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Shear rheology of non-Newtonian emulsions made of viscoelastic droplets in a Newtonian matrix Mohammad, Hosseini 2020

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Shear rheology of non-Newtonianemulsions made of viscoelasticdroplets in a Newtonian matrixbyMohammad HosseiniB.Sc., Amirkabir University of Technology, 2018A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Chemical and Biological Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2020© Mohammad Hosseini 2020AbstractIn the case of an oil spill in a marine environment, an important aspectof an early response is to confine the oil spill and prevent it from gettingdispersed. At the laboratory level, chemists are capable of designing newsolidifiers known as “gellants”, that when applied by spraying or injectionin the marine oil-spill, are capable of making the oil regroup and create agel-like material (gelled-oil). Even though gellants make the removing ofoil-spill from a marine environment easier, it is not clear whether gelled-oilis more stable in terms of break-up and how gellants affect the dispersionrate of the oil-spill.Gellants create a gelled-oil emulsion in the marine environment, whichcan be modelled as a dispersion of non-Newtonian droplets (gelled-oil) ina Newtonian matrix (sea water). In this study we develop a better under-standing of such a liquid/liquid system subjected to imposed shear. Boththe rheological nature of the oil and the oil/water surface tension are the keyparameters. We address this problem from a computational viewpoint us-ing an advanced open source academic code to perform parallel simulations.We analyze two main problems: 1) the deformation of a single droplet ina simple shear flow, and 2) the rheological behaviour of an emulsion in asimple shear flow. We show that, since applying gellants increase the sur-face tension and viscosity of the dispersed oil phase, the oil-water emulsionis more stable (i.e. the dispersed oil droplets deform less) and the dispersionrate of the oil in the marine environment is reduced. Moreover, we obtainresults showing that the elasticity of the dispersed gelled-oil phase has a non-monotonic impact on the flow features and a very limited influence on thegeneral behaviour of the emulsion. The conducted analysis in this projectand its outcome can help to provide recommendations on how these gelledmaterials behave.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Oil spills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Marine oil spills . . . . . . . . . . . . . . . . . . . . . 11.1.2 Responding to a marine oil spill . . . . . . . . . . . . 21.1.3 Gellants . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 Droplet deformation . . . . . . . . . . . . . . . . . . . 41.2.2 Emulsions . . . . . . . . . . . . . . . . . . . . . . . . 111.3 Summary of the literature review . . . . . . . . . . . . . . . 151.3.1 Single droplet . . . . . . . . . . . . . . . . . . . . . . 151.3.2 Emulsions . . . . . . . . . . . . . . . . . . . . . . . . 152 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Problem description and flow configuration . . . . . . . . . . 172.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . 182.2.1 Single fluid flow . . . . . . . . . . . . . . . . . . . . . 192.2.2 Two-fluid formulation with surface tension . . . . . . 212.2.3 Mixture formulation with surface tension written as avolumetric force . . . . . . . . . . . . . . . . . . . . . 232.2.4 Viscoelastic fluid . . . . . . . . . . . . . . . . . . . . . 25iiiTable of Contents2.2.5 Flow properties of interest and dimensionless numbers 262.3 Computational method and scientific software . . . . . . . . 282.4 Analysis of the computed results . . . . . . . . . . . . . . . . 303 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.1 Grid convergence . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Single droplet . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.1 Newtonian fluids . . . . . . . . . . . . . . . . . . . . . 353.2.2 Non-Newtonian fluids . . . . . . . . . . . . . . . . . . 393.3 Emulsion of droplets . . . . . . . . . . . . . . . . . . . . . . . 404 Results: Single Droplet Deformation . . . . . . . . . . . . . 434.1 Newtonian droplets . . . . . . . . . . . . . . . . . . . . . . . 444.1.1 Two-dimensional cases . . . . . . . . . . . . . . . . . 444.1.2 Three-dimensional cases . . . . . . . . . . . . . . . . 524.2 Two-dimensional case vs. three-dimensional case . . . . . . . 564.3 Viscoelastic droplets . . . . . . . . . . . . . . . . . . . . . . . 584.3.1 Two-dimensional cases . . . . . . . . . . . . . . . . . 584.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 Results: Emulsions . . . . . . . . . . . . . . . . . . . . . . . . . 685.1 Newtonian Emulsions . . . . . . . . . . . . . . . . . . . . . . 695.1.1 Two-dimensional emulsion . . . . . . . . . . . . . . . 695.1.2 Three-dimensional domain . . . . . . . . . . . . . . . 765.2 Non-Newtonian Emulsions . . . . . . . . . . . . . . . . . . . 815.2.1 Two-dimensional emulsion . . . . . . . . . . . . . . . 815.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.1 Single droplet . . . . . . . . . . . . . . . . . . . . . . . . . . 926.2 Emulsion of droplets . . . . . . . . . . . . . . . . . . . . . . . 936.3 Contribution to the industrial problem . . . . . . . . . . . . 946.4 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100ivList of Tables3.1 The results of this study and the results obtained by Zhouand Pozrikidis [97]. SD stands for standard deviation. . . . . 425.1 Results of this study together with Eilers’ correlation . . . . . 76vList of Figures1.1 (a) An oil boom confining the spreading of an oil slick (b)In situ burning of an oil spill on the water surface using fireresistant booms . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Schematic of a droplet immersed in an immiscible fluid un-dergoing simple shear flow between parallel plates. . . . . . . 51.3 Time evolution of the droplet deformation for λ = 10. Re-produced from Sheth and Pozrikidis [73]. . . . . . . . . . . . . 71.4 (a) Transient droplet deformation at Ca = 0.4 and R/H =0.07 (left sequence) and R/H = 0.5 (right sequence) at λ = 1(b) Major axis(L) relative to droplet diameter vs dimension-less time during flow startup (inset shows monotonous tran-sient in unbounded shear flow). Reproduced from Sibillo etal. [75]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Images of deformed droplets under steady shear flow. Theleft side is for a Newtonian system, the right side is for a non-Newtonian one. Both systems have the same Ca = 0.15 andλ = 0.1. Top images are the side views, bottom images areviews along the shear gradient direction. Reproduced fromGuido et al. [35]. . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Steady-state droplet deformation D (normalized against thevalue for Newtonian cases, De = 0) varying with De. Repro-duced from Aggarwal and Sarkar. [3]. . . . . . . . . . . . . . 101.7 Summary of some of the published rheological models andtheir range of applicability with respect to particle volumefraction ψ, viscosity ratio λ and capillary number Ca. Re-produced from Faroughi and Huber. [26]. . . . . . . . . . . . 121.8 Variation of viscoelastic shear stress with capillary numberCa. Reproduced from Aggarwal and Sarkar. [4]. . . . . . . . 142.1 Single-droplet and multiple-droplet systems examined. . . . . 192.2 Adaptive mesh refinement for the case of a falling droplet. . . 30viList of Figures2.3 Schematic of a deformed single droplet under the simple shearflow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1 Convergence study - time evolution of the droplet deforma-tion for Re = 0.05, Ca = 0.5, and λ = 1 at different gridresolutions. ∆x represents the smallest grid size. . . . . . . . 343.2 Convergence study - time evolution of droplet deformation forRe = 0.05, Ca = 0.5, and λ = 1 at different grid resolutions.∆x represents the smallest grid size. . . . . . . . . . . . . . . 353.3 Time evolution of the droplet deformation with λ = 1, (a)Re =1, Ca = 0.2, 0.4, 0.9;(b)Re = 10, Ca = 0.2, 0.4, 0.8;(c)Re =50, Ca = 0.2, 0.3, 0.4;(d)Re = 100, Ca = 0.05, 0.15, 0.2, 0.4 -dotted and solid lines correspond to the results from Shethand Pozrikidis [73] and this study, respectively. . . . . . . . . 373.4 Results of this study and of [28]. (a) breadth and (b) lengthof the deformed droplet. λ = 0.335, Re = 0.05 . . . . . . . . . 383.5 Results of this study on single droplet deformation in threedimensions and results obtained by [28]. λ = 0.335, Re = 0.05 393.6 Time evolution of the single droplet deformation when eitherthe dispersed or continuous phase is a viscoelastic fluid; Re =2.4, Ca = 1.2, De = 0.4(β = 0.5), λ = 1 . . . . . . . . . . . . 403.7 The evolution of the shear effective viscosity (µeff ) in time fora Newtonian emulsion and different random initial position ofthe droplets; φ = 0.2945, Ca = 0.5, λ = M = 1. . . . . . . . . 414.1 Effect of height(Ly) and length(Lx) of the channel on thedeformation of a single Newtonian droplet. . . . . . . . . . . 444.2 Effect of the initial condition (IC) on the deformation of asingle Newtonian droplet. . . . . . . . . . . . . . . . . . . . . 464.3 Effect of Capillary number Ca and viscosity ratio λ on the de-formation of a single Newtonian droplet - domain size: a)4×4, b)8× 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.4 Comparison of results with the correlation proposed by Tay-lor (equation 4.1) for the deformation parameter of a singleNewtonian droplet for different viscosity ratios λ and capil-lary numbers Ca - domain size: 8× 8 . . . . . . . . . . . . . 484.5 Effect of Capillary number Ca and viscosity ratio λ on theinclination angle of a single Newtonian droplet in a simpleshear flow - domain size: a)4× 4 , b)8× 8 . . . . . . . . . . 49viiList of Figures4.6 Snapshots of the steady-state deformed droplets. The firstand second rows correspond to λ = 0.5 and Ca = 1, re-spectively. The colour contour represents the adaptive meshrefinement such that the darker colours represent finer gridcells. domain size: 8×8 . . . . . . . . . . . . . . . . . . . . . 504.7 Effect of the Reynolds number Re at different capillary num-bers Ca on the deformation of a single Newtonian droplet ina Newtonian matrix - domain size: 8× 8 . . . . . . . . . . . . 514.8 Effect of Reynolds number Re at different Capillary numbersCa on the inclination angle of a single Newtonian droplet ina Newtonian matrix - domain size: 8× 8 . . . . . . . . . . . . 524.9 Effect of Lz on the deformation parameter D of a single three-dimensional Newtonian droplet. Lx = Ly = 16. . . . . . . . . 534.10 Effect of the height (Ly) and the length(Lx) of the channelon the deformation parameter of a single three-dimensionalNewtonian droplet. Lz = 2. . . . . . . . . . . . . . . . . . . . 544.11 Effect of the capillary number Ca on the deformation of sin-gle droplet in three-dimensional domain for different viscosityratios λ. Lx = Ly = Lz = 8. . . . . . . . . . . . . . . . . . . . 554.12 Effect of the Reynolds number Re and the capillary numberCa on the deformation of a single Newtonian droplet in three-dimensional domain. Lx = Ly = Lz = 8. . . . . . . . . . . . . 564.13 (a) Breadth B (b) Length L, in two and three dimensionstogether with the results obtained in [28]. λ = 0.335, Re = 0.05. 574.14 The deformation parameter D of a single droplet two andthree dimensions together with the results obtained in [28].λ = 0.335, Re = 0.05 . . . . . . . . . . . . . . . . . . . . . . . 584.15 Effect of height(Ly) and length(Lx) on the deformation of asingle viscoelastic droplet in a Newtonian matrix in a simpleshear flow. Re = 0.05, Ca = 0.2, De = 0.4, β = 0.5, λ = 0.335. 594.16 Effect of the capillary number Ca for different viscosity ratiosλ on the deformation parameter D of a single viscoelasticdroplet in a Newtonian matrix. Re = 0.05, De = 0.4, β = 0.5. 604.17 Effect of the capillary number Ca for different viscosity ratiosλ on the inclination angle θ of a single viscoelastic droplet ina Newtonian matrix. Re = 0.05, De = 0.4, β = 0.5. . . . . . . 614.18 Effect of the capillary number Ca for different Reynolds num-bers Re on the deformation parameter D of a single viscoelas-tic droplet in a Newtonian matrix. λ = 1, De = 0.4, β = 0.5. 62viiiList of Figures4.19 Effect of the capillary number Ca for different Reynolds num-bers Re on the deformation parameter D of a single viscoelas-tic droplet in a Newtonian matrix. λ = 1, De = 0.4, β = 0.5. 634.20 Effect of the Deborah number De for different values of β onthe deformation of a single viscoelastic droplet in a Newtonianmatrix in a simple shear flow. Re = 0.05, Ca = 0.2, λ = 1. . . 644.21 Effect of the Deborah number De for different values of β onthe deformation of a single viscoelastic droplet in a Newtonianmatrix in a simple shear flow. Re = 0.05, Ca = 1, λ = 1. . . . 654.22 Deformation parameter D from a work of Pengtao et al [96]together with the D obtained from this study as a functionof the Deborah number De. β = 0.5, Re = 0.05, Ca = 0.2,and λ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.1 Effective viscosity µeff of a two-dimensional Newtonian emul-sion as a function of the capillary number Ca for differentvolume fractions φ. Re = 0.05 and λ = 1. . . . . . . . . . . . 695.2 (a) Average deformation Davg and (b) average inclinationangle θavg of droplets in a two-dimensional Newtonian emul-sion as a function of the capillary number Ca for differentvolume fractions φ. Re = 0.05 and λ = 1. . . . . . . . . . . . 705.3 Snapshots of Newtonian emulsions at λ = 1, Ca = 0.5 anddifferent volume fractions φ. φ = (a) 5% (b) 10% (c) 40%(d) 50%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.4 Effective viscosity µeff of a two-dimensional Newtonian emul-sion as a function of viscosity ratio λ for different volumefractions φ. Re = 0.05 and Ca = 0.2. . . . . . . . . . . . . . . 735.5 (a) Average deformation Davg and (b) average inclinationangle θavg of the droplets in a two-dimensional Newtonianemulsion as a function of viscosity ratio λ for different volumefractions φ. Re = 0.05 and Ca = 0.2. . . . . . . . . . . . . . . 745.6 Eilers and Einstein correlation together with some of ourresults.(a) Ca = 0.2, λ = 2 (b) Ca = 0.2, λ = 5 (b)Ca = 0.05, 0.1, λ = 1. . . . . . . . . . . . . . . . . . . . . . . 755.7 Effective viscosity µeff of a three-dimensional Newtonian emul-sion as a function of the capillary number Ca for differentvolume fractions φ. Re = 0.05 and λ = 1. . . . . . . . . . . . 77ixList of Figures5.8 (a) Average deformation Davg and (b) average inclinationangle θavg of the droplets in three-dimensional Newtonianemulsion as a function of the capillary number Ca for dif-ferent volume fractions φ. Re = 0.05 and Ca = 0.2. . . . . . . 785.9 A snapshot of droplet position, velocity and grid cells for theemulsion with Ca = 0.7 and φ = 5%. . . . . . . . . . . . . . . 795.10 Effective viscosity µeff of a three-dimensional Newtonian emul-sion as a function of the viscosity ratio λ for different volumefractions φ. Re = 0.05 and λ = 1. . . . . . . . . . . . . . . . . 805.11 (a) Average deformation Davg and (b) average inclinationangle θavg of droplets in three-dimensional Newtonian emul-sion as a function of viscosity ratio λ for different volumefractions φ. Re = 0.05 and Ca = 0.2. . . . . . . . . . . . . . . 805.12 Effective viscosity of an iso-viscous viscoelastic emulsion (De =0.4 and β = 0.5) as a function of the capillary number Ca fordifferent volume fractions φ. . . . . . . . . . . . . . . . . . . . 825.13 (a) Average deformation Davg and (b) average inclinationangle θavg of iso-viscous viscoelastic emulsion (De = 0.4 andβ = 0.5) as a function of the capillary number Ca for differentvolume fractions φ. . . . . . . . . . . . . . . . . . . . . . . . . 835.14 Effective viscosity of a viscoelastic emulsion as a function ofviscosity ratio λ for different volume fractions φ. Re = 0.05,Ca = 0.2, De = 0.4, and β = 0.5. . . . . . . . . . . . . . . . . 845.15 (a) Average deformation Davg and (b) average inclinationangle θavg of a viscoelastic emulsion as a function of viscosityratio λ for different volume fractions φ. Re = 0.05, Ca = 0.2,De = 0.4, and β = 0.5. . . . . . . . . . . . . . . . . . . . . . . 845.16 Effect of the Deborah number De on the effective viscosityµeff of a viscoelastic emulsion for different volume fractionsφ. Re = 0.05, Ca = 0.2, and λ = 1. . . . . . . . . . . . . . . . 865.17 (a) Average deformation Davg and (b) average inclinationangle θavg of a viscoelastic emulsion as a function of the Deb-orah number De for different volume fractions φ. Re = 0.05,Ca = 0.2 and β = 0.5. . . . . . . . . . . . . . . . . . . . . . . 875.18 Effect of the Deborah number De on the effective viscosityµeff of a viscoelastic emulsion for different values of β. Re =0.05, Ca = 0.2, λ = 1, φ = 10%. . . . . . . . . . . . . . . . . . 88xList of Figures5.19 (a) Average deformationDavg and (b) average inclination an-gle θavg of a viscoelastic emulsion as a function of the Deborahnumber De for different values of β. Re = 0.05, Ca = 0.2,λ = 1, φ = 10%. . . . . . . . . . . . . . . . . . . . . . . . . . . 