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Dynamics and rheology of sheared two-dimensional foam Mohammadigoushki, Hadi 2014

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Dynamics and Rheology of ShearedTwo-dimensional FoambyHadi MohammadigoushkiB.Sc., Amirkabir University of Technology (Tehran Polytechnic), 2007M.Sc., Amirkabir University of Technology (Tehran Polytechnic), 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Chemical and Biological Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)February 2014c? Hadi Mohammadigoushki 2014AbstractUsing a shear cell device, we have studied four associated problems in foamby experiments: Bubble-bubble coalescence in sheared two-dimensional foam;lateral migration of a single large bubble in an otherwise monodisperse foam;size segregation of bubbles in sheared bidisperse foam; and the effect of non-Newtonian rheology of foam on lateral migration of bubble. For bubble-bubble coalescence in sheared two-dimensional foam, we observed a thresh-old of shear rate beyond which coalescence of bubbles happens. The mostpromising explanation was the model based on the centripetal force withqualitative agreement with experimental results.Next we studied the dynamics of monodisperse foam in the presence ofa single bubble whose size is different from the neighboring bubbles. Wereported the lateral migration of a larger single bubble away from the wall.We also reported thresholds of shear rate and bubble size ratio beyond whichmigration occurs. In this study we modified the Chan-Leal model and pre-dicted the experimental trajectories of migrating bubbles.For bidisperse foams, we reported evolution in foam structure to a sizesegregated structure, in which large bubbles accumulate at the middle of thegap whereas smaller ones close to walls. Then, we adopted a model basedon convection-diffusion equation to account for both lateral migration andshear induced diffusion.Finally, we extended the second work by widening the gap of Couettecoaxial cylinder geometry. Similar to the second work, we found that largebubble migrates laterally to an equilibrium position close to the inner wall.We believe this new mechanism is the non-Newtonian feature of foam. Wecharacterized our foam by measuring its degree of shear thinning and alsoestimated its elasticity based on the literature data on foam. Then, wefound out for a shear thinning fluid bubble migrated to position even closerto the inner wall than in the foam while a bubble in Boger fluid migratedto a position closer to the outer cylinder. Therefore, for a viscoselastic fluidwhich has the same feature one would expect to see bubble migration to aposition between these two for two fluids.iiPrefaceThis PhD thesis entitled ?Dynamics and Rheology of Sheared Two-dimensionalFoam? presents the main features of the research that I carried out during myPhD study under supervision of Professor James. J. Feng. In this preface,the contributions and collaborations to the papers published or submittedfor publication from current thesis are briefly explained.? A version of chapter 3 has been published. H. Mohammadigoushki, G.Ghigliotti, and J. J. Feng (2012), Anomalous coalescence in shearedtwo-dimensional foam. Physical Review E 85, 066301 (2012). Undersupervision of J. J. Feng, and collaboration with Giovanni Ghigliotti,I did a comprehensive experimental work with theoretical study ofcoalescence in sheared two-dimensional foam and drafted the paper.J. J. Feng put his ideas and helped me to prepare the final version ofpaper.? A version of chapter 4 has been published. M. H. Mohammadigoushki,and J. J. Feng (2012), Size-differentiated lateral migration of bubblesin Couette flow of two-dimensional foam. Phys. Rev. Lett. 109,084502. Under supervision of J. J. Feng, following first study I per-formed experiments on cross stream-line migration of a single largebubble inside a monodisperse foam in a Narrow gap Couette deviceand drafted the paper. J. J. Feng helped me to explore the effect of dif-ferent parameters in experiments as well as digging more into physicsof the problem and he also helped me to prepare the final version ofpaper.? A version of chapter 5 has been published. H. Mohammadigoushki,and J. J. Feng (2013), Size segregation in sheared two-dimensionalpolydisperse foam. Langmuir 29, 1370-1378. Through a systematicresearch, I studied the size-based segregation of bubbles in bidisperseand polydisperse two-dimensional foam and developed a model to ex-plain the experimental observation. I conducted this study under su-pervision of J.J. Feng. I prepared the draft of the paper with help ofJ. J. Feng.? A version of chapter 6 has been submitted for publication. H. Mo-hammadigoushki, P. Yue and J. J. Feng (2013), Bubble migration iniiiPrefacetwo-dimensional foam sheared in a wide-gap Couette device: effectsof non-Newtonian rheology. Through a systematic research, I stud-ied the lateral migration of single large bubble in monodisperse two-dimensional foam in wide-gap Couette co-axial cylinder device andexplained the experimental observations both by theoretical modelsand simulation. Professor Pengtao Yue at Virgina Tech helped us toget the simulation results for migration of a bubble in shear thinningfluid. I conducted this study under supervision of J.J. Feng. I preparedthe draft of the paper with help of J. J. Feng.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiNomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and application . . . . . . . . . . . . . . . . . . 11.2 Basic elements of a liquid foam . . . . . . . . . . . . . . . . . 11.3 Liquid foams, wet and dry . . . . . . . . . . . . . . . . . . . 21.4 Stability of liquid foams . . . . . . . . . . . . . . . . . . . . . 21.4.1 Quasi-static porcesses . . . . . . . . . . . . . . . . . . 21.4.2 Beyond the quasi-static limit . . . . . . . . . . . . . . 41.5 Rheology of liquid foams . . . . . . . . . . . . . . . . . . . . 51.5.1 Experimental methods . . . . . . . . . . . . . . . . . 61.6 Objectives and contributions of this research . . . . . . . . . 81.7 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Experimental setup and methodology of research . . . . . 102.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Material characterization . . . . . . . . . . . . . . . . . . . . 102.3 Shear cell device and foam making procedure . . . . . . . . . 112.4 Imaging techniques and bubble size measurement . . . . . . 123 Coalescence of bubbles in sheared two-dimensional monodis-perse foam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.1 Critical rotational velocity for coalescence . . . . . . 133.2.2 Effect of bubble size and liquid viscosity . . . . . . . 153.2.3 Interfacial shape and bubble distribution . . . . . . . 15vTable of Contents3.3 Potential mechanisms for anomalous coalescence . . . . . . . 183.3.1 Shear precludes surfactant-stabilized films . . . . . . 193.3.2 Surface remobilization due to surfactant transport . . 203.3.3 Bubble compression due to inertia . . . . . . . . . . . 213.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 Lateral migration of a single large bubble in monodispersetwo-dimensional foam . . . . . . . . . . . . . . . . . . . . . . . 274.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2.1 Migration of bubble in a Newtonian fluid . . . . . . . 284.2.2 Migration in two-dimensional monodisperse foam . . 294.3 A hydrodynamic model to explain the migration in foam . . 314.3.1 Model development . . . . . . . . . . . . . . . . . . . 314.3.2 Thresholds for migration . . . . . . . . . . . . . . . . 334.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Size segregation in sheared two-dimensional polydispersefoam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.2 Size segregation in bidisperse foam . . . . . . . . . . . . . . . 375.2.1 Effect of shear rate . . . . . . . . . . . . . . . . . . . 405.2.2 Effect of area fraction of large bubbles . . . . . . . . 405.2.3 Effect of bubble size ratio . . . . . . . . . . . . . . . . 425.3 A migration-diffusion model . . . . . . . . . . . . . . . . . . 435.4 Polydisperse foam . . . . . . . . . . . . . . . . . . . . . . . . 475.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 Effect of non-Newtonian rheology on bubble migration insheared foam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 546.2.1 Bubble migration in foam . . . . . . . . . . . . . . . . 546.2.2 Bulk rheology of 2D foam and polymer solutions . . . 