disappears.

Firstly a full presentation of the bulk droplet is drawn at different timesequences. The spreading of droplets is initialized at the same position on thesubstrate. The droplet profiles and the position of contact line are compared,see Fig.6.4.

The spreading history shows that a Giesekus droplet spreads faster thanits Newtonian counterpart. However it is not clear whether shear-thinning or

6.3. DYNAMIC WETTING OF VISCOELASTIC DROPLETS 33

(a) T=0.1

r/R0 1

z/R

0

0.5

1

1.5

2(b) T=0.5

r/R0 1

z/R

0

0.5

1

1.5

2

(c) T=1.0

r/R0 1

z/R

0

0.5

1

1.5

2(d) T=1.5

r/R0 1

z/R

0

0.5

1

1.5

2

Figure 6.4. Time sequence of droplets spreading on a flat homo-geneous surface. The red solid line represents a Giesekus dropletwith Wi = λσ/Rηl = 5, β = ηs/ηl = 0.2. The black dashed linerepresents a Newtonian droplet with the same ρl, ηl. The staticcontact angle θe = 43, T = tσ/Rηl. Figures are taken from Fig.2in Paper 2, Part II.

elasticity enhances the contact line motion. The flow field and stress distribu-tion near the contact line is then presented. Through analyzing the effectiveviscosity near the contact line, we found that it matches the viscosity variationof a Giesekus fluid under steady shear, see Fig.6.5. We also found that theincrease of Wi does not affect the contact line speed when the shear rate islow so that shear-thinning does not happen, see Fig.6.6(b). We conclude itis shear-thinning rather than elasticity that speeds up the spreading based onthese observations. The observation that shear-thinning enhances contact linemotion agrees with previous experiments. (Carre & Eustache 2000; Rafaı et al.2004).

To detect the role of elasticity, the first normal stress N1 in the droplet isexamined, shown in Fig.6.7. One can see that the stress concentrates in thenear the wall region, while in the bulk droplet the value is almost zero. Sincethe contact line has the largest advancing velocity and the wall is non-slip,one can expect large shear rate at the contact region. Thus N1 is large nearwall because it scales with γ2, where γ denotes the shear rate. To verify theeffect of elasticity, the effect of shear-thinning needs to be excluded. This isachieved by using the purely elastic Oldroyd-B model. As the Oldroyd-B modelcauses numerical difficulties on resolving the polymeric stresses while dropletspreading, the droplet is initialized at a position near its equilibrium stage.


Figure 6.5. Enlarged presentation of the contact line region whileGiesekus droplet spreading. The black solid lines are contour linesfor φ = −0.9 and φ = 0.9. The contours denote the shear rateγ = ∂ur/∂z normalized by σ/Rηl. The small chart plots the ef-fective viscosity on the red dashed line. In the small chart, theblue circles are calculated value from the computed shear rate andviscous stress. The solid line is obtained by substituting the com-puted shear rate into the analytical expression of Giesekus viscosityin steady shear. Figures are taken from Fig.5 in Paper 2, Part II.

Then the elastic effect is examined by comparing the results with Tanner-Cox-Voinov law, see Fig.6.6(a). The results show that both the Oldroyd-B fluid andthe Giesekus fluid obeys the Tanner’s law. The increase of Wi does not affectthe Giesekus model but changes the slope of the Oldroy-B fluid. This impliesthat the elastic effect can be hidden in a shear-thinning fluid. Comparing theslope of each case, the spreading speed in enhanced by increasing Wi. Similarobservations are made in the experiment by (Han & Kim 2013).

6.3.2. spreading droplet after impact

The study investigates the spreading, and receding of the contact line whilea droplet expands and contracts on a solid substrate. With an impact Webernumber We = ρlU

20R/σ 1, where U0 is the impact speed, the spreading

is dominated by inertia. Here one can expect that the effect of stress is notimportant and the elastic effect can easily be covered by inertia.