895.20 Effect of the Deborah number De on the effective viscosityµeff of a viscoelastic emulsion for different values of β. Re =0.05, Ca = 0.2, λ = 1, φ = 50%. . . . . . . . . . . . . . . . . . 905.21 (a) Average deformationDavg and (b) average inclination an-gle θavg of a viscoelastic emulsion as a function of the Deborahnumber De for different values of β. Re = 0.05, Ca = 0.2,λ = 1, φ = 50%. . . . . . . . . . . . . . . . . . . . . . . . . . . 906.1 Schematic of the gelation process. . . . . . . . . . . . . . . . . 946.2 Schematic of stratified flow of oil on the water surface expe-riencing oscillating shear stress. . . . . . . . . . . . . . . . . . 966.3 Schematic of an oil slick on the water surface experiencingextensional stresses. . . . . . . . . . . . . . . . . . . . . . . . 976.4 Schematic of droplets aggregation. (a) aggregated mode (b)broken-up mode after applying strong enough shear stress. . . 986.5 A snapshot of the aggregation of droplets used for modellingthe gelation process. The colour contour represents the x-component of the velocity. . . . . . . . . . . . . . . . . . . . . 98xiAcknowledgementsI would like to express my sincere gratitude to my supervisor, Professor An-thony Wachs. Without his never-ending support and help, I would not havebeen able to finish this master’s thesis. The door to Prof. Wachs’ office wasalways open whenever I had a question about my research or anything else.He always helped and steered me patiently in the right direction wheneverI needed it. I learned a lot from him; from critical thinking to how to bepassionate about the research and stay motivated when I got disappointed.I want to acknowledge Mitacs and BC-Research Inc. for supporting thismaster’s project financially.I would also like to appreciate the help from my colleagues, AashishGoyal, Arman Seyed-Ahmadi, Arthur Ghigo, Arun Rajendran, Can Selcuk,Damien Huet, Mona Rahmani and Zihao Cheng. I learned lots of new thingsfrom all the discussions I had with them, and they always listened to myproblems patiently.I owe particular thanks to my dearest friends who always have myback, Roza Vaez Ghaemi, Parisa Chegounian, Emma Moreside, ChristineBeaulieu, and many others who always supported and encouraged me inmany ways. Special thanks to my parents and my sisters, who have sup-ported me throughout my years of education, and for providing me withyour unconditional love.xiiDedicationTo My Lovely ParentsMy Beautiful SistersAndArta, My Sweet Nephew ;)xiiiChapter 1Introduction1.1 Oil spillsIn this section we introduce oil spills, discuss how they affect the environmentand list some of the methods to deal with this disastrous phenomenon. Oilspills can happen in different places, e.g., on land, on glaciers or in themarine environments. However, the interest of this study is limited to themarine environment.1.1.1 Marine oil spillsMarine oil spills are the most important threat to the ecosystem of the seaand the coastal environment. Not only they destroy the marine habitatsfor its inhabitants, but they also have a significant role in the loss of energyand its economy [9]. They are initiated mainly by unpredictable incidents ofsupertankers, oil rig drilling, natural disasters like hurricanes or floods, andmay happen due to war [18, 27]. For instance, in 2010 an unprecedentedamount of oil (∼4.0 million barrels) was released at a 1,500 meter depth intothe Gulf of Mexico after the blowout of the Deepwater Horizon drilling rig[69].This huge amount of oil in the seawater will remain for a long time andendanger the inhabitants’ life. Removing the oil from the oceans is reallycrucial and costly. Referring to the example of the Gulf of Mexico, the effortto recover the discharged oil reduced the oil in the marine environment to3.2 million barrels only (∼20% of the oil was removed)[49, 70]. Around 9%was evaporated or burnt and the residues of the burning remained floatingon the water surface for a while until dumped on the seafloor [1, 68, 79].The rest of the leaked oil formed surface slicks that were transported viathe formation of large marine snow or formed deep plumes of dissolved anddispersed hydrocarbons [12, 19, 49, 68, 70].Based on the numbers regarding the Gulf of Mexico, it is clear that alarge portion of oil spills will be resolved by sedimentation on the seabed.This is actually not a solution as it causes new and more difficult problems11.1. Oil spillsto be solved later. Since this sedimentation takes place gradually, we havea limited time to prevent it from occurring. There are different methodswhich are being used for remediating the marine oil spills. They are brieflydiscussed in the following section.1.1.2 Responding to a marine oil spillNowadays there are several methods for dealing with an oil spill incident.Generally these methods are classified into four major categories: (a) me-chanical/physical (b) chemical (c) thermal and (d) biological [20].Mechanical/PhysicalPhysical remediation methods are commonly used to confine the propaga-tion of oil in the water environment. They are mainly used as physicalbarriers without changing the oil chemical or physical characteristics. As anexample of these barriers, we can mention: 1) oil-booms (see figure 1.1-a)2) skimmers and 3) adsorbent materials [29, 84].ChemicalChemical methods for oil spill remediation are mostly used in combinationwith physical techniques and their goal is to change the physical and chem-ical properties of the oil, to slow down the spread of oil spill [84]. Thechemicals used to control oil spills include: 1) dispersants and 2) solidifiers[20].ThermalThermal remediation methods are techniques to remove the oil slick fromthe water environment using thermal energy. In situ burning (see figure 1.1-b) is a simple and quick thermal method for this purpose that proceeds withminimal specialized equipment (fire-resistant booms, igniters). Evaporationis another natural thermal means of oil spill remediation that is slower andcannot be relied on.BioremediationBioremediation is a process whereby micro-organisms degrade and metabo-lize chemical substances and restore environmental quality. The goal is to21.1. Oil spillsaccelerate the natural attenuation process through which micro-organismsassimilate organic molecules to cell biomass [5].(a) (b)Figure 1.1: (a) An oil boom confining the spreading of an oil slick (b) Insitu burning of an oil spill on the water surface using fire resistant booms1.1.3 GellantsAll of the aforementioned methods in the previous section for recovering amarine environment after an oil spill can improve the situation and thereare pros and cons with each of them. However, due to the interest of thisstudy, we focus on the solidifiers that in the present study will be referredto as “gellants”.Gelling agents, gellants, or gelators are considered as a subclass of so-lidifiers that convert the oil into a semi-solid viscoelastic state [38]. Afterapplying gellants to the oil, we have a system containing a gelled phase(gelled-oil) in a marine environment. Since gelled-oil has a higher viscosityand it is more like a rigid phase (in comparison with oil), removing it from amarine environment is easier. However, because gelled-oil has higher surfacetension and exhibits elastic properties, it is not clear that how these prop-erties affect the stability and the dispersion rate of the oil-spill. As stabilityof the disperssed phase and its dispersion rate are important parametersfor a response to an oil-spill, there is a need to make sure that gellants areable to improve the situation in terms of these parameters. The system ofgelled-oil and water can be modelled as an emulsion of viscoelastic particles(gelled-oil), in which particle interactions play an important role, and basedon the experimental observations, the probability of coalescence is low.31.2. Literature reviewIn this study, we want to get a better understanding on how a systemcontaining gelled-oil and water behave in a shear flow, to decide if gellantsare able to improve the situation in terms of increasing the stability andreducing the dispersion rate. Therefore, our interest is to investigate therheological behaviour of an emulsion of viscoelastic droplets (gelled-oil) in aNewtonian fluid (water).1.2 Literature reviewAs mentioned in the previous section, the main objective of this project is tostudy the rheological behaviour of emulsions made of viscoelastic dropletsin a Newtonian matrix. Before starting to study this problem, it would behelpful to look at the literature and published articles in this area to get abetter insight into this problem.One of the model problems in emulsion technology is investigating howa single droplet with specific properties behaves and deforms under a simpleshear flow [10]. These investigations are motivated by several fundamentaland practical applications in different fields. Understanding the type andmagnitude of droplet deformation is mandatory for providing thresholds forexcessive elongation and rupture of the droplets, leading to changes in theemulsion micro-structure that plays an essential role in the physical andrheological properties of the emulsion [15, 73, 83].1.2.1 Droplet deformationThe deformation of an isolated droplet under shear flow is a topic of fun-damental interest starting from the pioneering work of Taylor [82, 83].Even though looking at a single droplet simplifies the problem and doesnot account for droplet-droplet interactions and coalescence phenomena, itnonetheless provides valuable insight into the rheological behavior of emul-sions [90]; for instance by predicting the average droplet deformation andthe critical breakup conditions in the emulsion at a different volume concen-tration of the dispersed phase [13, 40]. In Taylor’s analysis [82, 83] there arefive physical parameters: shear rate γ˙, droplet radius R, matrix and dropletviscosities µc and µd, respectively, and interfacial tension σ. Taylor assumedthat the density of the two phases is the same and the fluids are Newtonianand immiscible with a clean interface (no interfacial agents).In Taylor’s studies, in which he assumed a Stokes flow (Re = 0), theaforementioned physical quantities are merged in two dimensionless num-bers, i.e., the capillary number Ca ≡ µdγ˙dpσ , representing the ratio of the41.2. Literature reviewdroplet deforming shear stress divided by the restoring action of interfacialtension, and the viscosity ratio λ ≡ µdµc , ranging from zero for bubbly flowsto infinity for rigid particle suspensions. Taylor also introduced a param-eter for droplet deformation, known as “Taylor deformation”, D ≡ L−BL+B ,where L and B are the longest diameter and smallest diameter of the de-formed droplet, respectively (see figure 1.2). When the droplet is deformed,it inclines toward the flow direction with an orientation angle θ betweenthe major axis L and the flow direction. Taylor’s approach is based ona small deformation perturbation procedure to first-order, with the capil-lary number as the expansion parameter. Taylor obtains a correlation forthe deformation parameter D = 19λ+1616λ+16Ca2λ , that is linearly dependent onthe capillary number Ca and has a weak dependence on the viscosity ra-tio λ. After Taylor’s investigation, other researchers improved his analysisby extending the perturbation procedure to the second-order in Ca. Thesecond-order perturbation solution does not change the correlation for thedeformation parameter but gives the following equation for the orientationangle: θ = pi4 − Ca (16+19λ)(3+2λ)160λ(1+λ) [7, 14, 32, 67].LBLyUϴLx(a)Figure 1.2: Schematic of a droplet immersed in an immiscible fluid under-going simple shear flow between parallel plates.For the case of highly deformable droplets, important progress has beenmade numerically through the use of the boundary integral method [41]and other methods such as volume-of-fluid [44]. Also by using an adaptive51.2. Literature reviewdiscretization of the droplet interface to have uniform grid resolution atincreasing deformation, other authors predicted a critical breakup point ofa droplet in agreement with experimental results [16, 17].The literature also features several articles studying droplet deformationand break up in two dimensions. The situation is different in two dimen-sions, and while they are not realizable directly in the laboratory experi-ments, these cases are more accessible by computer simulations [73]. Theinteresting part of two-dimensional studies is that beside breakup, dropletscould undergo breakdown which leads to the deformation of a long neckthat might in principle elongate and thin infinitely [88]. Wanger et al. [88]investigated the effect of inertia on the deformation and breakdown of atwo-dimensional fluid droplet in a fluid with a viscosity ratio λ = 1. Shethand Pozrikidis [73] conducted a study of shear flow past a one-dimensionalarray of two-dimensional viscous droplets at small and moderate Reynoldsnumbers (up to Re = 100), different capillary numbers and viscosity ratios.Figure 1.3 show their results on the evolution of droplet deformation withtime for a viscosity ratio λ = 10 and various pairs of (Re, Ca), from the ini-tial circular shape to the final deformed shape using a variation of Peskin’simmersed boundary method [63].61.2. Literature review(a)Figure 1.3: Time evolution of the droplet deformation for λ = 10. Repro-duced from Sheth and Pozrikidis [73].The majority of the conducted studies assumes an unbounded simpleshear flow, i.e., the distance between the droplet’s centre and the confiningwalls is high enough such that the wall-effect does not have any impact onthe deformation. However, because of the growing interest in microfluidictechnologies, studying the confined flow has also attracted attention in theliterature [37, 78].Shapira and Haber [72] theoretically addressed the problem of dropletdeformation under confined shear flow at first-order. They applied Lorentz’sreflection method up to the third reflection, which means that their resultsare only valid for a moderate degree of confinement. They concluded thatthe deformation of a droplet in a confined shear flow can be obtained bythe unbounded flow expression through a correction in the third power ofthe ratio between droplet radius and gap between the walls (RH )3. Sixteenyears later, Sibillo et al. [75] validated Shapira and Haber’s analysis ex-perimentally through a sliding plate apparatus coupled to a video opticalmicroscope. Figure 1.4 show their results on bounded vs unbounded dropletdeformation.71.2. Literature review(a) (b)Figure 1.4: (a) Transient droplet deformation at Ca = 0.4 and R/H = 0.07(left sequence) and R/H = 0.5 (right sequence) at λ = 1 (b) Major axis(L)relative to droplet diameter vs dimensionless time during flow startup (insetshows monotonous transient in unbounded shear flow). Reproduced fromSibillo et al. [75].The problem of droplet deformation becomes more complicated when atleast one of the phases is a non-Newtonian fluid. In order to distinguishelastic from shear-thinning contributions, most experiments have been per-formed with Boger fluids [11]. With all the simplifications, the theoreticalproblem is still challenging. A general solution to this problem has beenprovided with a slow flow assumption in the theoretical analysis of Greco[31]. The limitation of this analysis is that all the predictions are for steadystate flows [34]. Because of this limitation and the interest in transient flows,modified ellipsoidal models for non-Newtonian fluids have been developed[48, 52].Greco’s theory for the cases of a Newtonian droplet in a viscoelasticmatrix is that the deformation parameter is unchanged with respect to fullNewtonian cases, while the orientation angle is enhanced significantly [31].Later experimental investigations by Guido et al. [35, 36] supported thepredictions made by Greco’s analysis [31] for a Newtonian droplet in a vis-coelastic Boger fluid at small capillary numbers (see Figure 1.5). On theother hand, for higher values of the capillary number, the deformation pa-rameter was lower with respect to the Newtonian cases, and it contradictedformer experimental evidence [25, 51].81.2. Literature reviewFigure 1.5: Images of deformed droplets under steady shear flow. The leftside is for a Newtonian system, the right side is for a non-Newtonian one.Both systems have the same Ca = 0.15 and λ = 0.1. Top images arethe side views, bottom images are views along the shear gradient direction.Reproduced from Guido et al. [35].There are several experimental studies on the breakup problem of a sin-gle droplet under shear stress. Sibillo et al. [76] investigated the effect ofmatrix viscoelasticity on droplet breakup experimentally. Their main resultis that the matrix elasticity delays the breakup and the critical capillarynumber is an increasing function of the Deborah number De (ratio betweenthe fluid relaxation time and the so called emulsion time). Their result