576.2.3 Bubble migration in shear-thinning and Boger fluids . 616.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.3.1 Effect of elasticity . . . . . . . . . . . . . . . . . . . . 646.3.2 Effect of shear thinning . . . . . . . . . . . . . . . . . 666.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 Conclusion and recommendations . . . . . . . . . . . . . . . 727.1 Summary of key findings . . . . . . . . . . . . . . . . . . . . 737.1.1 Coalescence of bubbles in sheared two-dimensional monodis-perse foam . . . . . . . . . . . . . . . . . . . . . . . . 73viTable of Contents7.1.2 Cross stream-line migration of a single large bubble inmonodisperse foams. . . . . . . . . . . . . . . . . . . 737.1.3 Size-based segregation in sheared two-dimensional poly-disperse foam . . . . . . . . . . . . . . . . . . . . . . 747.1.4 Effect of non-Newtonian rheology on bubble migrationin sheared foam. . . . . . . . . . . . . . . . . . . . . . 747.2 Significance and limitations . . . . . . . . . . . . . . . . . . . 757.3 Recommendation . . . . . . . . . . . . . . . . . . . . . . . . 76Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78viiList of Figures1.1 Typical Structure of a liquid foam. . . . . . . . . . . . . . . . . . 11.2 Elements of a liquid foam in a magnified structure. . . . . . . . . 21.3 Typical Structure of a dry foam (A) and a wet foam(B). . . . . . . 31.4 Typical Structure of a two-dimensional foams. (b) Liquid foam withgas-liquid boundary condition (bubble raft), (b) Two-dimensionalfoam with glass-liquid boundary condition and (c) two-dimensionalfoam with glass-glass boundary condition (Hele-Shaw cell). . . . . 41.5 Illustration of T1 and T2 events in dry foam. . . . . . . . . . . . 52.1 Schematic of the shear cell (not to scale). (b) A top view snapshotof 2D foam at rest with bubble radius of 500 ?m. . . . . . . . . . 112.2 (a) A top view snapshot of 2D foam at rest with bubble radius of? 540 ?m. (b) Typical velocity field of foam obtained by PIV at? = 5 rpm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1 (a) A foam with bubble radius R = 500 ?m in liquid I shows no signof coalescence when sheared at 60 rpm. (b) Large bubbles appearafter about 20 seconds of shearing at 75 rpm. . . . . . . . . . . . 143.2 Critical rotation speed ?cas a function of bubble radius R forliquids I, II and III (see Table 3.1). . . . . . . . . . . . . . . . . . 163.3 Bubble velocity profiles at three rotation speeds with liquid I. Bub-ble size R = 250 ?m. The line represents the analytical solutionfor a Newtonian fluid. . . . . . . . . . . . . . . . . . . . . . . . 173.4 Shape of the free surface for liquid sheared at ? = 60 rpm. Theradial distance is measured from the axis of rotation and the innerand outer boundaries are at 93 mm and 100 mm, respectively. . . . 183.5 Bubble distribution in a foam 10 min after shearing at ? = 60 rpm.R = 500 ?m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6 Distribution of the film radius a in a sheared bubble raft withbubble size of R = 500 ?m undergoing rotational speed of ? = 75rpm. The curve shows a fitted normal distribution. . . . . . . 233.7 The critical condition for coalescence corresponds to ? = 8 forsolution I, ? being computed for the innermost layer of bubblesusing the measured bubble velocity profile. For solutions II andIII, the critical ? values are 12.5 and 28.5, respectively. . . . . . . 24viiiList of Figures4.1 Migration trajectories of a single bubble (R = 0.7 mm) at ? = 7rpm. The curves represent the Chan-Leal formula (Eq. 4.1). Thetop inset illustrates the migration schematically (not to scale) andthe bottom one depicts the liquid meniscus above the bubble cal-culated from the model of Ref. [54]. . . . . . . . . . . . . . . . . 284.2 Bubble migration in a 2D foam. (a) Effect of the bubble size ratio? at ? = 4 rpm. The bubble radii are (in mm): (r,R) = (0.35, 0.5)for ? = 1.43; (0.39, 0.6) for ? = 1.54; (0.435, 0.7) for ? = 1.61 and(0.35, 1) for ? = 2.86. The solid and dashed curves are predictionsof Eq. (4.3) for ? = 1.61 and 2.86. (b) Effect of the rotational rate? for fixed bubble sizes (r,R) = (0.35, 0.7) mm. The curves arepredictions of Eq. (4.3) for ? = 3 and 7 rpm. . . . . . . . . . . . 304.3 Bubble deformation in different environments at ? = 7 rpm. Thelarge bubble (R = 0.7 mm) in (a) deforms much more in a foamof smaller bubbles (r = 0.35 mm) than alone in (b). (c) A smallerbubble (R = 0.4 mm) is shielded by its neighbors (r = 0.58 mm). . 314.4 Deformation parameter of a larger bubble in a 2D foam as a func-tion of the bubble size ratio ?. The error bars indicate the variationamong 7 shear rates tested, and the curve is a quadratic fit to thedata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.5 Threshold for bubble migration. ?1, computed from Eq. (4.4) forthe first row of the bubbles next to the wall, is plotted as functionsof ? for various ? values. Hollow, half-filled and filled symbolsindicate no migration, partial migration and complete migration tothe center. The dashed line is ?1= 0.4. . . . . . . . . . . . . . . 345.1 Size segregation in sample D under shear (? = 5 rpm). The up-per row consists of snapshots of the foam at different times: (a)t = 0; (b) t = 2 min; (c) t = 5 min. The lower row shows thecorresponding large bubble distributions ?3(y)/?3. . . . . . . . . 385.2 Three different initial configurations of sample D (top row), withthe large bubbles randomly distributed (a), near the walls (b) andsegregated into azimuthal segments (c), lead to the same quasi-steady distribution in the lower row after shearing at ? = 7 rpmfor 10 min. The arrow indicates the direction of shearing. . . . . . 395.3 Steady state distribution of the large bubbles in Sample C aftershearing at (a) ? = 3 rpm and (b) ? = 7 rpm for 10 min. Thesolid lines are predictions of the migration-diffusion model to bediscussed in subsection III.E. . . . . . . . . . . . . . . . . . . . . 405.4 Temporal evolution of the half-width of the large-bubble distribu-tion, ye(t), for Sample D at ? = 3 and 7 rpm. The solid and dashedlines indicate predictions of the migration-diffusion model. . . . . . 41ixList of Figures5.5 Steady-state distribution of the large bubbles in Sample B, D andF after 10 min of shearing at 7 rpm. These samples have the samebubble sizes but different area fractions for the larger bubbles (seeTable 1). The solid lines are predictions of the migration-diffusionmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.6 Temporal evolution of ye(t) for Samples C (?3= 10%) and E(?3= 30%) undergoing shear at 7 rpm. The solid and dashedlines indicate predictions of the migration-diffusion model. . . . . . 425.7 Effect of the bubble size ratio ? on (a) the steady-state distributionof the large bubbles, and (b) the transient to the steady state interms of ye. Sample D has a bubble size ratio ? = 2 and SampleG has ? = 2.5. Both have 20% average area fraction for the largebubbles and are subject to ? = 7 rpm. The solid and dashed linesrepresent the model predictions. . . . . . . . . . . . . . . . . . . 435.8 Steady-state bubble distributions in the polydisperse foam samplesH, I, J and K, after 10 min of shearing at 7 rpm. The sampleshave the same three bubble sizes, a1= 350 ?m, a2= 500 ?m anda3= 700 ?m, at different average area fractions: (a) Sample H,(?1,?2,?3) = (90%, 5%, 5%); (b) Sample I, (80%, 10%, 10%); (c)Sample J, (60%, 20%, 20%); (d) Sample K, (40%, 30%, 30%). Thearea fractions are normalized for each species. . . . . . . . . . . . 475.9 Steady-state bubble distributions in the polydisperse foam Sam-ples L, M, N and O, after 10 min of shearing at 7 rpm. The sam-ples have the same three bubble sizes, a1= 350 ?m, a3= 700?m and a4= 875 ?m, at different area fractions: (a) Sample L,(?1,?3,?4) = (90%, 5%, 5%); (b) Sample M, (80%, 10%, 10%); (c)Sample N, (60%, 20%, 20%); (d) Sample O, (40%, 30%, 30%). Thearea fractions are normalized for each species. . . . . . . . . . . . 496.1 Migration trajectories of bubbles of two size R = 1 mm (? = 2.79)and R = 1.4 mm (? = 3.91), released from different positions inthe foam sheared at different shear rates. The bubble center isgiven by s, its distance from the inner cylinder scaled by the gapwidth d = Ro? Ri. The curve shows Chan-Leal?s prediction forthe bubble size R = 1.4 mm undergoing shear rate of ?? = 5.71 s?1. 