The simulation is set up in an axisymmetric plane. The droplet is initializedin a distance above the substrate with an imposed vertical velocity towards the


Ca0 0.005 0.01

θ3

/9-θ

e3/9

0

0.005

0.01

0.015(a)

Ca0 0.005 0.01

θ3

/9-θ

e3/9

0

0.005

0.01

0.015(b)

Figure 6.6. Measurement of dynamic contact angle θ at differentcontact line capillary number Ca = Ucηl/σ. Squares are data of theNewtonian droplet, triangles are data for VE droplets of Wi = 5,circles are data for VE droplets of Wi = 45, starts are data for VEdroplets of Wi = 450. Lines of different styles result from linearfitting of each case. (a)Oldroyd-B model (b)Giesekus model, figuresare taken from Fig.6 in Paper 2, Part II

substrate. Gravity is neglected because of the small scale below the capillarylength.

The spreading of Newtonian droplets after impact is analyzed before weexamine viscoelastic fluids. To validate the computation, the results from the


(a) T=0.1

r/R0.24 0.33 0.42

z/R

0

0.04

0.08

0.12

0.16

0

50

100

(b) T=0.5

r/R0.62 0.71 0.8

z/R

0

0.04

0.08

0.12

0.16

0

50

100

Figure 6.7. The first normal stress difference N1 = τrr − τzzscaled by σ/R, at the contact line region while Giesekus dropletspreading. The length of vectors scales with velocity magnitude.Figures are taken from Fig.3 in Paper 2, Part II.

phase field simulation are compared with several experiments predicting spread-ing diameters and a splashing/deposition phase diagram. The results show thatthe simulation agrees well with the experiments, details can be found in PartII, paper 3.

A phase diagram of impact Reynolds number and Ohnesorge number Oh =ηl/√ρlRσ is drawn, together with the plots of droplet shapes and the spreading

time taken to reach a reference spreading diameter. The purpose to use Ohinstead of We is that it highlights the importance of the fluid properties. Thephase diagram shows threshold parameters for different impact regions: theregions with or without gas entrapment and the regions where properties orinertia dominate. One observation made on Fig.6.8 is that when We 1, thespreading time is independent of fluid properties while the spreading time issignificantly influenced by Oh when We < 1.

Four representative cases in different impact regions are then separatelydemonstrated in detail, each case can correspond to a particular case in reality.The occurrence of some particular events such as the formation of a gas cushionand satellite droplets are discussed.

We then proceed to study the impact of viscoelastic droplets. The focus ison the deposition impact which is the subject of the most of previous exper-imental studies. Several studies have suggested viscoelastic effects on droplet


Figure 6.8. Visualizations of drop profiles for the time when itsfootprint reaches r/R = 1 on the substrate, for diverse combi-nations of Oh and Re. r stands for the contact line position onthe solid surface. The number written in white corresponds to thenon-dimensional time t∗ the droplet takes to reach r/R = 1 fromthe beginning of the simulation. The two values written in orangematches with the two cases explained below: (Re = 9, Oh = 1.05)and (Re = 9, Oh = 3.3× 10−1). The grey dashed denotes We = 1.Fig.4 in Paper 3, Part II

impact is not observed, especially in the expanding phase. That is becausethe experimental observation is usually made in situations where inertia dom-inates. For dilute polymeric solutions, the elastic effect only occur when theflow is sufficiently elongated or stretched, which is true in the contact regionrather than in the bulk droplet. This is to say that the viscoelastic effect isunder the cover of the macroscopic dynamics. This study intends to reveal itthrough detailed representation of small scale dynamics.

The simulations are made using parameters matching common rheologicalproperties of aqueous solution of polymers, i.e., shear-thinning viscosity andnormal stress, and impact Weber number We ranges from 70 to 200. TheGiesekus constitutive law is employed to represent dilute polymeric solutions.