isin agreement with the results obtained by Flumerfelt and Raymond [30] onshear-thinning fluids. In other studies, Lerdwijitjarud et al. [42, 43] andSibillo et al. [74] experimentally investigated the effect of a viscoelasticdroplet in a Newtonian matrix. They found that the deformation parame-ter and the orientation angle are reduced and increased, respectively. Theirresults are in agreement with the predictions of the Maffettone-Greco ellip-soidal model [48].91.2. Literature reviewFigure 1.6: Steady-state droplet deformation D (normalized against thevalue for Newtonian cases, De = 0) varying with De. Reproduced fromAggarwal and Sarkar. [3].Similar studies have been conducted numerically by several researchers.Yue et al. [96] performed an investigation of two-dimensional Newtoniandroplet deformation in a viscoelastic matrix, based on a diffuse-interfaceformulation and the Oldroyd-B constitutive equation for the viscoelasticphase. Their main conclusion is that the deformation of a Newtonian dropletin the viscoelastic system has a non-monotonic behaviour with respect tothe increasing Deborah number De. Following their studies, Aggarwal andSarkar [2, 3] supported their results and extended their work by using athree-dimensional front-tracking finite difference algorithm (see figure 1.6).For the cases of an immersed Newtonian droplet in a viscoelastic fluid, theydiscovered an increase in the droplet orientation compared to the Newtoniancase; this conclusion is in agreement with previous theoretical and experi-mental studies [31, 35, 36, 48, 85].In [2] the authors also investigated the effect of droplet viscoelasticity.They reported that droplet viscoelasticity acts to reduce the deformation pa-rameter at increasing Deborah number. Mukherjee and Sarkar [53] extendedthe same studies for different values of viscosity ratio λ. Like the cases ofa Newtonian droplet in the viscoelastic matrix, for viscosity ratio λ > 1, anon-monotonic dependence of the deformation parameter with the Deborah101.2. Literature reviewnumber, with a decrease followed by an increase, has been reported.1.2.2 EmulsionsIn the last century, since Sutherland [80] and Einstein [24] conducted stud-ies about the viscosity of a very dilute suspension of non-deformable solidspheres, the macroscopic rheological behaviour of multi-particle systems hasreceived remarkable attention. In 1932, Taylor [82] extended their theory tovery dilute emulsions, where he assumed that the particles remain spherical;i.e., the Capillary number is assumed Ca  1. Based on this model, themacroscopic shear viscosity of the system (effective viscosity of the emul-sion) is linearly proportional to the particle concentration φ. And the effectof the concentration increases as the particles are closer to solid spheres.Later, other studies have developed expressions for the rheology of diluteemulsions including highly deformable fluid particles (Ca  1) [6, 22, 46].Most of the proposed models to predict the effective viscosity of an emulsionwere limited to either very low deformation (Ca 1), or highly deformablewith viscosity ratio close to 0 (λ → 0). For instance, Mackenzie suggestedthe following correlation for predicting the effective viscosity of the emulsionµeff :µeff = µm(1− 53φ) (1.1)Where µm and φ are the viscosity of the matrix and volume fractionof dispersed phase, respectively. As most of the other models, Mackenzie’sprediction is limited to very dilute systems (limφ→0) with very low viscosityratios (limλ→0) and highly deformable particles (Ca 1).In the case of concentrated systems, by using the differential effectivemedium (DEM) theory, Pal [60, 62] obtained the relative viscosity for anelastic solid particle suspension (λ → ∞) and a bubble emulsion (λ → 0) .In another investigation, Pal [61] developed a generic model for concentratedemulsions with different viscosity ratios and deformable particles using theanalogy between shear modulus and shear viscosity. The effect of capil-lary number and viscosity ratio on the rheology and viscoelastic featuresof emulsions has been studied extensively by Pal [57–59, 61]. These exten-sive studies by Pal have given many experimental data on the viscosity ofemulsions later used by other researchers to validate their works.Figure 1.7, developed by Faroughi and Huber [26], shows a summary ofsome published rheological models and their range of applicability.111.2. Literature reviewFigure 1.7: Summary of some of the published rheological models and theirrange of applicability with respect to particle volume fraction ψ, viscosityratio λ and capillary number Ca. Reproduced from Faroughi and Huber.[26].Theoretical investigations by Oldroyd [54, 55] showed that dilute emul-sions with Newtonian phases show an elastic behaviour. Oldroyd explainedthis phenomenon as a result of interfacial tension. In another study, Batch-elor [8] developed a theory for the stress in homogeneous emulsions, and asa result, concluded that the effective stress in a Newtonian emulsion is theresult of two other components, 1) the contribution from the viscosity differ-ence between continuous and dispersed phase, 2) the contribution from thesurface tension at the interface between the two phases; and he expressedthe excess stress in terms of an interface tensor [50, 56].In case of dilute emulsions, since droplets are far from each other andthe droplet/droplet interaction is not significant, droplets usually behavelike a single droplet in the flow. Therefore, different researchers tried todescribe the evolution of a droplet in an emulsion with Newtonian phasesby assuming an ellipsoidal droplet-type morphology using a second-orderphenomenological (morphology) tensor [33, 39, 47, 89, 92–94].Contrary to dilute emulsions, droplets can undergo large deformationsin concentrated emulsion systems. They may experience breakup (for Cap-121.2. Literature reviewillary number Ca > Cacrit) or coalescence. Consequently, the morphologyof concentrated emulsions is more complicated. However, Doi and Ohta [21]suggested a simple phenomenological model for the evolution of the Batch-elor’s interface tensor, and based on their model, linear scaling for both theshear stress and the first normal stress difference were predicted. The Doi-Ohta scalings have been supported by experimental analyses for differentsystems [81, 86, 87] including systems with viscoelastic phases.Emulsions with non-Newtonian phases have not been investigated as ex-tensively as their Newtonian counterparts [4]. Yue et al. [95] extendedBatchelor’s formulation and developed a new perturbation theory for vis-coelastic systems. They predicted that viscoelasticity has no effect on steadydroplet deformation for small capillary numbers, and the elasticity of thedroplet tends to increase the initial deformation and relaxation rate, whilethe elasticity of the matrix has the opposite effect. Aggarwal and Sarkar [4]investigated the steady shear rheology of a dilute emulsion with viscoelasticdroplets numerically. They extended Batchelor’s formulation [8], and bymodelling viscoelasticity using the Oldroyd-B constitutive equation alongwith a front-tracking finite difference algorithm, they reported that the vis-coelasticity of the droplet does not contribute significantly to the effectiveshear viscosity of the emulsion. Figure 1.8 is one of their results showingthe variation of viscoelastic shear stress due to the droplet viscoelasticity asa function of Ca. The effect of the droplet viscoelasticity is not dominantat least in the range of Ca < Cacrit, and the magnitude of these stresses isnegligible in comparison with interfacial shear stress contribution.131.2. Literature reviewFigure 1.8: Variation of viscoelastic shear stress with capillary number Ca.Reproduced from Aggarwal and Sarkar. [4].Rheological experiments give a good insight into how the emulsions be-have. However, they are not sufficient to help to develop a constitutive equa-tion modelling the behaviour of emulsions [100]. Lowenberg and Hinch [45]performed three-dimensional numerical simulations of concentrated emul-sions in a shear flow at low Reynolds numbers and finite capillary numbers.They presented results for the transient and the steady-state rheology of con-centrated emulsions with volume fractions up to 30% and viscosity ratiosin the range 0 ≤ λ ≤ 5, and they reported a complex viscoelastic shear-thinning behaviour for the emulsion. Later, the first large-scale boundaryintegral multi-droplet simulations of emulsions with volume fractions of upto 55% were performed by Zinchenko and Davis [98, 99]. In another studyfor highly concentrated monodisperse emulsions, Zinchenko and Davis [100]constructed a general constitutive model. They conducted long-time, large-scale and high resolution simulations of deformable droplets with an insol-uble surfactant, assuming a linear model for the surface tension versus thesurfactant concentration.141.3. Summary of the literature review1.3 Summary of the literature review1.3.1 Single droplet• For the cases of a Newtonian droplet in a viscoelastic matrix at smallCa, the deformation parameter D is unchanged with respect to fullNewtonian cases, while the orientation angle is enhanced significantly[31, 35, 36]• The matrix elasticity delays the breakup and the critical capillarynumber is an increasing function of De [30, 76].• The deformation of a Newtonian droplet in the viscoelastic system hasa non-monotonic behaviour with respect to the increasing Deborahnumber De [2, 3, 96].• Studying the effect of a viscoelastic droplet in a Newtonian matrixshowed that the deformation parameter and the orientation angle arereduced and increased, respectively [42, 43, 48, 74].• For the cases of viscoelastic droplet in the Newtonian matrix at vis-cosity ratio λ > 1, the deformation parameter has a non-monotonicdependence with the Deborah number De, with a decrease followedby an increase as De increases [53].• Viscoelasticity has no effect on steady droplet deformation for smallCa, and the elasticity of the droplet tends to increase the initial defor-mation and relaxation rate, while the elasticity of the matrix has theopposite effect [95].1.3.2 Emulsions• As a result of interfacial tension, dilute emulsions with Newtonianphases show an elastic behaviour [54, 55].• The effective stress in a Newtonian emulsion is the result of two con-tributions:1. the contribution from the viscosity difference between continuousand dispersed phase,2. the contribution from the surface tension at the interface betweenthe two phases [8].151.3. Summary of the literature review• The viscoelasticity of the droplet does not contribute significantly tothe effective shear viscosity of the emulsion [4].• In the range of Ca < Cacrit the effect of the droplet viscoelasticityis not dominant and the magnitude of these stresses is negligible incomparison with interfacial shear stress contribution [4].In the current study we conduct numerical simulations for a single dropletand an emulsion subjected to an imposed shear rate. In these simulationswe cover a vast range of parameters including the capillary number Ca, theviscosity ratio λ, the Deborah number De and the volume fraction φ for botha Newtonian and a non-Newtonian dispersed phase. In all the simulationsthe matrix fluid is Newtonian. Some parts of the range of parameters weinvestigate were already reported in the literature, however, we report newresults in unexplored ranges of the Deborah number up to De = 10 and ofthe volume fraction up to φ = 50%.The key question addressed in this work is the influence of elas-ticity on the rheological properties of the gelled-oil/water emul-sion, and the impact of these rheological properties on the reme-diation process.16Chapter 2Numerical Model2.1 Problem description and flow configurationAs mentioned in the previous chapter, characterizing the rheological behav-ior of emulsions can give us a valuable insight into the problem of oil spills aswell as other industrial problems involving emulsions and blends. It is worthpointing out again that the problem of studying a single droplet deforma-tion in a simple shear flow is of fundamental relevance to many industrialproblems [77], at least when the volume fraction of droplets is low. Howa single droplet behaves in a shear flow sheds light on the properties andcharacteristics of emulsions containing multiple droplets of the same fluid.In the following, we consider either a single droplet or a collection ofdroplets, i.e., an emulsion, in a simple shear flow. The term simple shearflow refers to a configuration in which the shear rate is constant in theabsence of droplets and the velocity profile is one-dimensional and linear.The canonical problem of a single droplet in a simple shear flow has beenstudied by many researchers. The primary flow feature is the deformationof the droplet with respect to its reference shape assumed circular in twodimensions and spherical in three dimensions. The reference shape is hencerepresentative of a capillary number equal to 0. The problem of a collectionof droplets in a simple shear flow has received less attention at the numericallevel as computations are more challenging and one needs to address thedifficult question of physical versus numerical coalescence.To impose a simple shear flow, we design a model system with two par-allel plates moving in opposite directions. This is a very straightforwardway of imposing a simple shear flow but also raises the question of wall ef-fects, i.e., confinement, in the direction of shear. We will discuss the effectof confinement on our results in Chapter 4. Another solution is to employLees-Edwards boundary conditions that do not involve the presence of wallswith a no-slip condition in the system and model simple shear in the bulk ofthe emulsion, but Lees-Edwards boundary conditions are more complicatedto implement.Our systems have the following features:172.2. Governing equations• the geometry of the domain is a L∗x × L∗y rectangle in two dimensionsand a L∗x × L∗y × L∗z cuboid in three dimensions. In most cases weconsider squares in two dimensions, i.e., L∗x = L∗y, and cubes in threedimensions L∗x = L∗y = L∗z, and only a few select number of cases havean aspect ratio L∗y/L∗x or L∗z/L∗x different than 1.• the streamwise direction is x and the shear direction is y.• the boundary conditions are as follows:1. at the bottom wall y∗ = 0 we impose a Dirichlet boundary condi-tion on the velocity: (−U∗, 0) in two dimensions and (−U∗, 0, 0)in three dimensions,2. at the top wall y∗ = L∗y we impose a Dirichlet boundary conditionon the velocity: (U∗, 0) in two dimensions and (U∗, 0, 0) in threedimensions,3. all other walls satisfy periodic boundary conditions.• the two fluids are incompressible and immiscible.• the fluids are either Newtonian or viscoelastic.• we assume that the two fluids have the same density, i.e., the dropletsare neutrally buoyant.• in two dimensions and three dimensions droplets have an initial circu-lar and spherical shapes; respectively, and an initial diameter d∗p.