556.2 Shear flow curve of the two-dimensional foam measured in a rheome-ter with a bob-cup fixture. Two data sets are plotted along with abest-fitting curve to the Herschel-Bulkley equation (Eq. 6.4). . . . 576.3 Shear viscosity of xanthan gum solutions of various concentrations.The line indicates the foam viscosity. . . . . . . . . . . . . . . . . 586.4 Comparison of the power-law viscosity of xanthan solutions of dif-ferent concentrations (symbols) with that of the foam (horizontallines). (a) The power-law index n; (b) the consistency factor K. . 59xList of Figures6.5 Shear rheology of the Boger fluid, with open circles for the shearviscosity and filled diamonds for the first normal stress differenceN1. The straight line is a power-law fitting for N1with a slope closeto 2. The filled squares show the shear stress of the foam, which iscomparable in magnitude to N1of the Boger fluid, especially nearthe upper bound of the shear rate. . . . . . . . . . . . . . . . . 606.6 Migration trajectories of bubbles in the xanthan solution startingfrom different initial positions. The bubbles are of two sizes R = 0.6mm and R = 1 mm, and the shear rate is varied from ?? = 3.5 s?1to ?? = 8.62 s?1. . . . . . . . . . . . . . . . . . . . . . . . . . . 626.7 Migration trajectories of a single bubble in the Boger fluid. (a) Abubble of radius R = 1 mm released from two initial positions attwo shear rates. (b) The effect of bubble size at a fixed shear rate(?? = 4.77 s?1). The curves show the predictions of the Chan-Lealformula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.8 Calculated force for Boger fluid and Newtonian fluid vs radius ona bubble with radius of 1 mm in wide-gap Couette geometry usingChan-Leal formula. . . . . . . . . . . . . . . . . . . . . . . . . . 656.9 Dimensionless equilibrium position vs dimensionless radius of thebubble in Boger fluid sheared at ?? = 4.77 sec?1. Curve showsthe prediction of Chan-Leal?s model and the data are experimentalresults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.10 Bubble migration trajectories in shear thinning fluid for R/d = 0.05at different shear rates. . . . . . . . . . . . . . . . . . . . . . . . 676.11 Bubble migration trajectories in shear thinning fluid for ?? = 2.74s?1at different shear rates. . . . . . . . . . . . . . . . . . . . . . . . 686.12 Calculated force for shear thinning fluid and Newtonian fluid vsdimensionless position on a bubble with in wide-gap Couette ge-ometry. Newtonian force was evaluated using Chan-Leal formula. . 696.13 Normalized velocity versus normalized radius. Symbols show thelocal measurements by using PIV and curves show the correspond-ing predictions using global measurements. . . . . . . . . . . . . . 70xiNomenclatureSymbol Units (SI) DescriptionA J Hamaker constanta m Film radiusa ?? Short axes of deformed bubblea1,a2,a3m Bubble sizeCa ?? Capillary numberCae ?? Effective capillary numberc mol/m3 Bulk concentration of surfactantD m/sec?2 Surfactant diffusivityD ?? Deformation parameterDe ?? Dimensionless Deborah numberd m Gap widthF N ForceFb N Bumping forceFc N Capillary forceFw N Stokes wall repulsion forceh0m Initial thickness of filmhc m Critical film thickness for ruptureK Pa.secn Consistency factorkB m2kgs?2K?1 Boltzman constantl m Long axes of deformed bubbleMa ?? Dimensionless Marangoni numberM N.m Torque at the inner cylindern ?? Power law indexPe ?? Dimensionless Peclet numberR m Bubble sizeRi m Radius of inner cylinderRo m Radius of outer cylinderRe ?? Reynolds numberRg J/molK Universal gas constantr m Bubble sizers m Characteristic size of surfactantss ?? Dimensionless positionT K Absolute tempratureti sec Interaction time between two bubblestd sec Drainage time of the liquid film between two bubblesxiiNomenclatureu m/sec Tangential velocityvi m/sec Radial velocity at the inner cylindervm m/sec Migration velocityy ?? Dimensionless coordinate across the gapy? m Dimensional coordinate across the gapN1Pa First normal stress differencesGreek symbolsSymbol Units (SI) Description? Pa.s Shear viscosity of the fluid? Pa.s Shear viscosity of the fluid? kg.m?3 Density of the fluid? N.m?1 Surface tension of the fluid? N.m?1 Shear stress of the fluid?i N.m?2 Shear stress at the inner cylinder?o N.m?2 Shear stress at the outer cylinder?y N.m?2 Yield stress of the fluid? rpm Rotational velocity?c rpm Critical rotational velocity for coalescence?? sec?1 Shear rate? ?? Dimensionless time? mole/m2 Surfactant concentration at the interface of bubble?0mole/m2 Equilibrium surfactant concentration at the interfaceof bubble? ?? Coefficient of adsorption?v Pa Shear stress?M Pa Marangoni stress? m Thickness of boundary layer? ?? Bubble size ratio?0?? Threshold for bubble size ratio?0rpm Threshold for rotational velocity? ?? Dimensionless force?1?? Dimensionless force for the first innermost row ofbubbles in foam?1,?2,?3?? Average area fractions of the bubble species?1,?1,?1?? Local area fractions of the bubble species? ?? Dimensionless diffusion coefficientxiiiAcknowledgementsI completed this PhD thesis with the help and support of my supervisor,Professor James J. Feng, during which he provided me with his excellent andthoughtful supports and encouragements. He showed me how to be morecritic and to dig the problems thoroughly, and not to be disappointed whenthe hard times of research come. I also would like to thank Professor G. M.Homsy for giving me many valuable and insightful advices during my PhDstudies.xivChapter 1Introduction1.1 Background and applicationAqueous foams are highly concentrated dispersion of bubbles inside a sur-factant solution. Despite the fact that foams contain gas and liquid thatare simple fluids, their dynamics can be quite complex [13]. This complexbehavior finds unique applications in several industrial processes. For in-stance, their low density and high surface area make them good materialsfor flotation in which liquid foams are used to extract minerals from ore [40].Gases such as steam, carbon dioxide and hydrocarbon gases are injected intooil reservoirs to increase the recovery of oil. These gases are much less denseand less viscous than the oil they attempt to displace, so they tend to mi-grate to the top of the reservoir, leaving most of the oil behind. Foams canhelp these gases to sweep oil reservoirs more efficiently [40]. Liquid foamsare being used in daily life as well, in cosmetics and foods.1.2 Basic elements of a liquid foamFigure 1.1: Typical Structure of a liquid foam.A liquid foam is made up of some distinct structural elements (bubbles,films, and Plateau borders). This elegant structure is illustrated beautifullyin the images shown in figure 1.1.Films: In liquid foam bubbles are pressed together but are separatedby thin films. Although these are the most evident feature of the foam11.3. Liquid foams, wet and drystructure, they become significant only in stability of foam, since foamsbreak because of film rupture.Plateau borders: These are where films meet in threes along anedge. This region is a liquid-filled channel.Figure 1.2: Elements of a liquid foam in a magnified structure.1.3 Liquid foams, wet and dryFoam structure can be characterized by different parameters including thefoam quality and distribution of bubble size. Foam quality refers to thefraction of gas inside the sample of foam. With increasing of the foamquality, the shape of bubbles may change from spherical to polyhedral andfoam transforms from wet to dry.Fig.1.3 shows samples of a wet foam and a dry foam. Based on the sizedistribution of bubbles, foams can be monodisperse, bidisperse or polydis-perse. Liquid foams can also be categorized in terms of dimensionality to2D foam and 3D foam.Two-dimensional liquid foams: A so-called 2D foam is a monolayerof bubbles. Depending on how the monolayer is confined on the top and thebottom, there are three common configurations, as illustrated in Figure 1.4.Three-dimensional liquid foams: If the bubbles in foam are in con-tact with each other in three dimensions they form a three-dimensionalfoam.1.4 Stability of liquid foams1.4.1 Quasi-static porcessesA quasi-static process in a foam is one in which the relaxation of the struc-ture back to equilibrium is much faster than the time-scale at which the foam21.4. Stability of liquid foams Figure 1.3: Typical Structure of a dry foam (A) and a wet foam(B).is perturbed. Foams are unstable materials and therefore, surface activeagents or surfactants are added to the solution to stabilize them. Surfac-tant molecules have a hydrophobic and a