Firstly the droplet profiles are compared between the Newtonian fluid andthe Giesekus fluid at different spreading times. The normal stress difference N1

is drawn correspondingly. Fig.6.9 shows selected snapshots of Giesekus dropletsof different Wi against their Newtonian counterparts. The observation madeon these two cases is that the Giesekus droplets move faster than the New-tonian droplet in both the expanding and contracting phase. The discussion


attributes the enhanced spreading to the reduced viscosity in the contact re-gion. The normal stress difference is shown to concentrate in a thin layer abovethe substrate. Also, the macroscopic features in terms of the deformation of thedroplet is analyzed. The morphological deviation from the Newtonian dropletsis summarized as: the Giesekus droplet surface is more wavy, it has a largerspreading diameter and surface area, the film of formed lamella is thinner.These differences stem from shear thinning. The transient deformation ratio ofthe droplet is then measured, the results for different Wi are compared. Thecomparison shows that the deformation rate while droplet spreading reducesgiven larger Wi.

The impact with We = 216 occurs on a super hydrophobic surface. In thiscase the Newtonian droplet splashes at the end of spreading, but the Giesekusdoes not, see Fig.6.9(c)(d). The distribution of N1 in the droplet shows thatstrong tensile stresses exists in the neck area behind the spreading bulb. Thetensile stress eventually prevent breakup of the spreading tip. So far the anal-ysis lies on the macroscopic dynamics in which the elastic effect is reported tobe not visible. However the simulation reveals possible effects that are difficultto observe experimentally.

The contact line speed Uc and apparent contact angle θ are measured at awall distance between the microscopic inner layer and the intermediate layer.The results for Giesekus droplets with different Wi and a Newtonian droplet arecompared with Tanner-Cox-Voinov law, using θ and the contact line capillarynumber based on Uc, Cac = Ucηl/σ. It is found that the Newtonian dropletobviously follows Tanner-Cox-Voinov law in both spreading and contraction,but the VE droplets have multiple scalings at different phases, i.e., the fastspreading phase, the late spreading phase and the contraction phase. Thecontact line behavior in each spreading phase is then analyzed, along withthe rheological properties corresponding to Uc, see Fig.6.10. The observationmade in this figure is that the Giesekus fluid switches its rheological featureaccording to the contact line Weissenberg number Wic = λγ. The fluid behaveselasticity dominated from small contact speed (the black square) to an speedwhich gives Wec = 46. This range covers most of the late spreading phase.Nevertheless the fluid becomes shear-thinning dominated beyond Wic = 46,this range covers the entire fast spreading phase. We propose Wic = 46 as anapproximate turning point for the transition to occur. Connecting with thedata of θ, we see the reason why high Wi alters Tanner’s law is discussed, andthe role of elasticity at the contact line is revealed.

Interestingly the contact line behaves differently in the late spreading phaseand the contraction phase, although the contact line moves at a comparablespeed. To explain the hysteresis, the tensile stress on the contact line at dif-ferent Uc is recorded and compared. The results show that the tensile stress isevidently larger in the spreading phase than in the contraction phase, given thesame Uc. Further examining the flow field around the contact line, we foundthat an extensional flow is present in the late spreading phase, see Fig.6.11(a),


T=1.5

0 1.6 3.2r/R

0

0.63

1.26

z/R

10129

5

772.1 2.2 2.3 2.4

0

0.05

0.1

0.15

(a)

T=8.3

0 1.6 3.2r/R

0

0.26

0.52

z/R

(b)

T=1.8

0 2.265 4.53r/R

0

0.575

1.15

z/R

8553

6921

5

2.5 2.55 2.6 2.65 2.70

0.05(c)

T=6.84

0 2.265 4.53r/R

0

0.285

0.57

z/R

(d)

Figure 6.9. Droplet profiles while spreading on the substrate.(a)(b) correspond to the case of We = 72, the black solid linerepresents the Giesekus droplet of Wi = 0.2, the figures are part ofFig.3 in Paper 4, Part II. (c)(d) correspond to the case of We = 216,the black solid line represents Giesekus droplet of Wi = 0.18, thefigures are part of Fig.5 in Paper 4, Part II. The red dashed linerepresent the Newtonian droplet, contours represent the normalstress difference. The small charts are enlarged presentation of thespreading bulb, the digits in read are values for the contours

when the droplet approaches the maximum spreading radius. The extensionalflow turns to a shear flow in the contraction phase, see Fig.6.11(b). Thus thetensile stress grows much faster in the late spreading phase because of the ris-ing of extensional viscosity. The hysteresis proves that the elastic effect can bealtered by a transient flow geometry during droplet impact, which implies thatexperiments with different surface wettability and fluid properties can result inseemingly contradictory observations.