• in the case of emulsions, all droplets have the same initial diameter.Please note that with the aforementioned boundary conditions, the im-posed background shear rate is 2U∗/L∗y. The single droplet and multipledroplet systems are illustrated in Figure 2.1.2.2 Governing equationsWe present in this section a step by step derivation of the system of gov-erning equations that we later solve computationally together with theirnon-dimensionalization. We start with a single fluid flow, then derive thetwo-fluid flow formulation with surface tension, and finally introduce themixture formulation with surface tension. To be mathematically well posed,182.2. Governing equationsFluid 1Fluid 2dpULyLx****(a) Single dropletFluid 1Fluid 2dpULyLxdp*****(b) EmulsionFigure 2.1: Single-droplet and multiple-droplet systems examined.the sets of equations described in the following are complemented with clas-sical boundary conditions at the limit of the domain and initial conditionsat t = 0. Boundary conditions in the systems we examine were discussedin the previous sections and initial conditions are arbitrary but often takenas a quiescent flow or a fully developed flow in the absence of droplets. Fi-nally, please note that we write dimensional quantities with a ∗ superscriptto distinguish them from dimensionless quantities.2.2.1 Single fluid flowThe flow of a single fluid is governed by the following classical mass andmomentum conservation equations:∂ρ∗∂t∗+∇∗ · (ρ∗u∗) = 0 (2.1)ρ∗(∂u∗∂t∗+ u∗ · ∇u∗)= ∇∗ · τ∗ + ρ∗g∗ (2.2)where ρ∗ denotes the fluid density, u∗ the velocity, τ∗ the total stress tensor,g∗ the gravity acceleration and t∗ the time. τ∗ can be further decomposedintoτ∗ = −p∗I + Σ∗ (2.3)192.2. Governing equationswhere p∗ denotes the pressure and Σ∗ the stress tensor.Assuming that the flow is incompressible, i.e., the density ρ∗ is constant,and that the fluid is Newtonian, i.e., Σ∗ = 2µ∗D∗ with µ∗ the viscosity andD∗ = 12[∇∗u∗ +∇∗u∗T ] the deformation rate tensor, we obtain:∇∗ · u∗ = 0 (2.4)ρ∗(∂u∗∂t∗+ u∗ · ∇∗u∗)= −∇∗p∗ + µ∗∆∗u∗ + ρ∗g∗ (2.5)To non-dimensionalize equations (2.4)-(2.5), we introduce the followingscales: d∗p for length, U∗ for velocity, ρ∗U∗2 for pressure, d∗p/U∗ for time andg∗ = |g∗| for gravity acceleration. Please note that using ρ∗U∗2 for pressureand d∗p/U∗ for time correspond to an inertial scaling derived by assumingthat the pressure balances the advective term. This scaling is not valid inthe limit of zero Reynolds number, i.e., in viscosity dominated flows. Fora viscous flow, the pressure balances the viscous stresses and the properscaling is therefore µ∗U∗/d∗p for pressure and ρ∗d∗2p /µ∗ for time. The viscousscaling leads to a slightly different form of the dimensionless momentumconservation equation. The dimensionless equations read as follows:∇ · u = 0 (2.6)∂u∂t+ u · ∇u = −∇p+ 1Re∆u +1Fr2g (2.7)where Re and Fr denote the Reynolds number and the Froude numberrespectively, and are defined as:Re =ρ∗U∗d∗pµ∗(2.8)Fr =U∗√g∗d∗p(2.9)When the effect of gravity is negligible at the scale of the system or wouldsimply add a linear hydrostatic pressure term to the total pressure p∗, it iscustomary to absorb (subtract) the term ρ∗g∗ into (from) the pressure andto define a modified pressure p˜∗ as p˜∗ = p∗−ρ∗g∗ g∗g∗ x∗ where x∗ denotes theposition vector. p˜∗ hence represents the dynamic pressure, i.e., the pressuredue to fluid motion, and using p˜∗ the dimensionless momentum conservationequation reads as follows:∂u∂t+ u · ∇u = −∇p˜+ 1Re∆u (2.10)202.2. Governing equations2.2.2 Two-fluid formulation with surface tensionWe now assume that the system comprises two immiscible, incompressibleand Newtonian fluids. We name the first fluid fluid 1 with properties ρ∗1and µ∗1 and the second fluid fluid 2 with properties ρ∗2 and µ∗2. The two-fluidformulation involves writing a set of conservation equations for fluid 1 in thesub-domain occupied by fluid 1 and a set of conservation equations for fluid2 in the sub-domain occupied by fluid 2. The two sub-domains are separatedby an interface. The coupling between the two fluids relies on imposing thecontinuity of the velocity u∗ at the interface and the stress balance at theinterface.Since both fluids are incompressible, the mass conservation equationsimplifies to ∇∗ ·u∗ = 0 in both sub-domains, and hence ∇∗ ·u∗ = 0 is validin the whole domain. We now write the momentum conservation equationin each sub-domain, we get:ρ∗1(∂u∗∂t∗+ u∗ · ∇∗u∗)= −∇∗p∗ + µ∗1∆∗u∗ + ρ∗1g∗ (2.11)ρ∗2(∂u∗∂t∗+ u∗ · ∇∗u∗)= −∇∗p∗ + µ∗2∆∗u∗ + ρ∗2g∗ (2.12)Now we introduce a modified pressure using ρ∗1 as p˜∗ = p∗ − ρ∗1g∗ g∗g∗ x∗ andrewrite equations (2.11)-(2.12) as:ρ∗1(∂u∗∂t∗+ u∗ · ∇∗u∗)= −∇∗p˜∗ + µ∗1∆∗u∗ (2.13)ρ∗2(∂u∗∂t∗+ u∗ · ∇∗u∗)= −∇∗p˜∗ + µ∗2∆∗u∗ + (ρ∗2 − ρ∗1)g∗ (2.14)To non-dimensionalize equations (2.13)-(2.14), we introduce the same scalesas in section 2.2.1 and (arbitrarily) choose ρ∗1 as the reference density. Wealso define the Reynolds number using the properties of fluid 1, i.e., ρ∗1 andµ∗1 as Re =ρ∗1U∗d∗pµ∗1. After some simple algebra, the dimensionless form of(2.13)-(2.14) reads as follows:∂u∂t+ u · ∇u = −∇p˜+ 1Re∆u (2.15)∂u∂t+ u · ∇u = − 1M∇p˜+ λMRe∆u +Arλ2MRe2g (2.16)212.2. Governing equationswhereM denotes the density ratio, λ the viscosity ratio andAr the Archimedesnumber. These numbers are defined as follows:M =ρ∗2ρ∗1(2.17)λ =µ∗2µ∗1(2.18)Ar =ρ∗1(ρ∗2 − ρ∗1)g∗d∗3pµ∗22(2.19)The stress balance at the interface relies on the surface tension balancingthe jump in the stress. Assuming that the surface tension coefficient σ∗ isconstant along the interface, the general form of the interfacial stress balanceis:τ∗2 · n− τ∗1 · n = σ∗(∇∗ · n)n (2.20)where κ∗ = ∇∗ ·n represents the local curvature of the interface and σ∗(∇∗ ·n)n the normal curvature force per unit area. In case the two fluids areNewtonian, the interfacial stress balance becomes:(−p∗ + 2µ∗2D∗) · n− (−p∗ + 2µ∗1D∗) · n = σ∗κ∗n (2.21)Using the inertial scaling, we write the dimensionless interfacial stress bal-ance as:(−p+ 2 λReD) · n− (−p+ 2 1ReD) · n = 1Weκn (2.22)where We denotes the Weber number as the ratio between inertia force andsurface tension defined as:We =ρ∗1U∗2d∗pσ∗(2.23)Otherwise, we can introduce the capillary number Ca as the ratio betweenviscous force and surface tension as:Ca =µ∗1U∗σ∗(2.24)and recognize that we have:We =ρ∗1U∗2d∗pσ∗=ρ∗1U∗d∗pµ∗1µ∗1U∗σ∗= ReCa (2.25)Using Ca and Re, the dimensionless interfacial stress balance becomes:(−Re p+ 2λD) · n− (−Re p+ 2D) · n = 1Caκn (2.26)At this point, we would like to make the three following comments:222.2. Governing equations1. as for the momentum conservation equation, the limit of Re tendingto 0 is not captured by the dimensionless form (2.22) or (2.26) of theinterfacial stress balance because we assumed a scaling of the pressureas ρ∗1U∗2. Indeed when Re → 0, equation (2.26) mistakenly indicatesthat pressure is not involved in the balance while we know that ina static state, surface tension balances the pressure jump across theinterface. To get the limit of static state, we should assume the properscaling for the pressure. From the dimensional equation, we can seethat when the velocity goes to 0, the balance between pressure andsurface tension leads to assume a pressure scaling as σ∗/d∗p. Using sucha scaling for the pressure, the dimensionless form of the interfacialstress balance becomes:(−p+ 2λCaD) · n− (−p+ 2CaD) · n = κn (2.27)In this form, when Ca → 0, i.e., the limit of static state, the viscousstresses become negligible and we recover the proper balance betweenpressure and surface tension in a dimensionless form.2. we note that there is a jump in the normal direction only. In fact, if weproject equation (2.20) over the normal direction n and the tangentialdirection t, the projected equation along n results in a non-zero right-hand side as n · n = 1 while the projected equation along t results ina zero right-hand side, i.e., no jump, simply because n · t = 0. If thesurface tension coefficient σ∗ is assumed to vary along the interface,then there is an additional term on the right-hand side of equation(2.20) of the form ∇∗σ∗ called tangential Marangoni stresses that leadto a jump in the tangential direction as well.3. please note that in the two-fluid formulation, surface tension is notinvolved in the momentum conservation equation of each fluid.2.2.3 Mixture formulation with surface tension written as avolumetric forceThe interfacial stress balance (2.20) is actually obtained by writing a forcebalance over an elementary volume V ∗ containing a surface S∗ and thentaking the elementary volume size to 0. In this force balance, the surfacetension force is expressed as∫S∗ σ∗κ∗ndS∗. This force living on the interfaceS∗ can be written as a volumetric force by introducing a delta Dirac function232.2. Governing equationsδS∗ that is non-zero on S∗ only as follows:∫S∗σ∗κ∗ndS∗ =∫V ∗σ∗κ∗nδS∗dV ∗ (2.28)We can then extract the surface tension force per unit volume σ∗κ∗nδS∗ .Formulated as a volumetric force, we can now incorporate the surface tensionto the momentum conservation equation.We also formulate the problem for a mixture of two immiscible andincompressible fluids. For this, we introduce:• the mixture density ρ∗ as a linear combination of the fluid densitiesdefined as:ρ∗ = cρ∗1 + (1− c)ρ∗2 (2.29)where c is an indicator function that equals 1 in fluid 1 and 0 in fluid2.• the mixture momentum per unit volume as a linear combination of thefluid momenta defined as:ρ∗u∗ = cρ∗1u∗ + (1− c)ρ∗2u∗ (2.30)• the mixture viscosity as a linear combination of the fluid viscositiesdefined as:µ∗ = cµ∗1 + (1− c)µ∗2 (2.31)The mass and momentum conservation equations for the mixture areequivalent to the conservation equations for a single incompressible fluidflow with variable density and viscosity and an additional surface tensionterm. The mass conservation equation of the mixture hence reads as follows:∂ρ∗∂t∗+∇∗ · (ρ∗u∗) = 0 = ∂ρ∗∂t∗+ ρ∗∇∗ · u∗ + u∗ · ∇∗ρ∗ (2.32)Since both fluids are incompressible, we have ∇∗ · u∗ = 0 everywhere in themixture, and the mass conservation becomes:∂ρ∗∂t∗+ u∗ · ∇∗ρ∗ = 0 (2.33)The momentum conservation of the mixture reads as follows:ρ∗(∂u∗∂t∗+ u∗ · ∇∗u∗)= −∇∗p∗ +∇∗ · (2µ∗D∗) + σ∗κ∗nδS∗ + ρ∗g∗ (2.34)242.2. Governing equationsPlugging equation (2.29) into equation (2.33), we derive an equation forthe indicator function c that locates the two phases in the flow and theinterface between the two phases. In fact:0 =∂ρ∗∂t∗+ u∗ · ∇∗ρ∗=∂(cρ∗1 + (1− c)ρ∗2)∂t∗+ u∗ · ∇∗(cρ∗1 + (1− c)ρ∗2) (2.35)= (ρ∗1 − ρ∗2)∂c∂t∗+ c∂ρ∗1∂t∗+ (1− c)∂ρ∗2∂t∗+ (ρ∗1 − ρ∗2)u∗ · ∇∗c+ cu∗ · ∇∗ρ∗1 + (1− c)u · ∇∗ρ∗2Finally, given that ρ∗1 and ρ∗2 are constant in space and time and simplifyingby (ρ∗1 − ρ∗2), we establish the equation for the indicator function c as:∂c∂t∗+ u∗ · ∇∗c = 0 (2.36)To summarize, the mixture formulation of our immiscible fluid-fluidproblem with surface tension involves solving the following set of equations:∇∗ · u∗ = 0 (2.37)∂c∂t∗+ u∗ · ∇∗c = 0 (2.38)ρ∗(c)(∂u∗∂t∗+ u∗ · ∇∗u∗)= −∇∗p∗ +∇∗ · (2µ∗(c)D∗) + σ∗κ∗nδS∗ + ρ∗(c)g∗(2.39)where we emphasized the dependence of ρ∗ and µ∗ with c.The non-dimensionalization procedure is similar to that for the two-fluidformulation derived in section 2.2.2 and produces the same dimensionlessnumbers, except that the capillary number Ca or the Weber number Weenters the momentum conservation equation. We do not repeat it here forthe sake of conciseness.2.2.4 Viscoelastic fluidIf one of the phases is viscoelastic, its stress tensor is decomposed into aNewtonian part for the solvent and a viscoelastic part for the polymer as:Σ∗ = T∗ + 2µ∗sD∗ (2.40)252.2. Governing equationswhere µ∗s is the solvent Newtonian viscosity and T∗ is the extra stress tensor.In this study, we select the Oldroyd-B model and the extra stress tensor T∗hence satisfies the following constitutive equation:α∗Tˇ∗ + T∗ = 2µ∗pD∗ (2.41)where µ∗p denotes the polymer viscosity, α∗ the polymer relaxation time andTˇ∗ is the upper-convected derivative defined asTˇ∗ =∂T∗∂t∗+ (u∗ · ∇∗)T∗ − (∇∗u∗)T∗ −T∗(∇∗u∗)T (2.42)The total viscosity µ∗ is the sum of the solvent contribution µ∗s and thepolymeric contribution µ∗p as:µ∗ = µ∗s + µ∗p (2.43)For a Newtonian fluid, the polymeric contribution µ∗p is zero, leading to auniformly zero extra stress tensor T∗, and the total viscosity µ∗ is equal tothe solvent contribution µ∗s.Using the same inertial scaling for velocity and time and the total vis-cosity µ∗ = µ∗s + µ∗p to scale T∗ as (µ∗s + µ∗p)U∗/d∗p, the dimensionless formof the Oldroyd-B model reads as follows:DeTˇ + T = 2(1− β)D (2.44)where De denotes the Deborah number and β the solvent viscosity to totalviscosity ratio. The expression of these two dimensionless numbers is:De =α∗U∗d∗p(2.45)β =µ∗sµ∗p + µ∗s(2.46)2.2.5 Flow properties of interest and dimensionlessnumbersIn all the cases computed and examined in this thesis, we assume thatthe two fluids have the same density. From a dimensionless viewpoint, itmeans that the Archimedes number Ar is zero and the density ratio M is1. Therefore we solve the following set of equations:• the velocity divergence free equation (2.37) that guarantees that thetwo fluids are incompressible,262.2. Governing equations• the indicator function c equation (2.38) that locates the two phasesand their interface in the flow domain,• the momentum conservation equation (2.39) without the last term onthe right-hand side since the two fluids are assumed to be neutrallybuoyant,• the constitutive equation (2.42) for the extra stress tensor if one of thephases is viscoelastic,with appropriate initial and boundary conditions. The relevant dimension-less numbers in our study are then:• the viscosity ratio λ,• the Reynolds number Re,• the capillary number Ca,• the Deborah number De,• the viscosity ratio β.The formal definition arising from the non-dimensionalization procedure de-rived in section 2.2.2 implicity assumed that the shear rate scale is U∗/d∗p.However in our systems, we impose a background shear rate of 2U∗/L∗y.Leaving aside the factor 2, the discrepancy between the shear rate scale andthe actual shear rate underlines the presence of two length scales in thesystem: the scale of the droplet d∗p and the scale of the system L∗y. Oneway to include the actual shear rate in the definition of the dimensionlessnumbers is to define γ˙∗ = 2U∗/L∗y, which is the average shear rate seenby the droplets, and rewrite Ca and De as Ca =µ∗1 γ˙∗d∗pσ∗ and De = α∗γ˙∗,respectively.Finally, we use the subscripts “d” for the droplets and “c” for the contin-uous fluid (the matrix), and choose to use the droplet viscosity and densityas references. Assuming droplets are viscoelastic, this leads to the following272.3. Computational method and scientific softwareexpressions for the relevant dimensionless numbers in our study:λ =µ∗dµ∗c(2.47)Re =ρ∗dU∗d∗pµ∗d(2.48)Ca =µ∗dγ˙∗d∗pσ∗(2.49)De = α∗γ˙∗ (2.50)β =µ∗s,dµ∗p,d + µ∗s,d(2.51)2.3 Computational method and scientificsoftwareThe set of governing equations (2.37),(2.38),(2.39) and (2.42) are spatiallydiscretized by a second order accurate finite-volume scheme on a Cartesianadaptive grid. The solution is advanced in time by a second-order fractionalstep algorithm. The solution of equation (2.38) is representative of a classicalVolume-Of-Fluid (VOF) method. One of the challenging tasks in computingtwo-phase flows with surface tension is the proper computation on the gridof the curvature and the surface tension term such that it fully balancesthe pressure jump at the discrete level in a quiescent system, thus avoidingor minimizing the well documented issue of the presence of non-physicalspurious currents at the droplet surface. This is achieved through the useof height functions [64–66].The VOF method has the advantage of being a conservative methodbut cannot prevent coalescence when two droplets are getting very close toeach other as (i) the two droplets are described by the same phase indicatorc and (ii) on the grid the transition from one phase to the other phase isnot sharp (in the sense of a Heaviside function that would be physicallycorrect for two immiscible fluids) but is smoothed out over 2-3 grid cells.Consequently, when two droplets coalesce in the computations, it is notstraightforward to decide whether this coalescence is physical or numerical(and hence a physical artefact). A simple remedy to this problem exists if weare interested in systems in which droplets are assumed to never coalesce.This involves defining one phase indicator per droplet and treat them asseparate phases at the numerical level but with similar properties. Thisis the approach we employ in this thesis to investigate emulsions at high282.3. Computational method and scientific softwarevolume fraction with a constant number of droplets. When the number ofdroplets is high, i.e., O(100), this simple approach can be computationallyexpensive. But the computational cost can be significantly limited to fourphase indicators for any arbitrarily large number of droplets through the useof the Four colour theorem (note that the Four color theorem is theoreticallyvalid in two dimensions only, in three dimensions the minimum number ofphase indicators is a bit more complicated to determine) and constant re-assignment of colors to droplets. We have also utilized this more advancedmodel for high volume fraction emulsions without coalescence.All the numerical schemes are developed in an open-source environmentnamed “Basilisk”[91] for the solution of partial differential equations onadaptive Cartesian grids. Basilisk is widely used in the fluid mechanicscommunity and extensively validated for various problems involving multiplephases in incompressible fluid flows. The key features and advantages ofBasilisk that led us to select this software are listed below:• Ability to compute two- and three- dimensional flows in parallel.• Use of advanced VOF method algorithms based on height functions.• Dynamic adaptive mesh refinement (AMR) using octree grids.Dynamic adaptive mesh refinement in parallel is a really powerful featureof Basilisk. This feature allows for refining and coarsening the grid over thecourse of the simulation based on a given metric. Classical metrics are theHessian (spatial second derivatives) of physical quantities computed on thegrid such as, e.g., the phase indicator c or the velocity field u. Figure 2.2shows the adaptive mesh refinement for the simulation of a falling dropletand illustrates how the mesh is refined over the interface and in regions oflarge velocity gradient variations.292.4. Analysis of the computed resultsFigure 2.2: Adaptive mesh refinement for the case of a falling droplet.2.4 Analysis of the computed resultsThe strength of multi-dimensional computational fluid dynamics is thatcomputations generate detailed data in the core of the flow. However, thislarge amount of data needs to be analyzed such that we can extract meaning-ful, presumably macroscopic, features of the flow. Under the simple shearflows we are interested in, circular/spherical droplets deform and tend toadopt elliptical/ellipsoidal shape oriented with respect to the streamwisedirection as illustrated in Figure 2.3.302.4. Analysis of the computed resultsFluid 1Fluid 2ULyLxLB ϴU******Figure 2.3: Schematic of a deformed single droplet under the simple shearflow.In our study, we analyze our computed results in terms of:• the deformation parameter D of each droplet defined asD ≡ L∗ −B∗L∗ +B∗(2.52)• the orientation angle θ of each droplet defined as the angle betweenthe longest axis of the droplet and the streamwise direction.