hydrophilic part. When adsorbedto the gas-liquid interfaces, they reduce the surface tension and generateMarangoni stresses that would inhibit the tangential flow along the inter-face of bubbles and therefore, foam becomes more stable. It has been shownthat the foam is constantly evolving as soon as it is created. This compli-cates the measurements and therefore, alters experimental trends. Severalmechanisms result in foam structural evolution. The structural evolutionmay occur in mechanical equilibrium or under dynamic fields. These pro-cesses may act at the same time. At mechanical equilibrium, three processesmay lead to structural evolution , which are:i) Gravitational drainage: Due to the effect of gravity liquid flowsvertically and accumulates at the bottom of the column; this is gravitationaldrainage and leads to thinning of the film between bubbles and eventuallyrupture of the bubble.ii) Bursting of bubbles: Bubbles in foam may burst at the interfacewith free air. It has been shown that the burst of one bubble in a sample ofstatic foam might trigger some avalanches of bursting of neighboring bubblesas well (Vandewalle et al. 2002).iii) Coarsening or Ostwald Ripening. For polydisperse foams, theLaplace pressure in neighboring bubbles is different. In smaller bubbles31.4. Stability of liquid foamsFigure 1.4: Typical Structure of a two-dimensional foams. (b) Liquid foam withgas-liquid boundary condition (bubble raft), (b) Two-dimensional foam with glass-liquid boundary condition and (c) two-dimensional foam with glass-glass boundarycondition (Hele-Shaw cell).this pressure is higher than that in larger ones. This causes the gas todiffuse from small to large bubbles through liquid films; this process is calledcoarsening or Ostwald ripening. Consequently, strong coarsening may leadto topological changes called T1 and T2 events. T1 is a neighbor switchingevent in which bubbles switch their neighbors to gain a lower energy level(Weaire & Hutzler 1999). A typical schematic of T1 process is illustrated inFig. 1.5. T2 events happen when one small bubble completely vanishes (cf.Fig.1.5).1.4.2 Beyond the quasi-static limitSubject to shear, a liquid foam may undergo structure changes. For instance,in a flowing foam when two bubbles come into contact with each other, thefilm between two bubbles can thin into a critical thickness in which van derWaals forces can trigger film instability and consequently rupture. Hence,two bubbles coalesce with each other and form a larger one. In addition, afairly large bubble can break into smaller one in a flowing foam [27]. Gole-manov et al. (2008) sheared a three dimensional foam in a parallel disk41.5. Rheology of liquid foamsFigure 1.5: Illustration of T1 and T2 events in dry foam.geometry and observed bubble breakup above a critical shear stress that istwo orders of magnitude smaller than that for breakage of a single bubble.T1 events do not happen just as a result of coarsening; also shearing offoam may trigger T1 events [17, 46]. Finally, in flow of a confined foam,bubbles with different sizes might interact with each other and induce cross-streamline migration and an effective diffusion. Herzhaft (2002) sheared apolydisperse foam in a similar geometry and observed segregation of bub-bles across the gap according to the size; smaller bubbles were mostly foundcloser to the walls and bigger bubble at the middle of the gap. It is as yetnot clear whether this might be due to breakup, coalescence or migration ofbubbles or a combination thereof. Herzhaft (2002) also measured the shearstress in a parallel disk geometry and observed an overshoot in shear stressfollowed by reduction as a function of time in step strain tests. Interestingly,Golemanov et al. (2008) carried out similar experiments but observed thatthe shear stress increases as a function of time and attributed this to breakupof bubbles. Quilliet et al. (2005) observed migration of a bigger bubble in asheared two-dimensional foam toward the periphery sides (i.e. towards theside walls). The morphological changes mentioned above affect the rheologyand mechanical properties of the foam as a whole. Consequently, a funda-mental understanding of the structural and rheological evolutions of foam isof scientific as well as practical importance [30].1.5 Rheology of liquid foamsAlthough foams only contain fluids, they behave like viscoelastic solids orlike non-Newtonian liquids, depending on applied stress, liquid volume frac-tion and the time scale set by the inverse of frequency in oscillatory ex-periments or the time elapsed since the application of a transient stress or51.5. Rheology of liquid foamsstrain. The rheological behavior of foams, concentrated emulsions, pastesand many other soft materials is strikingly similar. Slow relaxations, agingand jamming phenomena are found in all of these forms of soft condensedmatter. Some might be due to generic mechanisms, acting on a mesosocopiclength scale, while others might arise from the phyico-chemical composi-tion of the materials. Subjected to small stresses, foam exhibits solid-likebehavior. Beyond a yield stress, it behaves as a liquid-like material. TheHerschel-Bulkley model is commonly used to describe foam rheology [62].Thus, the geometrical, hydrodynamical and rheological properties of gas liq-uid foam can be tuned to make it a uniquely versatile multiphase mixturefor a variety of process applications and product designs. It is therefore amaterial that is of broad interest to chemical engineers [72].1.5.1 Experimental methodsVarious experimental techniques are employed to characterize and analyzefoam deformation and flow [18, 30, 53]. The macroscopic response to ap-plied shear stress can be measured by conventional rheometers. For instance:Parallel plate, cone-plate and Couette cylinders have all been successfullyused as shear geometries. However, several precautions must be taken toobtain physically interpretable rheological results. The surfaces of the con-fining walls must be roughened to avoid wall slip. Alternatively, one canuse smooth surfaces and, in such cases, the foam-wall slip must be explicitlyconsidered in data analysis [63]. In addition, measures must be taken to en-sure foam stability during the experiment, with respect to liquid drainage,bubble coarsening and liquid evaporation at the contact with ambient atmo-sphere. Alternatively, one can study the coupling between foam aging (dueto bubble coarsening or size-based segregation) and the rheological foamproperties. In this case, the ageing process must be characterized for a foamsample, identical to that studied in the rheometer. For all these reasons,the rheological foam measurements are far from straightforward and the ex-perimental protocols should be designed carefully, depending on the specificsystem and aim of the study.Several methods have been used to characterize the bubble velocity pro-files and structural rearrangement dynamics in flowing foams. Magneticresonance imaging (MRI) detects the velocity distribution inside shearedfoam, while diffusing wave spectroscopy (DWS) provides statistical infor-mation about the rate of bubble rearrangements in strained and in flowingfoams [53, 60, 61]. Direct optical observations of bubble monolayers (2Dfoams) have provided rich information about the bubble shape and dynam-ics in flowing foams [39, 46, 70]. Direct observations of dynamics insidedry 3D foams have been carried out using optical tomography [69]. Theexperimental studies have clearly evidenced that the rheological response offoam involves processes in a wide range of length-scales. The deformation of61.5. Rheology of liquid foamsindividual bubbles creates the elastic stress of foams, while the yielding andplastic flow are the consequence of rearrangements in the bubble packing,and the viscous friction in the liquid films between neighbouring bubbles is asource of energy dissipation. At present, one of the most challenging and ex-citing research problems in foam rheology is to explain and predict the linksbetween the macroscopically observed foam behavior and the microscopicprocesses that govern this behavior.Our focus in this thesis will be on the mechanisms for structural evolutionof foams in dynamic state and on the correlation between their rheology andstructure.Due to the opacity of the three-dimensional foam, it is hard to directlyobserve the changes in its structure during flow and correlate them to