10 -4 10 -2 10 0 10 2 10 4

λγ

10 -1

10 0

η/η

l

10 -4

10 -2

10 0

10 2

N1R/(U0ηl)

Figure 6.10. Viscoelastic properties of the Giesekus fluid ofWi = 0.5, α = 0.3, β = 0.1. The dashed dotted line represents thenormalized shear viscosity, the dashed line represents N1R/(U0ηl).λγ is the scaled shear rate and also the contact line Weissenbergnumber Wic. The black squares correspond to Uc = 0.01U0, theblue diamonds correspond to Uc = 0.46U0, the red triangles corre-spond to Uc = 1.26U0. Fig.7 in Paper 4, Part II

3 3.05 3.1 3.15r/R

0

0.02

0.04

0.06

0.08

0.1

z/R

(a)

3 3.05 3.1 3.15r/R

0

0.02

0.04

0.06

0.08

0.1

z/R

(b)

Figure 6.11. Transient flow fields in the spreading tip. Vectorsare scaled by the magnitude of Uc. The black solid line repre-sents the droplet profile. Case parameters are We = 72,Wi =0.5, β = 0.1. (a)T = 2.8, Uc = 0.036U0 in the late spreading phase(b)T = 3.1, Uc = −0.019U0 in the contraction phase. The enlargedpresentation of the contact region corresponding to these figures arein Fig.10 of Paper 4, Part II.

CHAPTER 7

Conclusions

Two interfacial phenomena regarding droplet dynamics of viscoelastic(VE) flu-ids are studied in this work, droplet splitting in a capillary y-shaped junctionand spreading of droplets on a wetted substrate. For the latter, both spreadingwithout inertia and spreading after droplet impact are considered. The purposeis to explore the effects of general non-Newtonian features on similar issues ofNewtonian fluids and answer remaining questions from the literature.

The first problem arises from droplet-based microfluidic applications whichutilize droplet splitting/non-splitting behavior. In a microfluidic environment,capillarity and elastic stress is important for the evolution of the interface be-tween the droplet fluid and the media fluid. A Newtonian droplet with weakcapillarity is forced to stretch and eventually split at the junction tip, formingtwo smaller droplets that migrate into each daughter channel separately there-after; while stronger capillarity makes the droplet entirely slips into one of thedaughter channels. With the presence of elasticity, the deformation and split-ting of a droplet is suppressed, due to the elastic tensile stress in the droplet.The critical capillary number for a droplet to split increases with the degreeof elasticity. Elasticity also enhances the unstable nature of a non-splittingdroplet at the junction tip, the time period for a droplet to hang on the junc-tion tip is reduced, compared to the Newtonian droplet. This viscoelasticdroplet behavior can initialize a new approach for cell sorting.

Dynamic wetting of dilute polymeric solutions is not well understood sofar, and the role of elasticity on droplet spreading is not yet clear. Whetherelasticity accelerates or hinders the contact line, there are experimental evi-dence for both sides (Guemas et al. 2012; Han & Kim 2013; Smith & Bertola2010). The simulation in this work indicates that the contact line motion isenhanced by the degree of elasticity for both Boger fluids and shear-thinningfluids. Nevertheless the elastic effect can be restricted in different ways duringdroplet spreading. Shear-thinning as another common feature of polymeric flu-ids can overtake the elastic effect. Also the presence of the elastic effect requiresa matching of polymeric relaxation time with the the characteristic time of theflow near a contact line, in agreement with (Yue & Feng 2012). Both the shear-thinning fluids and the Boger fluids follow Tanner-Cox-Voinov law for dropletreceding and spreading without inertia. Deviation from Tanner-Cox-Voinovlaw is observed in the late spreading during droplet impact. This deviation isattributed to a transient variation of the flow geometry in the contact region.