• in the case of emulsions, the effective shear viscosity µeff of the emul-sion defined as the ratio of the shear viscosity of the emulsion to theviscosity of the matrix. We calculate the effective shear viscosity of theemulsion as the ratio of the actual average shear stress exerted on thetop and bottom walls (S∗top and S∗bottom, respectively) to the referenceshear stress calculated with the matrix viscosity and the backgroundimposed shear rate:µeff =12(1S∗top∫S∗top|Σ∗xy|dS∗ + 1S∗bottom∫S∗bottom|Σ∗xy|dS∗)2U∗L∗yµ∗c(2.53)312.4. Analysis of the computed resultsPlease note that all results presented from now on are in a dimensionlessform.32Chapter 3ValidationIn this chapter we validate our numerical model by comparing our computedresults with existing results from the literature. We first discuss the con-vergence of the computed solution with grid refinement and then presentresults for both the single droplet problem and the emulsion problem.3.1 Grid convergenceIn this section we study the grid convergence in order to determine whatlevel of resolution we should use such that our results are independent ofthe grid size. Therefore, we measure the deformation of a single Newtoniandroplet in a simple shear flow with the following properties:Parameter Re Ca λ Lx LyValue 0.05 0.5 1 8 8Figure 3.1 shows the time evolution of the droplet deformation for sixincreasingly smaller values of the grid size.333.1. Grid convergenceFigure 3.1: Convergence study - time evolution of the droplet deformationfor Re = 0.05, Ca = 0.5, and λ = 1 at different grid resolutions. ∆xrepresents the smallest grid size.Figure 3.1 clearly shows that having four or eight grid cells over thedroplet initial diameter is not enough to capture the right dynamics andresults are significantly off. However, with a finer grid size, results convergeto the correct value of the deformation parameter. Since properly comparingresults in figure 3.1 is not straightforward, we show a magnified view of ourresults in figure 3.2.343.2. Single dropletFigure 3.2: Convergence study - time evolution of droplet deformation forRe = 0.05, Ca = 0.5, and λ = 1 at different grid resolutions. ∆x representsthe smallest grid size.While a spatial resolution of 16 cells per initial droplet diameter is enoughto predict the right level of deformation, the time evolution of the deforma-tion features fluctuations combined to non-physical sharp jumps. From 32grid cells per initial droplet diameter, we can confidently claim that the spa-tial convergence is reached. Therefore, from now on, we use a grid size ofeither 32 or 64 cells per initial droplet diameter.3.2 Single dropletWe consider two types of system to validate the single droplet cases: 1)both phases are Newtonian and 2) one phase is viscoelastic.3.2.1 Newtonian fluidsAs computations with Newtonian fluids are rather straightforward and rel-atively fast, we consider both two- and three-dimensional cases.3.2.1.1 Two-dimensional single dropletIn order to validate our method and code for Newtonian fluids in two di-mensions, we selected the work of Sheth and Pozrikidis [73] and comparedour results to theirs. The properties of the system are as follows:353.2. Single dropletParameter Re Ca λ Lx LyValue 1 ≤ Re ≤ 100 0.05 ≤ Ca ≤ 0.9 1 2 2We plot in figure 3.3 our results and the results obtained by Sheth andPozrikidis [73], and observe that the two sets of results are comparable. It isworth mentioning that [73] used 12 grid cells over the initial droplet diame-ter which is considered under-resolved compared to our computations with64 grid cells per initial droplet diameter. We attribute the small discrep-ancy between the results to the difference in spatial resolution. The spatialresolution is an important parameter in cases where the droplet experienceslarger deformations, because in those cases, the number of grid cells in thesmallest diameter of the deformed droplet may not be sufficient to measurethe right flow dynamics. Moreover, the radius of curvature at the tips ofthe highly deformed droplet is smaller than the less deformed droplet. Inorder to track this small curvature, there is a need to have a sufficientlylarge number of grid cells at the tips.363.2. Single droplet0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0t0.00.10.20.30.40.50.60.70.8DRe=1 =1Ca=0.2 - This StudyCa=0.4 - This StudyCa=0.9 - This StudyCa=0.2 - Sheth & PozrikidisCa=0.4 - Sheth & PozrikidisCa=0.9 - Sheth & Pozrikidis(a)0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0t0.00.10.20.30.40.50.60.70.80.9DRe=10 =1Ca=0.2 - This StudyCa=0.4 - This StudyCa=0.8 - This StudyCa=0.2 - Sheth & PozrikidisCa=0.4 - Sheth & PozrikidisCa=0.8 - Sheth & Pozrikidis(b)0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0t0.00.10.20.30.40.50.60.70.8DRe=50 =1Ca=0.2 - This StudyCa=0.3 - This StudyCa=0.4 - This StudyCa=0.2 - Sheth & PozrikidisCa=0.3 - Sheth & PozrikidisCa=0.4 - Sheth & Pozrikidis(c)0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0t0.00.10.20.30.40.50.60.70.8DRe=100 =1Ca=0.05 - This StudyCa=0.15 - This StudyCa=0.2 - This StudyCa=0.4 - This StudyCa=0.05 - Sheth & PozrikidisCa=0.15 - Sheth & PozrikidisCa=0.2 - Sheth & PozrikidisCa=0.4 - Sheth & Pozrikidis(d)Figure 3.3: Time evolution of the droplet deformation with λ = 1, (a)Re =1, Ca = 0.2, 0.4, 0.9;(b)Re = 10, Ca = 0.2, 0.4, 0.8;(c)Re = 50, Ca =0.2, 0.3, 0.4;(d)Re = 100, Ca = 0.05, 0.15, 0.2, 0.4 - dotted and solid linescorrespond to the results from Sheth and Pozrikidis [73] and this study,respectively. 373.2. Single droplet3.2.1.2 Three-dimensional single dropletHere we compare our results to the results from the work of Feigel et al.[28]. [28] investigated the behaviour of a three-dimensional viscous dropletwith and without surfactant in a simple shear flow using simulations andexperiments in the Stokes flow regime. The flow parameters of this studyare listed in the following table:Parameter Re Ca λ Lx Ly LzValue 0.05 0 ≤ Ca ≤ 0.16 0.335 32 32 32Figure 3.4 shows our results and the results from Feigl et al. [28]. The de-formation of the droplet is reported by the length (L = L∗/d∗p) and breadth(B∗/d∗p) of the droplet; where L and B correspond to the largest diameterand the smallest diameter of the deformed droplet (nearly ellipsoidal shape),respectively.0.00 0.03 0.06 0.09 0.12 0.15 0.18Ca0.750.800.850.900.951.001.05BRe=0.05  |  =0.335 This StudyExperimentalNumerical(a)0.00 0.03 0.06 0.09 0.12 0.15 0.18Ca0.91.01.11.21.31.41.5LRe=0.05  |  =0.335 This StudyExperimentalNumerical(b)Figure 3.4: Results of this study and of [28]. (a) breadth and (b) length ofthe deformed droplet. λ = 0.335, Re = 0.05Our results match fairly well both the experimental results and the nu-merical results from [28]. Since all the results for the deformation of a singledroplet are usually reported in terms of the Taylor deformation parameterdefined in equation 2.52, we reproduced this parameter and analyze it aswell. To calculate the Taylor deformation from dimensionless length and383.2. Single dropletdimensionless breadth, we use equation 3.1.D =L∗d∗p− B∗d∗pL∗d∗p+ B∗d∗p=L−BL+B(3.1)Figure 3.5 shows the Taylor deformation parameter extracted from ourcomputed results and from the work of Feigl et al. [28]. Our results al-most perfectly overlap with their numerical results and also reasonably wellcapture the trend of the experimental results.0.00 0.03 0.06 0.09 0.12 0.15 0.18Ca0.000.050.100.150.200.250.30Dlow Re (0.05)  |  =0.335 This StudyExperimentalNumericalFigure 3.5: Results of this study on single droplet deformation in threedimensions and results obtained by [28]. λ = 0.335, Re = 0.053.2.2 Non-Newtonian fluidsIn the previous section we showed that our numerical model works for two-and three-dimensional Newtonian problems. Here we do the same analysisfor non-Newtonian, and specifically viscoelastic, fluids.3.2.2.1 Two-dimensional caseWe compare our results to the work of Chinyoka et al. [15]. [15] studied thedeformation of a single droplet with different configurations such as: 1) aviscoelastic droplet in a Newtonian matrix, and 2) a Newtonian droplet ina viscoelastic matrix. For these simulations, please recall that the Oldroyd-B constitutive model is used to model the viscoelastic phase. The flowproperties are as follows:393.3. Emulsion of dropletsParameter Re Ca λ De β Lx LyValue 2.4 1.2 1 0.4 0.5 8 4Figure 3.6 demonstrates a very satisfactory agreement between our re-sults (solid line) and the results obtained by Chinyoka et al.[15]. In thetwo-letter notation in this figure, N and V stands for “Newtonian” and“Viscoelastic”, respectively; the first letter gives the droplet fluid and thesecond represents the matrix fluid; e.g., NV denotes a Newtonian droplet ina viscoelastic fluid.Figure 3.6: Time evolution of the single droplet deformation when eitherthe dispersed or continuous phase is a viscoelastic fluid; Re = 2.4, Ca = 1.2,De = 0.4(β = 0.5), λ = 13.3 Emulsion of dropletsSo far we showed that our model for Newtonian and viscoelastic fluids per-forms well for cases involving a single droplet. In this section we validateour model for system of emulsions.For validating our model, we reproduce the results obtained by Zhouand Pozrikidis [97]. The parameters of this system are listed in the followingtable:403.3. Emulsion of dropletsParameter φ (%) Re Ca λ Lx LyValue 29.45 0.01 0.5 1 8 4For the sake of removing the effect of droplets’ initial positions, [97]repeated their simulations with two different sets of randomly positioneddroplets.0 5 10 15 20 25t1.01.11.21.31.41.51.6effRe=0.01 | Ca=0.5 | =1 Random1 - This StudyRandom2 - This StudyRandom1 - Zhou et al.Random2 - Zhou et al.Figure 3.7: The evolution of the shear effective viscosity (µeff ) in time fora Newtonian emulsion and different random initial position of the droplets;φ = 0.2945, Ca = 0.5, λ = M = 1.Figure 3.7 shows our results together with the results from the work ofZhou and Pozrikidis [97]. In order to remove the fluctuations in the results,we applied a noise filter known as “Savitzky–Golay filter”[71]. As shownin figure 3.7 results are fairly comparable. In table 3.1 results from thiscomparison is presented quantitatively in terms of time-averaged effectiveviscosity.413.3. Emulsion of dropletsThis Study Zhou and Pozrikidis [97]µeff SD µeff SDRandom 1 1.419 0.0351 1.446 0.0265Random 2 1.412 0.0291 1.425 0.0488Average 1.415 0.0321 1.436 0.0376Table 3.1: The results of this study and the results obtained by Zhou andPozrikidis [97]. SD stands for standard deviation.42Chapter 4Results: Single DropletDeformationIn this chapter, we present our results on the deformation of a single dropletin a simple shear flow. We consider both a Newtonian droplet and a non-Newtonian droplet in a Newtonian matrix. We start with the Newtoniandroplet cases and then turn our attention to the viscoelastic droplet cases.The parameters controlling the deformation and the inclination angle of thedroplet can be classified into two categories:• The relevant dimensionless numbers: Ca, Re, λ and De.• Geometric parameters and initial conditions.The objective of this chapter is to gain insight into how the above con-trolling parameters quantitatively affect the deformation of a single dropletin a simple shear flow. From experimental evidence, we assume that theapplication of gellants leads to a more viscous (larger λ) and more elastic(larger De) dispersed phase and a higher surface tension coefficient betweenthe two phases (lower Ca), hence increasing the stability of the droplets. Weput this assumption to the test and also investigate the effect of the otherparameters listed above.The majority of the results presented are in the Stokes (viscous) regimes.We select Re = 0.05 to represent a viscous regime in which inertia is quasi-negligible, hence modelling a quasi-zero Re flow. We consider larger Re upto 100 only when we investigate the effect of Re on the droplet deformation.In the case of Newtonian droplets, we present both two-dimensional andthree-dimensional results, while for viscoelastic droplets we only presenttwo-dimensional results.434.1. Newtonian droplets4.1 Newtonian droplets4.1.1 Two-dimensional cases4.1.1.1 Effect of confinementTo analyze the effect of the confinement on the deformation D, we set theparameters to the following values:Parameter Re Ca λ Lx LyValue 0.05 0.2 0.335 2, 4, 8, 16 2, 4, 8, 16Figure 4.1 shows the effect of Lx and Ly on the deformation parameterD. Note that the x-axis is not linear. As a reminder, Lx is the distancebetween two consecutive droplets in the streamwise periodic direction andLy is the height of the channel.21 22 23 24Lx0.240.260.280.300.320.340.360.380.40DRe=0.05 Ca=0.2 =0.335Ly = 2Ly = 4Ly = 8Ly = 16Figure 4.1: Effect of height(Ly) and length(Lx) of the channel on the defor-mation of a single Newtonian droplet.At fixed Lx, a decrease of Ly leads to larger local shear stresses exertedon the droplet, translating into a larger D. In other words, at fixed Lx, Dincreases when Ly decreases.444.1. Newtonian dropletsAt fixed Ly, a decrease of Lx brings a droplet and its periodic imagecloser to each other. As a result, its periodic image creates a disturbance ofthe flow field around the droplet and the droplet does not experience the fullamount of shear stress that it would experience in an unbounded domain, i.e.without the disturbance created by its periodic image. Consequently, thedroplet deforms less when its periodic image gets closer. In other words, atfixed Ly, D decreases when Lx decreases. Obviously, there exists a thresholdin Lx above which two consecutive droplets are sufficiently far away fromeach other such that one droplet does not disturb the flow field around thenext droplet, i.e., its periodic image. This threshold is Lx = 4 when Ly = 2,Lx = 8 when Ly = 4 and Lx = 16 when Ly ≥ 8.4.1.1.2 Effect of initial condition on the deformationIn chapter 3 we presented the results from Sheth and Pozrikidis [73]. In [73],the authors used the linear profile of a simple shear flow without dropletsas the initial condition on velocity. In some cases, especially the cases at ahigh Reynolds number and a low capillary number, we observed overshootsin the time evolution of the deformation parameter. To examine if the initialcondition is responsible for this behaviour, we recompute two sets of thosecases with an uniformly zero velocity field (quiescent flow) as the initialcondition on velocity.We set the parameters to the following values:Parameter Re Ca λ Lx LyValue 100 0.05, 0.4 1 2 2and plot in figure 4.2 the effect of the initial condition on velocity on thetime evolution of D.454.1. Newtonian droplets(a) (b)Figure 4.2: Effect of the initial condition (IC) on the deformation of a singleNewtonian droplet.As presented in figure 4.2, the linear velocity profile as the initial con-dition results in multiple overshoots of D before D reaches its steady statevalue. However, if the system starts from quiescence, D grows monotonicallyuntil reaching its steady state value without any overshoot. While these os-cillations are not straightforward to interpret from a physical viewpoint andmay simply be the result of wrong initial conditions on the velocity field, wesatisfactorily observe that the steady state value of D is the same regardlessof the initial condition on velocity.4.1.1.3 Effect of viscosity ratioTo analyze the effect of the viscosity ratio λ on the deformation D, we setthe parameters to the following values:Parameter Re Ca λ Lx LyValue 0.05 0.05 ≤ Ca ≤ 1 0.25 ≤ λ ≤ 5 4, 8 4, 8Figure 4.3 shows the deformation parameter D as a function of capillarynumber Ca for different viscosity ratios λ and for two domain sizes. Themissing data points are because of the high deformation that makes thedroplet and its periodic images coalesce such that defining the D is notpossible anymore.464.1. Newtonian droplets(a) (b)Figure 4.3: Effect of Capillary number Ca and viscosity ratio λ on thedeformation of a single Newtonian droplet - domain size: a)4× 4 , b)8× 8Figure 4.3 shows that D increases when λ decreases for all values of Ca.This is an expected result that confirms that the droplet deforms more whenits viscosity is small, i.e., its bulk offers less viscous resistance. For most Caand in particular at small Ca, D varies as the inverse of λ, and as expectedwhen λ tends to infinity D goes to zero as the droplet is similar to a rigidparticle.For large λ and for all λ at small Ca, D is a linearly increasing functionof Ca. The domain size has close to no effect on the deformation, except forthe lowest λ and largest Ca. As expected, the extrapolation of all the linescrosses the origin, in line with the fact that at asymptomatically zero