therheological properties. To connect the rheology to the local behavior ofbubbles, therefore, researchers have studied flow of two-dimensional foamwhich are monolayers of 3D bubbles. Experiments on two-dimensional foamflow have been performed in different geometries. For example some authorshave confined foam bubbles in a Hele-Shaw cell in Couette geometry androtated the inner disk while having the outer cylinder stationary [17]. Theyreported localized flow profiles in which the velocity profiles show fast decayaway from the driving boundary. In the liquid-glass case, the foam is con-stricted from the top by a glass plate and at the bottom is in contact with asolution. For the third case, the bubble monolayer floats on the liquid and isexposed to air on top. This configuration is also known as the bubble raft. InCouette geometries, Dennin and co-workers have sheared bubble rafts with afixed inner disk and a rotating outer cylinder [46]. More recently Katgert etal. (2010) studied both bubble rafts and two-dimensional foam with glass-liquid boundary condition and showed that the normalized velocity profileis shear dependent in the presence of the upper wall and is independent ofshear rate for a freely floating bubble raft [39]. They concluded that theboundary condition plays an important role on the localization of velocityprofile.All studies so far on shearing two-dimensional foams have been focusedon low shear rates and the only topological changes observed is T1 events.For three dimensional foams Herzhaft (2002) and Golemanov et al. (2008)reported structural changes, but there is at present little understanding ofthe mechanism or mechanisms responsible for that structural evolution. Inaddition apparently there is contradiction in the evolution of shear stressin their measurements [27, 28]. Therefore, it is of high importance to mea-sure the rheological properties of foam flow and see the effect of structuralevolution on rheological properties. These are our motivations for carryingout a series of experiments to identify the mechanisms involved in structuralevolution for the flow of two-dimensional foam. In the following we will beexplaining the objectives of this work in detail.71.6. Objectives and contributions of this research1.6 Objectives and contributions of this researchWe propose a series of experiments to identify and investigate the mecha-nisms responsible for structural evolution of freely floating 2D foams (bubbleraft) in a Couette co-axial cylinder geometry undergoing shear. Details ofthe geometry will be given later. The objectives of the thesis are to answerthese fundamental questions:(1) Will there be breakup or coalescence in a sheared two-dimensionalmonodisperse foam?(2) Does a bubble whose size is larger than the neighboring bubbles mi-grates laterally with respect to the flow stream-lines?(3) What if we introduce more than one large bubble in a sea of smallerbubbles? Would we see the size-based segregation in sample of bidis-perse two-dimensional foam?(4) What is the contribution of non-Newtonian behavior of two-dimensionalfoam to the structural evolution?As will be seen, these have been accomplished to a good degree in theresearch described in Chapters 3-6. The main contributions of the thesiscan be summarized into four general items:? We have discovered an anomalous coalescence mechanism, wherebybubbles coalesce for shear rates above a threshold, as opposed to be-low a critical rate, which is the normal scenario for bubble and dropcoalescence. We also proposed an explanation for the anomalous coa-lescence.? We reported for the first time, the lateral migration of a single largebubble in an otherwise monodisperse foam. This cross-streamline mi-gration pushes the large bubble away from the walls. We developed acontinuum model to account for the migration.? We reported the size-based segregation in bidisperse and polydispersetwo-dimensional foams and developed a model based on migrationand shear induced diffusion to explain this process. We developed amigration-diffusion model that accurately predicts the size segregation.? We studied the effect of non-Newtonian behavior of foam on bubblemigration in a wide gap Couette device. We found that non-Newtonianrheology of foam changes the migration process and then we strivedto explain it by separating the viscous and elastic contributions inmigration process.81.7. Thesis outline1.7 Thesis outlineChapter 2 presents details of the experimental setup and research methodol-ogy. We describe a shear cell Couette device used for all the experiments tobe presented in the thesis. Additionally, we explain the characterization ofthe material used in this research and at the end we present the visualizingtechnique that was utilized in this research.Chapters 3-6 deal with the four research projects in turn. Chapter3 presents detailed results of bubble-bubble coalescence in sheared two-dimensional foam. We observed a threshold above which coalescence occurs.This threshold depends on the bubble size and liquid viscosity. Then, weoffered several mechanisms to explain this anomalous coalescence. The mostpromising one is the model based on the centripetal force.Chapter 4 presents new results on migration of a single large bubble ina sea of smaller bubbles. We report lateral migration of this single largebubble across the gap to a final equilibrium position which turned out to bethe middle of the gap. The migration occurs above some thresholds for shearrate and bubble size ratio. We modify the Chan-Leal formula to explain themigration in foam and also used a force balance to explain the presence ofthresholds for migration.Chapter 5 presents experimental results on bidisperse and polydispersetwo-dimensional foams. We report the size based segregation above thresh-olds for bubble size and shear rate, similar to migration study. Then, wedevelop a model to account for shear induce diffusion as well as lateral mi-gration.Chapter 6 presents an extension to chapter 4 in which we have widen thegap of the Couette cell device. We showed that this seemingly naive changein geometry elicits the non-Newtonian behavior of foam. Again, migrationof a single bubble in an otherwise monodisperse foam was investigated, butthis time bubble migrates to a final equilibrium position which is not at themiddle of the gap anymore. Modification of Chan-Leal formula no longerpredicts the experimental observations. Hence, the viscoelastic nature offoam comes to play. We rationalize the bubble migration experiments infoam by studying the migration in shear thinning and Boger fluids whichmimic the foam behavior.Chapter 7 summarizes the key results of the thesis, outlines the limita-tions of the current work, and makes recommendations for future work.9Chapter 2Experimental setup andmethodology of research2.1 MaterialsSince the experiments are performed on the two-dimensional liquid foam,the materials that we need to make the foam include: distilled water, glyc-erol (Fisher Scientific), dish washing liquid (Unilever, Sunlight) and ni-trogen, which will be provided by Praxair company in compressed steelcylinders. In addition to that for other experiments we have used xan-than gum (West Point Naturals) and Polyacrylamide (Sigma-Aldrich, Mw= 5,000,000-6,000,000).2.2 Material characterizationPrior to the actual experiments on the foam flow, we have performed someexperiments to characterize the materials that are present in the experi-ments.? Surface tension: Surface tension was measured by a tensiometer(Cole-Parmer, Surface Tensiomat 21 WU-59951-14) which is a Du NoyRing type Tensiometer. We measured surface tension for different liq-uids, including the pure liquids and soap solution at room temperature.This type of tensiometer uses a platinum ring which is fully submergedin a liquid. As the ring is pulled out of the liquid, the tension requiredis precisely measured in order to determine the surface tension of theliquid.? Density: Density of different fluids is measured by using a DensityMeter (Anton- Paar DMA 35N) at room temperature.? Rheology: Rheological properties such as shear viscosity and shearstress of all pure materials and soap solutions as well as two-dimensionalfoam are measured by using a rheometer (Malvern, Kinexus) and MCR(502) with a co-axial cylinder and cone-plate geometry.102.3. Shear cell device and foam making procedureFigure 2.1: Schematic of the shear cell (not to scale). (b) A top view