41

42 7. CONCLUSIONS

The change of flow geometry causes elastic stresses to grow differently duringdroplet expansion and contraction, which brings a hysteresis of the elastic effecton a contact line.

These multiphase problems are investigated using numerical simulationwhich involves solving Navier-Stokes (NS) equation, viscoelastic constitutivelaws and Cahn-Hilliard (CH) equation introduced by the phase field method.Advanced numerical schemes for computing a Maxwell type of constitutive lawis integrated into the NSCH system. The combination of a diffuse interfaceand stabilization scheme allows us to obtain detailed rheological informationin a divergent geometry and near a contact line, where numerical instabilityusually takes place for viscoelastic flow. The phase field simulation in this workprovides good predictions on droplet deformation and droplet impact dynamics.

We expect more interesting results of interfacial dynamics with viscoelasticrheology, and the accumulation of exciting results will never stop. There area few suggestions on direct extension of this study. The viscoelastic dropletin a viscoelastic medium or or a Newtonian droplet in a viscoelastic mediumcan be observed in the same setup of bifurcation channel or other geometriessuch as a T junction. Rich possibilities can arise when the elastic stress fromthe ambient flow acts on a soft body. Inertia seems to be able to flip over theelastic effect on the contact line motion. This can be further investigated bylooking into the hysteresis mentioned above. One hypothesis is that if the timescale of the extensional flow is much shorter than the polymeric relaxation timeTanner-Cox-Voinov law will be recovered in the late spreading stage. In thefast spreading phase, the droplet does not spread as fast as expected for shear-thinning fluids. This implies that elasticity hinders the contact line motion.How the elastic drag weighs over capillarity and viscosity on will decide theacceleration or deceleration of a contact line, some universality may be devel-oped accordingly, using common assumptions such as the lubrication theory inthe contact region.

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Acknowledgements

I’m very grateful to Prof. Gustav Amberg for his supervision and initializationon this work and the profound benefits obtained from his deep intuition onphysics, mathematics and sparking research ideas. The greatest skill learnedfrom Gustav is to keep an optimistic attitude to new challenges and alwaysdiscover the positive aspects from outcomes, which not only helps me to ac-celerate on the road of science but also brings happiness to life. My gratitudealso goes to Dr. Minh Do-Quang, who serves as the co-supervisor of my PhDeducation, for the countless discussions on numerical implementation and hisdedicated guide for me to learn FemLego. Minh is always problem-solving andefficient when I encountered technical difficulties in simulating complex fluids.

I would like to thank my former and current colleagues in Linne FLOWCenter, Dr. Andreas Carlson and Dr. Lailai Zhu for their assistance and valu-able materials provided in the initial stage of this project, Prof. Laszlo Fuchsand Dr. Mihai Mihaescu for their supervision efforts in the beginning of myPhD study, Dr. Lisa Prahl Wittberg for her feedbacks on this thesis, AnthonyGratadeix for his contribution to the third article, Dr. Anders Dahlkild for hisadministrative assistance on my study.

I wish to thank Prof. Minguan Yang at Jiangsu University for his continu-ous support on my PhD education at KTH. I also appreciate the contributionfrom Dr. Bo Gao and Mr. Chen Gong to the project that I’m responsible forin China. Their dedicated work spares me more time to complete this thesis.Besides, I also would like to acknowledge funding from the China ScholarshipCouncil, National Nature Science Foundation of China and KTH.

It has been a long journey since I started my postgraduate study. I owedmy deepest gratitude to my family, my parents for their selfless support thatmakes every achievement of mine possible, my wife, Caijuan Zhan for all theseyears we’ve been together, it’s her love that strengthes me and stimulates meto overcome any hardship in life.

Part II

Papers

Documents

Capillarity and wetting of non-Newtonian droplets915139/... · 2016. 3. 29. · non-Newtonian uids,(b) the Weissenberg e ect and (c) the Barus e ect. and inkjet printing. For emerging