Ca,the droplet does not deform.In figure 4.4 we present the plot b in figure 4.3 along with the deformationparameter calculated by the correlation proposed by Taylor (equation 4.1).Taylor’s theory is valid for flows at low capillary number (Ca 1).D =19λ+ 1616λ+ 16Ca2λ(4.1)We note that equation 4.1 confirms the scaling we observed in our re-sults, i.e., linear with Ca and inversely proportional to λ. The quantitativeagreement is also very satisfactory at low Ca as confirmed by Figure 4.4.474.1. Newtonian droplets0.0 0.2 0.4 0.6 0.8 1.0Ca0.00.20.40.60.81.0DRe=0.05 = 0.25 | This Study = 0.25 | Theory = 0.5 | This Study = 0.5 | Theory = 1.0 | This Study = 1.0 | Theory = 2.0 | This Study = 2.0 | Theory = 5.0 | This Study = 5.0 | TheoryFigure 4.4: Comparison of results with the correlation proposed by Taylor(equation 4.1) for the deformation parameter of a single Newtonian dropletfor different viscosity ratios λ and capillary numbers Ca - domain size: 8×8We plot in figure 4.5 the inclination angle θ for the same sets of parametervalues as in figure 4.3.484.1. Newtonian droplets(a) (b)Figure 4.5: Effect of Capillary number Ca and viscosity ratio λ on theinclination angle of a single Newtonian droplet in a simple shear flow -domain size: a)4× 4 , b)8× 8From figures 4.3 and 4.5, it is clear that the more deformed a dropletis, the smaller its inclination angle is. Figure 4.6 displays snapshots of thesteady state shape of the droplet as a function of Ca at λ = 0.5 and as afunction of λ at Ca = 1. It can be seen that when deformation is high, forinstance at Ca = 1 and λ = 0.5, the steady state droplet is not ellipsoidaland adopts an S-like shape.494.1. Newtonian dropletsCa = 0.05 Ca = 0.2 Ca = 0.5 Ca = 1λ = 0.5 λ = 1 λ = 2 λ = 5Figure 4.6: Snapshots of the steady-state deformed droplets. The first andsecond rows correspond to λ = 0.5 and Ca = 1, respectively. The colourcontour represents the adaptive mesh refinement such that the darker coloursrepresent finer grid cells. domain size: 8×84.1.1.4 Effect of Reynolds numberTo analyze the effect of the Reynolds number Re on the deformation pa-rameter D, we set the parameters to the following values:Parameter Re Ca λ Lx LyValue 0.05 ≤ Re ≤ 100 0.05 ≤ Ca ≤ 1 1 8 8Figure 4.7 shows the deformation parameter D as a function of capillarynumber Ca at different Reynolds numbers Re. The missing data points arebecause of the high deformation that makes the droplet breakup.504.1. Newtonian dropletsFigure 4.7: Effect of the Reynolds number Re at different capillary numbersCa on the deformation of a single Newtonian droplet in a Newtonian matrix- domain size: 8× 8Figure 4.7 shows that D increases when Re increases for all values ofCa. For low Ca, the interfacial forces are dominant and the deformationparameter is almost an independent function of Re. At fixed Ca, D increaseswith Re. At Re larger than 1, D is determined by a balance between inertiaand surface tension, and as expected viscous forces do not play a big roleanymore. In the inertial regime, it would then be more appropriate to usethe Weber number than the Capillary number.Figure 4.8 shows the inclination angle for the same sets of parametervalues as in figure 4.7. Figure 4.8 shows that highly deformed dropletshave a smaller inclination angle. At low Ca, the deformation parameterD is small and the droplet has a shape close to a perfect circle/sphere.Therefore, its inclination angle is not such an important parameter. At afixed Ca, increasing Re increases the inclination angle θ and orients thedroplet closer to the vertical axis.514.1. Newtonian dropletsFigure 4.8: Effect of Reynolds number Re at different Capillary numbersCa on the inclination angle of a single Newtonian droplet in a Newtonianmatrix - domain size: 8× 84.1.2 Three-dimensional cases4.1.2.1 Effect of confinementIn order to study the effect of confinement on D, we first study the effect ofLz and then study the effect of Lx and Ly. To analyze the effect of Lz onthe deformation parameter D, we set the parameters to the following values:Parameter Re Ca λ Lx Ly LzValue 0.05 0.2 0.335 16 16 2, 4, 8, 16Figure 4.9 shows the deformation parameter D as a function of Lz.524.1. Newtonian droplets21 22 23 24Lz0.3200.3250.3300.3350.3400.3450.350DRe=0.05  |  Ca=0.2  |  =0.335 Lx= Ly=16Figure 4.9: Effect of Lz on the deformation parameter D of a single three-dimensional Newtonian droplet. Lx = Ly = 16.Figure 4.9 shows that D increases when Lz increases up to a thresh-old above which Lz does not impact D but in general the effect of Lz isvery minor and it seems that even above Lz = 2 the effect of Lz is almostnegligible.To reduce the computational cost of analyzing the effect of confinementin x and y directions on D, we use a fixed value Lz = 2. To analyze theeffect of Lx and Ly on the deformation parameter D, we set the parametersto the following values:Parameter Re Ca λ Lx Ly LzValue 0.05 0.2 0.335 2, 4, 8, 16 2, 4, 8, 16 2Figure 4.10 shows the effect of Lx and Ly on the deformation parameterD.534.1. Newtonian droplets21 22 23 24Lx0.300.320.340.360.380.40DRe=0.05  |  Ca=0.2  |  =0.335 Ly = 2Ly = 4Ly = 8Ly = 16Figure 4.10: Effect of the height (Ly) and the length(Lx) of the channel onthe deformation parameter of a single three-dimensional Newtonian droplet.Lz = 2.At fixed Lx, a decrease of Ly results in larger local shear stresses exertedon the droplet, translating into a larger deformation parameter D. And atfixed Ly, increasing Lx increases D up to a point above which D does notchange with increasing Lx. These results are very similar to those obtainedin two dimensions.4.1.2.2 Effect of viscosity ratioTo analyze the effect of the viscosity ratio λ on the deformation parameterD, we set the parameters to the following values:Parameter Re Ca λ Lx Ly LzValue 0.05 0.05 ≤ Ca ≤ 1 0.25 ≤ λ ≤ 5 8 8 8Figure 4.11 shows the deformation parameter D as a function of capillarynumber Ca for different viscosity ratios λ.544.1. Newtonian droplets0.0 0.2 0.4 0.6 0.8 1.0Ca0.00.10.20.30.40.50.6DRe=0.05 = 0.25 = 0.5 = 1.0 = 2.0 = 5.0Figure 4.11: Effect of the capillary number Ca on the deformation of singledroplet in three-dimensional domain for different viscosity ratios λ. Lx =Ly = Lz = 8.Figure 4.11 shows that D increases when λ decreases for all values ofCa. This is an expected result that confirms that the droplet deforms morewhen its viscosity is small.For all λ at small Ca, D is a linearly increasing function of Ca. Asexpected, the extrapolation of all the lines crosses the origin, confirmingthat at asymptomatically zero Ca, the droplet does not deform. Again,these results are very similar to those obtained in two dimensions.4.1.2.3 Effect of Reynolds numberTo analyze the effect of Reynolds number Re on the deformation parameterD, we set the parameters to the following values:Parameter Re Ca λ Lx Ly LzValue 0.05 ≤ Re ≤ 100 0.05 ≤ Ca ≤ 0.75 1 8 8 8and plot in figure 4.12 the deformation parameter D as a function of Ca fordifferent Re.554.2. Two-dimensional case vs. three-dimensional caseFigure 4.12: Effect of the Reynolds number Re and the capillary numberCa on the deformation of a single Newtonian droplet in three-dimensionaldomain. Lx = Ly = Lz = 8.We plot in figure 4.12 the deformation parameter D as a function of Cafor various Re. Again, this plot is very similar to figure 4.7 obtained in twodimensions and the same comments apply.4.2 Two-dimensional case vs. three-dimensionalcaseIn chapter 3, we compared our results for a single three-dimensional dropletto the results obtained by Feigl et al. [28] (figures 3.4 and 3.5). Here weexamine the same flow configuration but in a two-dimensional domain. Toanalyze the effect of dimension on the deformation parameter D, we set theparameters to the following values:Parameter Re Ca λ Lx Ly Lz(3D)Value 0.05 0 ≤ Ca ≤ 0.16 0.335 32 32 32and plot in figure 4.13 the breadth B and the length L in two and threedimensions.564.2. Two-dimensional case vs. three-dimensional case0.00 0.03 0.06 0.09 0.12 0.15 0.18Ca0.750.800.850.900.951.001.05BRe=0.05  |  =0.335 This Study-3DThis Study-2DExperimentalNumerical(a)0.00 0.03 0.06 0.09 0.12 0.15 0.18Ca0.91.01.11.21.31.41.5LRe=0.05  |  =0.335 This Study -3DThis Study -2DExperimentalNumerical(b)Figure 4.13: (a) Breadth B (b) Length L, in two and three dimensionstogether with the results obtained in [28]. λ = 0.335, Re = 0.05.As shown in figure 4.13, the purple and green lines represent the two-dimensional case and the three-dimensional case, respectively. Results oftwo-dimensional and three-dimensional simulations are very similar and itseems that at low Re and low Ca, it is possible to predict the shape ofa three-dimensional droplet from two-dimensional simulations. This is animportant conclusion because simulations in two dimensions are drasticallyfaster than the simulations in three dimensions.Since all the results for the deformation parameter are usually reportedin terms of the Taylor deformation D, we reproduce this parameter andanalyze it as well. Figure 4.14 shows the deformation parameter D as afunction of Ca. Figure 4.14 confirms that D has almost the same value intwo dimensions and in three dimensions.574.3. Viscoelastic droplets0.00 0.03 0.06 0.09 0.12 0.15 0.18Ca0.000.050.100.150.200.250.30DRe=0.05  |  =0.335 This Study - 3DThis Study - 2DExperimentalNumericalFigure 4.14: The deformation parameter D of a single droplet two and threedimensions together with the results obtained in [28]. λ = 0.335, Re = 0.054.3 Viscoelastic droplets4.3.1 Two-dimensional cases4.3.1.1 Effect of confinementTo analyze the effect of the confinement on the deformation parameter D,we set the parameters to the following values:Parameter Re Ca λ De β Lx LyValue 0.05 0.2 0.335 0.4 0.5 2, 4, 8, 16 2, 4, 8, 16and plot in figure 4.15 the effect of Lx and Ly on D.584.3. Viscoelastic droplets21 22 23 24Lx0.240.260.280.300.320.340.360.380.40DRe=0.05 Ca=0.2 De=0.4 ( = 0.5) = 0.335Ly = 2Ly = 4Ly = 8Ly = 16Figure 4.15: Effect of height(Ly) and length(Lx) on the deformation of asingle viscoelastic droplet in a Newtonian matrix in a simple shear flow.Re = 0.05, Ca = 0.2, De = 0.4, β = 0.5, λ = 0.335.The effect of confinement is very similar to the case of a Newtoniandroplet. Indeed, figure 4.1 and figure 4.15 are very similar. We can thenconclude that the viscoelastic properties of the droplet do not have anyimpact on the effect of the confinement.4.3.1.2 Effect of the viscosity ratioTo analyze the effect of viscosity ratio λ on the deformation parameter D,we set the parameters to the following values:Parameter Re Ca λ De β Lx LyValue 0.05 0.05 ≤ Ca ≤ 1 0.25 ≤ λ ≤ 5 0.4 0.5 8 8We plot in figure 4.16 and in figure 4.17 the deformation parameter Dand the inclination angle θ, respectively, as a function of Ca for different λ.594.3. Viscoelastic dropletsFigure 4.16: Effect of the capillary number Ca for different viscosity ra-tios λ on the deformation parameter D of a single viscoelastic droplet in aNewtonian matrix. Re = 0.05, De = 0.4, β = 0.5.As shown in figure 4.16, an increase in viscosity ratio λ leads to lowerdeformation parameter D at all Ca. This result confirms that a dropletwith higher viscosity deforms less, because its bulk offers higher viscousresistance. For most Ca and in particular at small Ca, the deformationparameter D varies as the inverse of λ. Moreover, for large λ and for all λat small Ca, D is linearly increasing function of Ca.We plot in figure 4.17 the inclination angle θ for the same sets of param-eter values as figure 4.16. In all λ the inclination angle θ decreases as Caincreases, and at fixed Ca, θ increases when λ increases.604.3. Viscoelastic dropletsFigure 4.17: Effect of the capillary number Ca for different viscosity ratiosλ on the inclination angle θ of a single viscoelastic droplet in a Newtonianmatrix. Re = 0.05, De = 0.4, β = 0.5.4.3.1.3 Effect of the Reynolds numberTo analyze the effect of the Reynolds number Re on the deformation pa-rameter D, we set the parameters to the following values:Parameter Re Ca λ De β Lx LyValue 0.05 ≤ Re ≤ 100 0.05 ≤ Ca ≤ 1 1 0.4 0.5 8 8and plot in figure 4.18 and in figure 4.19 the deformation parameter D andthe inclination angle θ, respectively, as a function of Ca for different Re.These two plots are very similar to the plots obtained for the Newtoniancases (see figures 4.7 and 4.8) and the same comments apply. Comparisonof these figures to their Newtonian counterparts show that the deformationparameter D has slightly larger value for the Newtonian cases. Therefore,as expected, the inclination angle θ has a slightly smaller value for theNewtonian cases. These results are in line with the results reported in[42, 43, 48, 74].614.3. Viscoelastic dropletsFigure 4.18: Effect of the capillary number Ca for different Reynolds num-bers Re on the deformation parameter D of a single viscoelastic droplet ina Newtonian matrix. λ = 1, De = 0.4, β = 0.5.624.3. Viscoelastic dropletsFigure 4.19: Effect of the capillary number Ca for different Reynolds num-bers Re on the deformation parameter D of a single viscoelastic droplet ina Newtonian matrix. λ = 1, De = 0.4, β = 0.5.4.3.1.4 Effect of viscoelasticityTo analyze the effect of viscoelasticity (De and β) on the deformation pa-rameter D, we set the parameters to the following values:Parameter Re Ca λ De β Lx LyValue 0.05 0.2, 1 1 0.1 ≤ De ≤ 10 0.1 ≤ β ≤ 0.9 8 8and plot in figures 4.20 and 4.21 the deformation parameter D as a functionof De for different values of β at Ca = 0.2 and Ca = 1, respectively.Both figures 4.20 and 4.21 show a non-monotonic behaviour of D for allvalues of β with a decrease followed by an increase as De increases.634.3. Viscoelastic dropletsFigure 4.20: Effect of the Deborah number De for different values of β onthe deformation of a single viscoelastic droplet in a Newtonian matrix in asimple shear flow. Re = 0.05, Ca = 0.2, λ = 1.As expected, for high values of β, i.e., more solvent contribution thanpolymer contribution, the deformation parameter does not change signif-icantly with De. At low values of De, D is a monotonically decreasingfunction of β. Then at De ≈ 1, D reaches a minimum and then increaseswith De for De ≥ 1. Note however that the magnitude of the variations ofD with De is quite small. In fact, for 0 ≤ De ≤ 10, D varies between 0.097and 0.102 for Ca = 0.2, i.e., a relative variation of roughly 5%, and between0.38 and 0.48 for Ca = 1, i.e., a relative variation of roughly 25%. Whilea 25% variation is substantial, comparing figure 4.16 to figures 4.20-4.21indicates that the viscosity ratio λ has a much stronger influence than theDeborah number on D.644.3. Viscoelastic dropletsFigure 4.21: Effect of the Deborah number De for different values of β onthe deformation of a single viscoelastic droplet in a Newtonian matrix in asimple shear flow. Re = 0.05, Ca = 1, λ = 1.Such non-monotonic behaviour of D with De is reported by Yue et al.[96] for a Newtonian droplet in a viscoelastic matrix. In the case of vis-coelastic droplets as considered in our study, for λ > 1, Mukherjee andSarkar [53] showed the same non-monotonic behaviour of D with De. Onthe other hand, Pengtao et al. [96] reported a monotonic decrease of thedeformation parameter with respect to the increasing Deborah number butrestricted their study to De ≤ 2. In figure 4.22, we present results of [96] andour results with the parameters listed in the following table. In figure 4.22,the deformation parameter D is normalized with respect to the deformationparameter of the case with De = 0. Inspecting figure 4.22, the results from[96] seem to reach a minimum at De ≈ 2 and could very well increase forDe > 2 if the authors had reported results for a larger range of De. So themajor difference between the results from [96] and our results might be thatour D versus De plot attains at minimum at De ≈ 1 while results from [96]attain a minimum at De ≈ 2. Please also note that the difference in D at theminimum is pretty low (0.97 compared to 0.96). Consequently, accountingfor the differences between our work and [96] in the computational methodsand the mesh resolution, the agreement between our results and the resultsfrom [96] is deemed to be satisfactory.654.4. SummaryParameter Re Ca λ De β Lx LyValue 0.05 0.2 1 0 ≤ De ≤ 2 0.5 8 80.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00De( = 0.5)0.950.960.970.980.991.001.01D/D(De=0)Re=0.05 | Ca=0.2 | =1 This StudyPengtao et al.Figure 4.22: Deformation parameter D from a work of Pengtao et al [96]together with the D obtained from this study as a function of the Deborahnumber De. β = 0.5, Re = 0.05, Ca = 0.2, and λ = 1.4.4 SummaryIn this chapter we investigated the effect of the following parameters:• Capillary number Ca,• Reynolds number Re,• Viscosity ratio λ,• Deborah number De,• Viscosity ratio β,• Confinement Lx/dp, Ly/dp,Lz/dp,• and initial condition.on the deformation parameter D and the inclination angle θ of a singledroplet subjected to an imposed shear rate. We obtained results showingthat:• an increase of Ca or Re leads to higher D and lower θ.