snapshot of2D foam at rest with bubble radius of 500 ?m.2.3 Shear cell device and foam making procedureThe migration experiments are carried out in a modified Couette cell device.This cell consists of two cylinders: a stationary outer cylinder with theinner radius of R0= 10 cm and a rotating sharp-edged inner disk with twodifferent radii, Ri = 9.3 and 8.1 cm (Fig. 1a). The static liquid level isflush with the top surface of the inner and outer cylinders such that theinterface is pinned at the sharp corners. Furthermore, triangular teeth aremachined onto the solid surfaces to prevent slippage of first row of bubbles.Bubbles are produced by blowing nitrogen through an immersed capillarytube in a soapy solution using a pneumatic PicoPump (WPI, model PV-820). This method allows us to make an extremely uniform bubble size thatcan be fine-tuned by the nitrogen pressure. Moreover, the inner cylinder isattached to a servomotor that can be rotated from 0.1rpm to 100rpm usinga motor controller.112.4. Imaging techniques and bubble size measurement Figure 2.2: (a) A top view snapshot of 2D foam at rest with bubble radius of? 540 ?m. (b) Typical velocity field of foam obtained by PIV at ? = 5 rpm.2.4 Imaging techniques and bubble sizemeasurementMultiple cameras have been used for the experiments. We used a high-speedcamera (Megaspeed, MS 70K) to capture the evolving foam structure. Theframe rate ranges from 25 fps to 20,000 fps. In addition to that, we usedtwo other high-resolution cameras (Watec model 902B) to directly observethe structure of the foam and to measure the bubble velocity profile acrossthe gap by using Particle Image Velocimetry (PIV). For PIV we used anopen source Graphical User Interface (GUI) code in MATLAB developedby William Thielicke to track the position of bubbles in consecutive expo-sures [74]. By measuring the displacement of bubbles between consecutiveimages we can calculate the foam velocity profile across the gap. Fig. 2.2shows top view snapshots of foam at rest ( Fig. 2.2 (a)) and typical veloc-ity vectors obtained PIV method for sheared foam (Fig. 2.2 (b)). We alsomeasured the bubble size and bubble size distribution using microstructuralmeasurement software developed by Nahamin Pardazan Asia Co. [80]12Chapter 3Coalescence of bubbles insheared two-dimensionalmonodisperse foam3.1 IntroductionAs discussed in chapter 1, foams are fragile soft matter, with a microstruc-ture which is thermodynamically and mechanically unstable [2, 6, 68, 76].For three-dimensional foams there have been number of studies which inves-tigated the structural evolution in flowing foam. For instance, Golemanovet al. [27] observed breakup of bubbles sheared between parallel disks. Onthe other hand, in a similar geometry, Herzhaft [28] reported size-based seg-regation of bubbles in a polydisperse foam in which smaller bubbles foundto be close to the top and bottom disks while larger ones mainly accu-mulated in the middle. Conspicuously missing, however, is any report ofshear induced coalescence, a common occurrence in sheared emulsions [58].This has motivated us to study the possibility of bubble-bubble coalescencein a sheared foam. The most important difficulty with understanding thestructure-flow coupling in a 3D foam emanates from its opacity; Therefore,recently researchers have mainly focused on experimentation with 2D foams,i.e. monolayers of 3D bubbles. This way, they can easily see and correlatethe microstructure to flow properties. The only structural changes observedso far are T1 events [46]. Our main goal in this study is to investigate thestructural changes in 2D foams under more vigorous shearing. The mainfinding is a new type of bubble coalescence unexpected at the start. Weconsider several models for the anomalous coalescence, the most promis-ing one is the one based on inertia of the fluid which can explain most ofexperimental observations.3.2 Results3.2.1 Critical rotational velocity for coalescenceIn this work we found that there is a threshold for rotational velocity ?cabove which large bubbles start to appear quickly after the start of shearing133.2. ResultsFigure 3.1: (a) A foam with bubble radius R = 500 ?m in liquid I shows no signof coalescence when sheared at 60 rpm. (b) Large bubbles appear after about 20seconds of shearing at 75 rpm.(Fig. 3.1). At shear rates below ?c, no large bubbles appear during the longperiod of experiments and the foam morphology remains the same as thebeginning(? 30 min). The threshold is observed for all bubble sizes, liquidcompositions and surfactant concentrations that we tested.Our optical setup offers a 2 cm ? 2 cm viewing window that is fixed inspace. Thus, we capture only a small portion of the circular trajectory of thebubbles. The coalescence takes place very quickly after shearing above ?c.Therefore, we cannot capture the actual process of the coalescence as Ritaccoet al. [68] did for bursting of bubbles in static bubble raft. Coalescence isthus inferred from the appearance of large bubbles.This surprising result is not in line with the conventional wisdom thatcoalescence happens for gentler collisions, with an upper bound on the shearrate and a corresponding maximum capillary number [e.g. 10, 12, 19, 75,81]. The coalescence between two freely suspended bubbles or drops isdetermined by the competition between two time scales, the interactiontime ti and the drainage time td. The former is the time that the twobubbles spend to interact with each other, and scales with the inverse shearrate, while the latter is the time required for the liquid film between themto drain to a critical thickness such that van der Waals forces can trigger143.2. ResultsSolution Glycerin c ? (mPa?s) ? (mN/m)I 10 wt.% 5 wt.% 1.0? 0.1 27.0? 1.0II 30 wt.% 5 wt.% 1.8? 0.2 27.0? 1.0III 50 wt.% 5 wt.% 4.2? 0.4 27.0? 1.0Table 3.1: Composition and properties of the solutions.the rupture of film [10]. The requirement of ti  td for coalescence leadsto an upper critical capillary number. Such a criterion has been verified byextensive studies that examined various parameters in the process, includingdrop size, viscosity of the fluids, lateral offset of the colliding drops, andsurfactant concentration [e.g. 32, 33]. This apparently does not apply in ourcase.3.2.2 Effect of bubble size and liquid viscosityWe have examined the effects of the bubble size R and liquid viscosity ?on coalescence process. Fig. 3.2 shows that the critical angular velocity ?cincreases with both R and ?. This is again surprising: it implies that theanomalous coalescence cannot be analyzed in the conventional frameworkof a capillary number, i.e. in terms of viscous forces competing with surfacetension. There must be a new mechanism at play that was absent in theconvention scenario of collision and coalescence.If we draw straight lines through the data points in this log-log plot,their slopes give the scaling ?c ? R0.27?0.02. The dependence of ?c on theliquid viscosity ? is rather weak: ?c ? ?0.1.3.2.3 Interfacial shape and bubble distributionWe have also recorded the shape of the foam-air interface and spatial redis-tribution of the bubbles under shear. These may offer potential clues to thecause of the anomalous coalescence.Intuitively one may expect the centripetal force to deform the interfaceon the rotating liquid. This is not the case; the interface exhibits no observ-able variation in its elevation across the gap even at the highest ? tested.This is largely due to the effect of pinning of the interface at the solid walls.In the experiment, we fill the gap between the cylinders such that the staticliquid surface is flush with the tops of the cylinders. Once shearing starts,the free surface is subject to the centripetal force as well as anchoring onthe sharp edges of the inner and outer walls. Using the velocity profilesof Fig.3.3, we have computed the shape of the interface, shown in Fig. 3.4for a rotational speed of 60 rpm. Thus, the anchoring of the surface limitsits undulation to negligible amounts (< 0.2 mm; one order of magnitudesmaller than without anchoring). Furthermore, the bubbles are held mostly153.2. ResultsFigure 3.2: Critical rotation speed ?cas a function of bubble radius R for liquidsI, II and III (see Table 3.1).163.2. ResultsFigure 3.3: Bubble velocity profiles at three rotation speeds with liquid I. Bubblesize R = 250 ?m. The line represents the analytical solution for a Newtonian fluid.173.3. Potential mechanisms for anomalous coalescence93 94 95 96 97 98 99 100?0.1?0.0500.050.1Radial distance (mm)Free surface profile (mm)Figure 3.4: Shape of the free surface for liquid sheared at ? = 60 rpm. The radialdistance is measured from the axis of rotation and the inner and outer boundariesare at 93 mm and 100 mm, respectively.underwater by surface tension [54].At relatively low rotation speed, the bubbles slide past each other in rows.At higher ?, however, there appears to be radial motion of the bubbles thatdisrupts the layers. As a result, bubbles tend to be more tightly packed inthe inner half of the gap than in the outer half. In fact, voids of clear liquidstart to appear in the outer region (Fig. 3.5), which quickly disappear afterthe shearing stops. The most plausible cause of this spatial inhomogeneityis the centripetal force exerted by the rotating liquid on the bubbles. Thismay be the cause of this anomalous coalescence as explained below.3.3 Potential mechanisms for anomalouscoalescenceOne should not that there are some differences between our experimentsand previously reported drop- or bubble-coalescence experiments. In shear-induced drop collision, the shear brings into contact two freely suspendeddrops that would otherwise not interact with each other at all. In our bub-ble raft, on the other hand, bubbles are in close contact with each othereven without shear. Why should the static bubbles be immune to coales-cence while the sheared ones are not? Moreover, our bubbles are covered bysurfactants, and there is also ample supply of it in the surrounding liquid.Finally, our coalescence occurs at relatively high flow rates, much higher183.3. Potential mechanisms for anomalous coalescenceFigure 3.5: Bubble distribution in a foam 10 min after shearing at ? = 60 rpm.R = 500 ?m.than typical of drop-coalescence experiments [83]. In the following, we willexplore these differences for clues to the anomalous coalescence.3.3.1 Shear precludes surfactant-stabilized filmsThe first idea is to investigate difference between the stability of static foamand the coalescence in a sheared one. Static foams are stabilized by sur-factants because the latter form regular structures in liquid films that aresufficiently thin [64]. If the bulk surfactant concentration is below CMC,bilayers of surfactants form the so-called ?black films? [14]. At higher con-centrations, micelles arrange themselves into a more or less regular colloidalstructure in the liquid film, producing thick stable films [64]. In either case,the surfactant structure contributes a disjoining pressure that prevents liq-uid drainage and stabilizes the static foam. Conceivably, vigorous shearingmay disrupt such surfactant structures or prevent them from forming in thefirst place. This could be a mechanism for the observed coalescence.In a recent study, Denkov et al. [20] demonstrated how the black filmmay cause jamming in flowing foams. In essence, they assume that for lowenough shear rates, there is enough time for the film between neighboringbubbles to thin down to a critical thickness where attractive forces act toproduce black films. Then the bubbles are locked into a rigid structurethat resists the shearing, and the foam is jammed. In our experiment, thesurfactant concentration is above CMC and the stable structure should bethe thick stable film instead of the black film [64].At low ?, we observe nonhomogeneous shearing with large domains ofjammed bubbles. Around ? = 3 rpm, all such domains unjam and thebubble raft starts to shear more or less uniformly. We thus take this to bethe threshold for the destruction of the thick stable films. However, largerbubbles only start to appear at a much higher rotational speed of ? ?193.3. Potential mechanisms for anomalous coalescence60 rpm. Therefore, the unjamming cannot be the cause of the anomalouscoalescence, which requires much more vigorous shearing.3.3.2 Surface remobilization due to surfactant transportThe second mechanism which might be responsible for coalescence in foamis surfactant transport. Drainage in liquid films carries surfactants alongthe interface, and creates a gradient in surfactant concentration along theinterface of bubbles. This in turn produces a tangential Marangoni stressthat resists the interfacial flow. Conceivably, sufficiently strong shearingmay produce a viscous stress ?v that overpowers the Marangoni stress ?M ,thereby remobilizing the bubble surfaces. Once bubble surface is mobilizedthe film drainage will be facilitated and so will coalescence. This suggestsusing the Marangoni number Ma = ?M/?v ? 1 as a criterion for the observedanomalous coalescence. In studying pairwise collision of surfactant-covereddrops, Yoon et al. [83] used this argument to rationalize the appearance of a?transition capillary number? for lower bulk surfactant concentrations suchthat coalescence occurs above it but not below. This seems to be consistentwith our anomalous coalescence.Therefore, we will study the antagonism between Marangoni stress andviscous stress as a potential explanation for the anomalous coalescence ob-served in our experiment. For soluble surfactants, the surface concentrationis determined by two steps: bulk diffusion of surfactants toward the inter-face and adsorption onto the interface [49]. For our commercial detergent,it is not possible to estimate the relative rates of these two steps. We willexamine the cases of either one being the limiting step by adapting the clas-sical analysis of Levich on falling drops [49]. In our problem, the liquid flowoutside the bubbles is due to shear instead of sedimentation. Thus, we needto replace the characteristic liquid velocity in Levich?s calculations by ??R,?? being the local shear rate.If adsorption is the limiting step that dictates the surfactant distribution? on the bubble surface, one can estimate the surface concentration gradientas [49]:|??| ??0??? R , (3.1)where ?0is the equilibrium concentration and ? is the coefficient of adsorp-tion. This implies that the Marangoni stress?M = |??| =(????)?0|??| (3.2)is proportional to the shear rate. Since the shear stress ?v on the surface isalso proportional to ??, the ratio ?M/?v will be independent of the shear rate.This cannot explain the fact that coalescence happens above a thresholdrotational speed.203.3. Potential mechanisms for anomalous coalescenceWhen the bulk diffusion determines the surfactant distribution on thebubble surface, Levich [49] estimated |??| and hence |??| based on a bound-ary layer thickness ? ? (DR/??)1/3, D being the bulk diffusivity:?M = |??| ??0???DR(???c), (3.3)where c is the bulk concentration of the surfactant, and ??/?c = ?0RgT/c byvirtue of the Gibbs equation, Rg and T being the gas constant and absolutetemperature. Now the stress ratio can be written as?M?v??20RgTc??DR. (3.4)From the Stokes-Einstein relationship, the surfactant diffusivity D is in-versely proportional to the liquid viscosity ?: D = kBT/(6??rs), rs beingthe characteristic size of the surfactants and kB the Boltzmann constant.Plugging this and the estimation of ? into the above equation, we obtain?M?v? C (???R2)?1/3, (3.5)where C contains factors including T and c, and is a constant in our exper-iment. The prediction that ?M/?v decreases with ?? allows the possibilitythat the Marangoni stress be overpowered by the viscous shear stress atsufficiently high ??, which would be consistent with the proposed mechanismof bubble-surface remobilization. However, the prediction of a critical shearrate that scales with ??1 and R?2 contradicts the observations in Fig. 3.2.In view of the above analysis, we are driven to the conclusion that theremobilization of bubble surface by shear stress overcoming Marangoni stresscannot be the cause of the anomalous coalescence.3.3.3 Bubble compression due to inertiaThe photo in Fig. 3.5 indicates a tendency for the bubbles to be pushedradially inward. The only plausible agent for such an effect is the centripetalforce of the rotating liquid. As the spinning liquid generates an inwardpressure gradient, the bubbles, having a much lower density than the liquid,are pushed inward towards the inner cylinder. Thanks to pinning on thewalls, the liquid surface rises little (Fig. 3.4). The radial pressure gradientis thus maintained not by hydrostatic head but by surface tension in theliquid meniscus. Conceivably the squeezing between bubbles accelerates thedrainage in the liquid film. If the film drains down to a critical thicknesswithin the interaction time between two bubbles, coalescence would occur[10]. Thus, one may be able to adapt ideas from conventional drop-dropcollision to explain the anomalous coalescence. In the following we test thismechanism through a scaling model.213.3. Potential mechanisms for anomalous coalescenceFor a pair of bubbles pushed into each other by a constant force F , wemay estimate the drainage time from an initial film thickness h0to the finalcritical one hc using the rigid parallel disk model [10, 52]:td =3??a44F(1h2c?1h20), (3.6)where a is the radius of the liquid film. In our geometry, the radial pressuregradient due to the spinning liquid is dp/dr = ?u2/r, ? being the liquiddensity and u the tangential velocity of the liquid at distance r. This exertsa force (dp/dr) ? 