• an increase of λ decreases D and increases θ.664.4. Summary• the Deborah number affects D and θ non-monotonically and its influ-ence is low.• decreasing Ly/dp increases D, however, decreasing Lx/dp or Lz/dpdecreases D.In the current work we extended the range of De in [4] to a larger range(up to De = 10). We showed that even up to De = 10 the viscoelasticity ofthe droplet has a non-monotonic and very limited impact on the deforma-tion parameter D. Therefore, it may only slightly increase or decrease thestability of the droplet that is primarily controlled by the capillary numberCa and the viscosity ratio λ.67Chapter 5Results: EmulsionsIn this chapter, we present our results on the rheological behaviour of andispersion of droplets in a simple shear flow. We consider both Newtoniandroplets and non-Newtonian droplets in a Newtonian matrix. We startwith the Newtonian droplet cases and then present our results for the non-Newtonian droplet cases. We analyze the effect of Ca, λ, De, and β on thefollowing properties of the emulsion:• The effective viscosity µeff• The average deformation parameter Davg• The average inclination angle θavgThe objective of this chapter is to extend the analysis conducted inChapter 4 on a single droplet to a dispersion of droplets and understandhow the above parameters affect the rheological properties of the emulsionin order to infer the impact of the application of gellants on the stability ofthe emulsion and the dispersion rate of the droplets.Since in emulsions, droplets do not deform homogeneously, defining thelength and breadth for them is not straightforward. Therefore, we used thelargest and smallest distance between the centre of mass and the interfaceof each droplet as its length and breadth, respectively.All the results presented thereafter are in the Stokes (viscous) regimes.As in chapter 4, we select Re = 0.05 to represent a viscous regime in whichinertia is quasi-negligible, hence modelling a quasi-zero Re flow. Moreover,we only consider a fixed domain size in each case, and the effect of thegeometric parameters are not covered in this study.In the case of Newtonian droplets, we present both two-dimensional andthree-dimensional results, while for viscoelastic droplets we only presenttwo-dimensional results.685.1. Newtonian Emulsions5.1 Newtonian Emulsions5.1.1 Two-dimensional emulsion5.1.1.1 Effect of the capillary numberTo analyze the effect of the capillary number Ca on the effective viscosityµeff , we set the parameters to the following values:Parameter Re φ (%) Ca λ Lx LyValue 0.05 5 ≤ φ ≤ 50 0.05 ≤ Ca ≤ 0.7 1 16 16Figure 5.1 shows the effective viscosity µeff as a function of the capillarynumber Ca for different volume fractions φ.Figure 5.1: Effective viscosity µeff of a two-dimensional Newtonian emulsionas a function of the capillary number Ca for different volume fractions φ.Re = 0.05 and λ = 1.As shown in figure 5.1, at all volume fractions φ, the effective viscosityµeff decreases as the capillary number Ca increases. The effect of Ca is veryweak at low volume fractions φ ≤ 10%. As the volume fraction increases,Ca plays a more significant role. At φ = 50% the ratio between µeff atCa = 0.05 and µeff at Ca = 0.7 is about 2, which is deemed to be significant.695.1. Newtonian EmulsionsThe extrapolation of all plots at Ca = 0 correspond to the effectiveviscosity of non-deformable droplets. However, the values of µeff at Ca =0 do not correspond to the values of suspensions of rigid particles as thebulk of the droplets does not move as a rigid body but as a viscous fluidwith the same viscosity as that of the matrix. To impose a rigid bodymotion and recover the effective viscosity of a suspension of rigid circularcylinders, we would need to set a very high value to λ such that the bulk ofdroplets is so viscous that it moves as a rigid body. We will re-discuss thisin subsubsection 5.1.1.2In figure 5.2, we plot Davg and θavg for the same sets of parameter valuesas in figure 5.1.(a) (b)Figure 5.2: (a) Average deformation Davg and (b) average inclination angleθavg of droplets in a two-dimensional Newtonian emulsion as a function ofthe capillary number Ca for different volume fractions φ. Re = 0.05 andλ = 1.As expected, at fixed φ, Davg increases and θavg decreases when Caincreases. We also observe that at a fixed Ca, Davg increases and θavgdecreases when φ increases. When φ increases, each droplet is more confinedand consequently experiences higher local stresses compared to a droplet ina lower concentrated emulsion, i.e., at a lower φ. This result is coherentwith the results obtained for the effect of confinement on the deformationparameter of a single droplet.Figure 5.3 displays snapshots of the emulsion at Ca = 0.5 for four differ-ent φ. In figure 5.3, colours represent the spatial resolution of the grid cells.705.1. Newtonian EmulsionsYellow cells correspond to the coarsest resolution and dark brown cells tothe highest resolution. At high volume fractions, most of the grid would beat the highest resolution as a result of the adaptive mesh refinement (AMR)algorithm that uses interfaces and flow kinematics to locally refine the grid.Once the grid is almost uniformly refined everywhere, there is no point inusing an AMR algorithm whose computational overhead is not negligible.In order to reduce the computational cost, we perform the computations athigh volume fractions on a simple regular Cartesian grid.715.1. Newtonian Emulsions(a) Ca = 0.5 — φ = 5% (b) Ca = 0.5 — φ = 10%(c) Ca = 0.5 — φ = 40% (d) Ca = 0.5 — φ = 50%Figure 5.3: Snapshots of Newtonian emulsions at λ = 1, Ca = 0.5 anddifferent volume fractions φ. φ = (a) 5% (b) 10% (c) 40% (d) 50%.5.1.1.2 Effect of the viscosity ratioTo analyze the effect of the viscosity ratio λ on the effective viscosity µeff ,we set the parameters to the following values:Parameter Re φ (%) Ca λ Lx LyValue 0.05 5 ≤ φ ≤ 50 0.2 0.25 ≤ λ ≤ 5 16 16725.1. Newtonian EmulsionsFigure 5.4 shows µeff as a function of λ for different volume fractionsφ. As shown in figure 5.4, µeff increases when λ increases. This is anexpected behaviour since higher viscosity ratios mean that the droplets aremore viscous and closer to rigid particles, and consequently µeff is closer tothe value we would obtain for a suspension of rigid circular cylinders at thesame φ.Moreover, the trends of the lines in figure 5.4 indicate that for all φ, theeffective viscosity µeff reaches a plateau for λ larger than a threshold. Thisthreshold depends on φ.Figure 5.4: Effective viscosity µeff of a two-dimensional Newtonian emulsionas a function of viscosity ratio λ for different volume fractions φ. Re = 0.05and Ca = 0.2.As figure 5.4 shows, at low volume fractions, the effective viscosity is notnoticeably sensitive to changes of λ when λ ≥ 1. Moreover, comparing µeffat low volume fractions in figure 5.4 and figure 5.1 indicates that µeff hasroughly the same value. Therefore, it can be concluded that for low volumefractions, µeff is almost only a function φ.In figure 5.5 we plot Davg and θavg for the same sets of parameter valuesas in figure 5.4. It shows that at fixed φ, Davg decreases and θavg increaseswhen λ increases. Moreover, at a fixed λ, Davg decreases and θavg increaseswhen φ decreases.735.1. Newtonian Emulsions(a) (b)Figure 5.5: (a) Average deformation Davg and (b) average inclination angleθavg of the droplets in a two-dimensional Newtonian emulsion as a functionof viscosity ratio λ for different volume fractions φ. Re = 0.05 and Ca = 0.2.One of the well-known correlations in the literature for the effective vis-cosity of emulsions is proposed by Eilers in 1943 [23] (equation 5.1).µeff =1 + Kφ2(1− φφM)2 (5.1)Where φM is the maximum random close packing, andK = (1 + 2.5λ)/(1 + λ).Eilers’ correlation is suggested for flows at low Ca and quasi non-deformabledroplets. We plot Eilers’ correlation together with our results at low Ca andsmall Davg in figure 5.6.745.1. Newtonian Emulsions(a) (b)(c)Figure 5.6: Eilers and Einstein correlation together with some of our re-sults.(a) Ca = 0.2, λ = 2 (b) Ca = 0.2, λ = 5 (b) Ca = 0.05, 0.1, λ = 1.Figure 5.6 confirms that when Ca decreases and λ increases, our resultsget closer to Eilers’ correlation. Figure 5.6 also shows that at low φ and forλ ≥ 1, our results match Eilers’ correlation but also Einstein’s correlation[24] µeff = 1 + 2.5φ equivalently well. Note that Einstein’s correlation wasdeveloped for a suspension of rigid spheres. However, at low φ, if Ca is nottoo small and λ ≥ 1, the presence of a few droplets in the matrix is only a1st order perturbation of the flow, hence the linear dependence of µeff withφ.755.1. Newtonian EmulsionsIn order to prove that we can properly recover Eilers’ correlation byconsidering droplets that do almost not deform, we run two additional sim-ulations with the following sets of parameter values:Parameter Re φ (%) Ca λ Lx LyValue 10 60 0.1 10, 500 8 8and obtain µeff . Since λ has a very large value, the particles are almostnon-deformable and results presented in table 5.1 are very close to Eilers’correlation as expected.µeffλ Eilers This Study10 16.03 15.62500 17.42 16.72Table 5.1: Results of this study together with Eilers’ correlation5.1.2 Three-dimensional domain5.1.2.1 Effect of capillary numberTo analyze the effect of the capillary number Ca on the effective viscosityµeff , we set the parameters to the following values:Parameter Re φ (%) Ca λ Lx Ly LzValue 0.05 5, 20 0.05, 0.2, 0.7 1 8 8 8and plot in figure 5.7 µeff as a function of Ca for different volume fractionsφ. We also plot Davg and θavg in figure 5.8. Comparing the results in figures5.7-5.8 with their two-dimensional counterparts in figures 5.1-5.2 shows that:• µeff in three dimensions has a larger value with the same behaviour,• Davg in three dimensions has a larger value with the same behaviour,i.e., the droplets experience larger deformations,• θavg in three dimensions has almost the same value and the samebehaviour,765.1. Newtonian EmulsionsSince the behaviour of the two-dimensional cases and three-dimensionalcases is similar, the same comments apply to the three-dimensional case.Figure 5.7: Effective viscosity µeff of a three-dimensional Newtonian emul-sion as a function of the capillary number Ca for different volume fractionsφ. Re = 0.05 and λ = 1.775.1. Newtonian Emulsions(a) (b)Figure 5.8: (a) Average deformation Davg and (b) average inclination angleθavg of the droplets in three-dimensional Newtonian emulsion as a functionof the capillary number Ca for different volume fractions φ. Re = 0.05 andCa = 0.2.Figure 5.9 displays a snapshot of the emulsion with φ = 5%, Ca = 0.7and λ = 1. This figure also shows the grid cells with AMR, and the z-component of the velocity.785.1. Newtonian EmulsionsFigure 5.9: A snapshot of droplet position, velocity and grid cells for theemulsion with Ca = 0.7 and φ = 5%.5.1.2.2 Effect of viscosity ratioTo analyze the effect of the viscosity ratio λ on the effective viscosity µeff ,we set the parameters to the following values:Parameter Re φ (%) Ca λ Lx Ly LzValue 0.05 5, 20 1 0.25, 1, 5 8 8 8We plot µeff in figure 5.10 as a function of the viscosity ratio λ fordifferent volume fractions φ. Plots of Davg and θavg are presented in figure5.11. By comparing these figures with their two-dimensional counterpartsin figures 5.4-5.5, the following conclusions can be made:• µeff in three dimensions has a larger value with the same trend,• Davg in three dimensions has a larger value with the same trend,• θavg in three dimensions has smaller value and the same behaviour,The plots in figures 5.10 and 5.11 are similar to the plots in figures 5.4and 5.5, respectively, and the same comments apply here.795.1. Newtonian EmulsionsFigure 5.10: Effective viscosity µeff of a three-dimensional Newtonian emul-sion as a function of the viscosity ratio λ for different volume fractions φ.Re = 0.05 and λ = 1.(a) (b)Figure 5.11: (a) Average deformationDavg and (b) average inclination angleθavg of droplets in three-dimensional Newtonian emulsion as a function ofviscosity ratio λ for different volume fractions φ. Re = 0.05 and Ca = 0.2.Overall, the differences between two-dimensional and three-dimensional805.2. Non-Newtonian Emulsionssimple shear flows of emulsions are primarily quantitative, but qualitativelywe observe the same trend. Three-dimensional simulations are hence valu-able in case we intend to compare numerical predictions to experimental dataand to derive enhanced correlations to be later used in coarse-grained three-dimensional models utilized in studies of industrial or real-life problems.However, three-dimensional computations of emulsions containing O(100)droplets are time consuming and we only had a chance to run a few of thesecomputations on the supercomputer Cedar.5.2 Non-Newtonian Emulsions5.2.1 Two-dimensional emulsionThe whole analysis in this section is conducted in a square domain with thefollowing size:Parameter Lx LyValue 16 165.2.1.1 Effect of the capillary numberTo analyze the effect of the capillary number Ca on the effective viscosityµeff , we set the parameters to the following values:Parameter Re φ Ca λ De βValue 0.05 5 ≤ φ ≤ 50 0.05 ≤ Ca ≤ 0.7 1 0.4 0.5and plot µeff as a function of Ca in figure 5.12.815.2. Non-Newtonian EmulsionsFigure 5.12: Effective viscosity of an iso-viscous viscoelastic emulsion (De =0.4 and β = 0.5) as a function of the capillary number Ca for different volumefractions φ.Figure 5.12 shows that an increase of Ca leads to a decrease of µeff . Theeffect of Ca on µeff is more significant in high volume fractions φ. In thedilute flows, µeff is a very weak function of Ca and almost only a functionof φ.For the same sets of parameter values as in figure 5.12 we plot Davgand θavg in figure 5.13. This plot is very similar to figure 5.2 obtained forNewtonian droplets and the same comments apply.825.2. Non-Newtonian Emulsions(a) (b)Figure 5.13: (a) Average deformation Davg and (b) average inclinationangle θavg of iso-viscous viscoelastic emulsion (De = 0.4 and β = 0.5) as afunction of the capillary number Ca for different volume fractions φ.5.2.1.2 Effect of viscosity ratioTo analyze the effect of viscosity ratio λ on the effective viscosity µeff , weset the parameters to the following values:Parameter Re φ Ca λ De βValue 0.05 5 ≤ φ ≤ 50 0.2 0.25 ≤ λ ≤ 5 0.4 0.5Figure 5.14 shows µeff as a function of λ for different volume fractionsφ. We also plot Davg and θavg as a function of the viscosity ratio λ in figure5.15. These plots show that an increase of λ results in an increase in µeffand θavg, and a decrease in Davg. Moreover, at a fixed λ, increasing thevolume fraction φ leads to an increase in µeff and Davg, but to a decreaseof θavg. Again, these plots are very similar to figures 5.4-5.5 and the samecomments apply.835.2. Non-Newtonian EmulsionsFigure 5.14: Effective viscosity of a viscoelastic emulsion as a function ofviscosity ratio λ for different volume fractions φ. Re = 0.05, Ca = 0.2,De = 0.4, and β = 0.5.0 1 2 3 4 50.00.10.20.30.40.5DavgRe=0.05 | Ca=0.2 | De=0.4( = 0.5) = 5% = 10% = 20% = 40% = 50%(a)0 1 2 3 4 5202530354045avgRe=0.05 | Ca=0.2 | De=0.4( = 0.5) = 5% = 10% = 20% = 40% = 50%(b)Figure 5.15: (a) Average deformation Davg and (b) average inclinationangle θavg of a viscoelastic emulsion as a function of viscosity ratio λ fordifferent volume fractions φ. Re = 0.05, Ca = 0.2, De = 0.4, and β = 0.5.845.2. Non-Newtonian Emulsions5.2.1.3 Effect of Deborah numberTo analyze the effect of the Deborah number on the effective viscosity µeff ,we set the parameters to the following values:Parameter Re φ Ca λ De βValue 0.05 5 ≤ φ ≤ 50 0.2 1 0.1 ≤ De ≤ 10 0.5and plot in figure 5.16 µeff as a function of De for different volume fractionsφ. Figure 5.16 shows that an increase of De results in a small decrease ofµeff . In dilute flows φ ≤ 10%, the effective viscosity is a very weak functionof the Deborah number De but even in the concentrated flows φ > 10%,only at the low Deborah numbers, µeff is affected by De in a slightly moresignificant way. However, the largest change of µeff occuring at φ = 50%and low De does not exceed about 10%. This behaviour is supported by theresults of [4]. In [4] authors examined the effect of De for De ≤ 3 in a verydilute flow and they reported that the viscoelasticity of the droplet does notcontribute significantly to the effective viscosity of the emulsion.Considering Ca = 0.2, the results presented in figure 5.1 can be comparedwith the results presented in figure 5.16. This comparison indicates thatµeff of an emulsion with Newtonian droplets is higher than the counterpartemulsion with viscoelastic droplets. Since De = 0 represents the Newtoniancase, the same conclusion can be made by extrapolating the lines in figure5.16.855.2. Non-Newtonian EmulsionsFigure 5.16: Effect of the Deborah number De on the effective viscosityµeff of a viscoelastic emulsion for different volume fractions φ. Re = 0.05,Ca = 0.2, and λ = 1.At all volume fractions φ except 50%, µeff is a slightly decreasing func-tion of De that reaches a plateau for De approximately larger than 5. Atφ = 50%, µeff features a faint minimum at De = 5 and then seems toslightly increase. But we only have a single data point to highlight thetrend of µeff reaching a minimum and then increasing, and the trend is notvery strong. Consequently it is hard to conclude at that stage.We plot in figure 5.17 Davg and θavg for the same sets of parametervalues as in figure 5.16. Figure 5.17 shows that Davg as a function of Defollows the same trend as µeff as a function of De. Both Davg and θavgare non-monotonic functions of De. Because of the scaling of y−axis, thenon-monotonicity of Davg and θavg is not as visible as in the single dropletanalysis (figures 4.20 and 4.21). Looking at figures 5.16 and 5.17-a, weobserve that µeff and Davg decrease together when De increases. The si-multaneous decrease of µeff and Davg is partly counter-intuitive or at leastcontrary to what we observe in the case of Newtonian droplets. Indeed, theincrease of µeff was always associated with droplets that would deform less.In fact, the ability of droplets to deform enables them to adapt to the sur-rounding flow field and lower the stress field around them, and in turn theviscous energy dissipation, resulting in a lower µeff than for less deformable865.2. Non-Newtonian Emulsionsdroplets. Conversely, rigid particles maximize energy dissipation as theyforce the flow field to go around them and hence increase stress magnitude.So in short, high deformation is associated to a lower equivalent viscosity.When droplets become more elastic, i.e., when De increases, we expect thatdroplets deform less, and this is confirmed by figure 5.17-a. However, we donot observe in figure 5.16 the corresponding increase of µeff , and insteadobserve a slight decrease. This counter-intuitive result remains unexplained.(a) (b)Figure 5.17: (a) Average deformation Davg and (b) average inclinationangle θavg of a viscoelastic emulsion as a function of the Deborah numberDe for different volume fractions φ. Re = 0.05, Ca = 0.2 and β = 0.5.5.2.1.4 Effect of viscosity ratio βTo analyze the effect of β on the effective viscosity µeff , we set the param-eters to the following values:Parameter Re φ Ca λ De βValue 0.05 10, 50 0.2 1 0.1 ≤ De ≤ 10 0.1, 0.5, 0.9For the case of φ = 10%, we plot in figure 5.18 the effective viscosityµeff as a function of De for different values of β. Figure 5.18 shows thatfor all the values of β, µeff decreases when De increases up to De = 5 andthen reaches a plateau, except for β = 0.1. Lower values of β correspond toless contribution from the Newtonian solvent and more contribution fromthe non-Newtonian polymer. Consequently, when β tends to 0, µeff shows875.2. Non-Newtonian Emulsionsa stronger dependence on De, and conversely when β tends to 1, µeff isalmost independent of De. This behaviour is confirmed in figure 5.18.0 2 4 6 8 10De1.101.121.141.161.181.20effRe=0.05 | Ca=0.2 | = 1.0 | = 10% = 0.1 = 0.5 = 0.9Figure 5.18: Effect of the Deborah number De on the effective viscosity µeffof a viscoelastic emulsion for different values of β. Re = 0.05, Ca = 0.2,λ = 1, φ = 10%.We plot Davg and θavg as a function of De for different values of β infigure 5.19. As with µeff , figure 5.19 confirms that Davg and θavg are moredependent on De when β decreases. A smaller value of β leads to a smallerdeformation parameter and a higher inclination angle. This is in line withthe results obtained for a single viscoelastic droplet (figures 4.20 and 4.21).However, in the plots of a single droplet, a clear non-monotonicity is ob-served. Results obtained here do not show a clear non-monotonic behaviourfor Davg . Results on θavg show a better defined behaviour with θavg increas-ing withDe before reaching a pseudo-plateau forDe ≥ 5. As already pointedout concerning variations with De, variations with β are counter-intuitive.In fact, figures 5.18 and 5.19-a reveal that when viscoelastic droplets deformless, µeff is lower, while we would expect µeff to be larger.885.2. Non-Newtonian Emulsions0 2 4 6 8 10De0.1000.1050.1100.1150.1200.1250.130DavgRe=0.05 | Ca=0.2 | = 1.0 | = 10% = 0.1 = 0.5 = 0.9(a)0 2 4 6 8 10De38.038.539.039.540.040.541.041.542.0avgRe=0.05 | Ca=0.2 | = 1.0 | = 10% = 0.1 = 0.5 = 0.9(b)Figure 5.19: (a) Average deformation Davg and (b) average inclinationangle θavg of a viscoelastic emulsion as a function of the Deborah numberDe for different values of β. Re = 0.05, Ca = 0.2, λ = 1, φ = 10%.To observe the effect of the volume fraction φ on the results in figures5.18 and 5.19, we plot the same data for φ = 50% in figures 5.20 and 5.21.Qualitatively, we observe the same behaviour for φ = 50% as for φ = 10%,except that the amplitude of the variations with De and with β is larger atφ = 50% than at φ = 10%.895.2. Non-Newtonian Emulsions0 2 4 6 8 10De1.61.71.81.92.02.12.22.32.4effRe=0.05 | Ca=0.2 | = 1.0 | = 50% = 0.1 = 0.5 = 0.9Figure 5.20: Effect of the Deborah number De on the effective viscosity µeffof a viscoelastic emulsion for different values of β. Re = 0.05, Ca = 0.2,λ = 1, φ = 50%.0 2 4 6 8 10De0.120.140.160.180.200.220.24DavgRe=0.05 | Ca=0.2 | = 1.0 | = 50% = 0.1 = 0.5 = 0.9(a)0 2 4 6 8 10De3334353637383940avgRe=0.05 | Ca=0.2 | = 1.0 | = 50% = 0.1 = 0.5 = 0.9(b)Figure 5.21: (a) Average deformation Davg and (b) average inclinationangle θavg of a viscoelastic emulsion as a function of the Deborah numberDe for different values of β. Re = 0.05, Ca = 0.2, λ = 1, φ = 50%.905.3. Summary5.3 SummaryIn this chapter we studied emulsions to understand how applying gellants tothe oil-spill change the properties of the oil/water emulsion. We investigatedthe effect of the following parameters:• Capillary number Ca,• Viscosity ratio λ,• Deborah number De,• and viscosity ratio βat different volume fractions φ (up to φ = 50%) on the effective viscosityµeff of the emulsion, the average deformation parameter Davg of the dis-persed phase and the average inclination angle θavg of the dispersed phase.We obtained results showing that:• an increase of Ca or a decrease of λ leads to a lower µeff , a lower θavg,and a higher Davg.• the effect of the Deborah number on µeff , Davg and θavg is limited.• increasing φ increases µeff and Davg and decreases θavg.In the current work, we studied a wide range of parameters for emulsionsof different volume fractions (5% ≤ φ ≤ 50%). We showed that even thoughthe elasticity can decrease the effective viscosity of the gelled-oil/water emul-sion, since its influence on the properties is low in comparison with Ca andλ, the properties of the emulsion are mostly controlled by Ca and λ. Be-cause the gelled-oil has a higher viscosity (higher λ) and a stronger surfacetension (lower Ca) than the oil, the effective viscosity of the gelled-oil/wateremulsion is higher. In practice, applying gellants to the oil-spill increasesflow resistance and decreases the dispersion rate of gelled-oil droplets..91Chapter 6ConclusionIn this project we investigated two main problems:1. A single droplet in a simple shear flow2. An emulsion of droplets in a simple shear flowWe carried out the numerical simulations using an open source softwarecalled “Basilisk” that is designed to solve partial differential equations on anadaptive Cartesian grid. To track the interface in a two-phase system, weused the volume-of-fluid method along with AMR, which allows for a moreaccurate prediction of the flow features.In the case of emulsions, to see the droplet-droplet interactions withoutthe effect of coalescence, and to have the same number of droplets through-out the simulations, a modification of the volume-of-fluid method has beendeveloped. Moreover, the Oldroyd-B model is used as the constitutive modelto describe the viscoelastic fluid.6.1 Single dropletAlong with the sets of validation simulations on the deformation parameterD of a single droplet in a simple shear flow, we analyzed the effect of thefollowing parameters on D:1. The relevant dimensionless numbers: Ca, Re, λ and De.2. The geometric parameters and the initial conditions.and the following results were obtained:• For all the cases (Newtonian, non-Newtonian, two-dimensional, andthree-dimensional), increasing the wall confinement (decreasing Ly)increases D of a single droplet. Also the smaller the distance betweentwo consecutive droplets in the streamwise direction (Lx) is, the lessdeformed the droplets are. Finally in three-dimensional cases, increas-ing Lz slightly increases D.926.2. Emulsion of droplets• The initial condition on the velocity only changes the time evolution ofD and can lead to fluctuations and overshoots of D before reaching itssteady-state value. However, the steady-state value of D is the sameregardless of the initial condition.• For flow regimes at low Ca and low Re, we showed that D has thesame value in both two-dimensional and three-dimensional flows.• Increasing Ca increases D, while increasing λ decreases D. D is amonotonically increasing function of Re, and at relatively high Re,depending on Ca (particularly at small Ca), the inclination angle θmay exceed 45◦.• For the case of viscoelastic droplets, D has a non-monotonic depen-dence on De, with a decrease followed by an increase when De in-creases. At lower values of β, De plays a more significant role in theflow characteristics.6.2 Emulsion of dropletsWe showed that the effective viscosity µeff of the dilute emulsions are almostonly a function of the volume fraction φ when the viscosity ratio λ is largeror equal to 1. The effect of the capillary number Ca, the viscosity ratio λ,the volume fraction φ, the Deborah number De and the viscosity ratio β areas follows:• For any set of parameters, increasing φ leads to a higher µeff , a higherDavg and a lower θavg.• When Ca increases µeff and θavg decrease while Davg increases.• When λ increases µeff and θavg increase while Davg decreases. Whenλ is very large, µeff tends to a plateau that corresponds to the valueobtained for a suspension of rigid spheres, provided Ca is small enough.• In emulsions of viscoelastic droplets, for all values of β, increasingDe only marginally decreases µeff for De ≤ 2. For De > 2, µeff isindependent of De.• At lower values of β, the effect of De on µeff becomes more significant.Moreover, at a fixed value of De, µeff increases with β.936.3. Contribution to the industrial problem6.3 Contribution to the industrial problemApplying a gellant on the oil spill creates units of aggregated oil dropletsglued together, which we refer to as “gelled-oil droplets”. In other words,applying gellants creates several gelled-oil droplets such that each gelled-oildroplet is actually a bundle of oil droplets glued together. In comparisonwith oil droplets, gelled-oil droplets have higher viscosity and higher surfacetension. Moreover, gelled-oil droplets exhibit viscoelastic properties. Thegelation process is shown schematically in figure 6.1.Figure 6.1: Schematic of the gelation process.Since the viscosity and the surface tension of the gelled-oil droplets arelarger than the viscosity and the surface tension of oil droplets, the resultsobtained in the current study allow to make the following conclusions:1. the viscosity of the gelled-oil/water emulsion is higher than the oil/wateremulsion,2. under the same conditions, gelled-oil droplets experience less deforma-tion than oil droplets.As a result, we can expect the following behaviour from an oil spill afterapplying a gellant compared with an oil spill with no treatment:1. due to the lower level of deformation, droplets in the gelled-oil/wateremulsion are more stable,2. due to the higher viscosity of the gelled-oil/water emulsion, the dis-persion rate of the oil spill is decreased.946.4. Future workThe level of viscoelasticity of the gelled-oil can increase or decrease thelevel of deformation and the viscosity of the gelled-oil/water emulsion (dueto the non-monotonic behaviour of µeff and Davg with respect to De).However, for the range of parameters studied in the current work and in thesimple shear flow, we did not observe a strong effect of the viscoelasticityon the deformation or the emulsion viscosity. Therefore, we observed thatthe properties of an emulsion in a simple shear flow are controlled by theviscosity ratio and the surface tension. Note that we only studied simpleshear flow configurations and thus the aforementioned observation is validin a simple shear flow only. The key feature of viscoelastic properties is thelevel of normal stresses in the flow compared to Newtonian properties. Theviscoelastic properties of the droplets might manifest in a more remarkableway in extensional flow configurations that were not covered in the presentwork.In conclusion, the obtained results in the current study confirm that thegelled-oil/water system is hydrodynamically more stable than the oil/watersystem and therefore favourable for the industrial applications.Key Contributions• ComputationsUsing an advanced volume-of-fluid method, emulsions can be stud-ied without coalescence at any volume fraction.• ComprehensionThe range of parameters studied in [4] is extended to a larger rangeof volume fraction φ (up to 50%) and of Deborah number De (upto 10). Our results confirm that viscoelasticity has a very minoreffect on the effective shear viscosity, and that this minor effect ispresent up to De = 10.6.4 Future workTo get more insight into the oil spill incident and into using gellants as aresponse method, there are different topics that constitute future areas ofresearch. Here we list some of these topics:956.4. Future workStratified flowStratified flows can mimic the oil slick on the water surface. By applying anoscillatory shear stress to the system, the reaction of the system can be stud-ied to analyze whether a mixture of oil/water will be created or the systemcan stay stable and stratified. Figure 6.2 shows a schematic of a strati-fied flow configuration. We studied this problem to some extent, but moreinvestigations are needed for a deeper understanding of the phenomena.Figure 6.2: Schematic of stratified flow of oil on the water surface experi-encing oscillating shear stress.Furthermore, for the case of an oil-slick on the water surface, due tothe presence of waves, the oil slick experiences strong extensional strains.Figure 6.3 shows a schematic of an oil (or gelled-oil) slick on a wavy sur-face. As mentioned, viscoelastic properties play a more significant role in anextensional flow, so it is important and of a fundamental interest to studyextensional flow problems to shed some light on the effect of the gelled-oilviscoelasticity on the behaviour of the system.966.4. Future workFigure 6.3: Schematic of an oil slick on the water surface experiencing ex-tensional stresses.Gelation process and its breakageTo model the gelation process, where the oil droplets attract and glue toeach other, we developed a model using an attractive/repulsive potentialfunction equation. We used equation 6.1 to define the forces between everytwo droplets.Force = 4[(qr)12 − (qr)6](6.1)Where , q, and r are constant values controlling the maximum atractionforce, the distance between two droplets where the force is zero, and thedistance between two droplets, respectively. Please note that the two powers12 and 6 are arbitrary at this stage, we could have used 8 and 4 or 4 and2, for instance. After droplets attracted each other and formed one largeunit, i.e., the gelled-oil droplet, we can study that under what shear stresslevel the gelled-oil droplet will break into individual droplets. Figure 6.4shows the schematic of the gelled oil and what can be expected upon theapplication of strong enough shear stress. Figure 6.5 displays a snapshot ofa preliminary simulation with this model. The colours in figure 6.5 representthe x-component of the velocity.976.4. Future work(a)(b)Figure 6.4: Schematic of droplets aggregation. (a) aggregated mode (b)broken-up mode after applying strong enough shear stress.Figure 6.5: A snapshot of the aggregation of droplets used for modellingthe gelation process. The colour contour represents the x-component of thevelocity.986.4. Future workThree-dimensional viscoelastic solverOne of the computational limitations in this project was the solver for vis-coelastic cases that is implemented in two dimensions only. Accordingly, itwill be helpful to be able to extend this solver to three dimensions in orderto examine the viscoelastic cases of this project in three dimensions.Elastoviscoplastic fluidThe gelled-oil droplets may also feature some viscoplastic properties. There-fore, modelling the gelled-oil droplets using an elastoviscoplastic constitutiveequation can be more realistic to predict the dynamics of the flows contain-ing gelled-oil droplets.Three-phase flowIn this thesis, we only studied two-phase systems. However, since largeportions of an oil spill is on top of the water surface and in contact with air,it is more realistic to model an oil spill using a three-phase (air/water/oil)system. The flow configuration we have in mind is the one depicted infigure 6.3 where a film of oil is sandwiched between water at the bottom andair at the top. 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