2R ? ?R2 on each bubble. Since the bubbles are in closecontact with each other, they transmit the centripetal force onto their innerneighbors in a sort of force chain, resulting in the largest cumulative forceon the innermost layer of bubbles:F = 2??R3N?1?i=1u2r , (3.7)with the summation over the outer layers of bubbles. In comparison with F ,the squeezing force ?a2(?/R) due to capillary pressure is at least an orderof magnitude smaller, and has thus been neglected.Chesters and Bazhlekov [11] have proposed an empirical relation for thecritical film thickness hc for rupture due to van der Waals force:hc =23(A4??)0.3(aR)0.2, (3.8)A being the Hamaker constant taken here to be A = 3?10?19 J [35]. We neednow to estimate a. For pairwise collisions in a shear flow, the classical theorygives a/R ? Ca1/2 [10]. We have measured a directly by using ImageJ [67],and found it relatively insensitive to shear. In the static foam, a ? 0.17R, inclose agreement with previous computations [16]. With shearing, a tends toincrease with ? but quickly saturates to an average value a ? 0.2R at about25 rpm. Apparently the close packing constrains the bubble movement anddiminishes the role of shearing. Measuring a among hundreds of pairs ofbubbles reveals moderate variations in any given foam, and Fig. 3.6 showsa typical distribution of a in a sheared foam. Since smaller a gives fasterfilm drainage, and we are concerned with the onset of coalescence, we usethe smallest a = 0.14R. Inserting this value along with Eqs. (3.7) and (3.8)into Eq. (3.6), the ratio between drainage and interaction times is:? = tdti= 1.64? 10?3(??A)0.6 ?R0.2????N?1i=1u2r. (3.9)where we have neglected h?20relative to h?2c , and taken the interaction timebetween neighboring rows of bubbles to be ti ? ???1 as in previous analysis223.3. Potential mechanisms for anomalous coalescenceFigure 3.6: Distribution of the film radius a in a sheared bubble raft with bubblesize of R = 500 ?m undergoing rotational speed of ? = 75 rpm. The curve showsa fitted normal distribution.233.3. Potential mechanisms for anomalous coalescenceFigure 3.7: The critical condition for coalescence corresponds to ? = 8 for solutionI, ? being computed for the innermost layer of bubbles using the measured bubblevelocity profile. For solutions II and III, the critical ? values are 12.5 and 28.5,respectively.243.4. Conclusion[10]. We argue that ?  O(1) should give the critical condition for theanomalous coalescence observed here.The validity of the scaling theory can now be tested against the keyexperimental observations. First, note that ?? and u are both proportional to?. Thus ? ? ??1 and ? < 1 does yield a minimum critical rotational speedas observed. Quantitatively, however, the critical condition corresponds to? = 8, 12.5 and 28.5 for solutions I, II and III, respectively (Fig. 3.7). Thesenumbers are one order of magnitude too large. Second, the ? criterionpredicts a scaling for the critical rotational speed ?c ? R0.2, in reasonableagreement with the power-law scaling observed in Fig. 3.2. Third, it alsopredicts ?c to increase linearly with the liquid viscosity ?. While the trendis correct, the experimentally observed dependence on ? is much weaker:?c ? ?0.1 (cf. Fig. 3.2). Finally, the large bubbles appear more often inthe inner part of the gap than the outer. Given that the smallest a can beanywhere in a particular experiment, this provides indirect support for theaccumulation of the inward force in Eq. (3.7).Thus, the inertia-based mechanism explains the qualitative trends ob-served. But quantitatively it overestimates the drainage time as well as theeffect of liquid viscosity. The latter recalls the study of Yoon et al. [82]on freely suspended droplets, where the viscosity effect is also weaker thanexpected. In our case, the numerical discrepancies have many potentialcauses. For example, the Hamaker constant [35] is not known for the flu-ids used here, and possibly the bubble surface may develop dimples duringthinning [75] that would compromise the calculation above. Since our bulksurfactant concentration is 100 times CMC, the abundance of surfactantsmay introduce additional effects. Rapid adsorption onto the bubble surfacemay partially mitigate the Marangoni stress and locally remobilize the sur-faces [64]. Though this has been dismissed as a critical condition for theanomalous coalescence, it might explain the fact that the drainage rate isunderestimated in our model, producing too large a critical ? value. More-over, the later stage of drainage is probably influenced by the presence ofmicelles, which may form layers that hinder film thinning below h ? 100 nm[64]. This non-viscous effect may reduce the overall dependence on ?. Un-fortunately, not knowing the chemical properties of the surfactant mixturein the detergent, it is difficult to formulate these ideas quantitatively.3.4 ConclusionIn this chapter we reported an anomalous type of bubble coalescence ina monolayer sheared in a Couette device, which occurs above a criticalrotational speed ?c. This contrasts the conventional wisdom about bub-ble and drop coalescence that it occurs below a critical capillary number.Our coalescence cannot be characterized by a critical capillary number; the253.4. Conclusioncritical ?c increases with bubble size and the viscosity of the suspendingliquid. To rationalize the experimental observations, we have consideredthree potential mechanisms for the coalescence: shear preventing the forma-tion of surfactant-stabilized films between bubbles, shear stress overcomingMarangoni stress to remobilized the bubble surface, and centripetal forcepressing the bubbles radially inward into each other. None of these accountsquantitatively for all the experimental results.The third is the most promising. According to this model, the anoma-lousness of the scenario arises from two factors: the film drainage is drivenby a centripetal force instead of a viscous one, and the bubble deformationis determined by geometric constraints rather than shearing. The apparentreversal in the coalescence criterion, from the conventional maximum cap-illary number to a minimum shear rate, is similar in spirit to that demon-strated recently by Ramachandran and Leal [66] for collision between vesi-cles. Though clearly not a complete theory for the anomalous coalescence,the inertia-based model captures the qualitative trends of the experiment,and may serve as a starting point for further investigations.26Chapter 4Lateral migration of a singlelarge bubble in monodispersetwo-dimensional foam4.1 IntroductionFoams rheology and hydrodynamics are intimately coupled to its microstruc-ture, i.e. the shape and spatial organization of the bubbles [30, 78]. Aparticularly intriguing phenomenon is size-based segregation of bubbles ina polydisperse foam [28]. After shearing between rotating parallel plates,smaller bubbles appear predominantly near the top and bottom plates whilethe larger ones are in the middle. The cause is unclear, but one possibilityis that the bubbles have migrated across streamlines based on their size. Ina more recent experiment on a two-dimensional (2D) foam under oscillatoryshear [65], a bubble larger than its monodisperse neighbors migrates towardone of the four borders confining the foam. This seems to contradict theobservations of Herzhaft on size-based segregation of bubbles [28]. Morecuriously, the migration does not distinguish between the flow direction andthe direction of the velocity gradient. These two studies hint at some rulegoverning lateral migration of bubbles in sheared foam, but little is known atpresent. In contrast, lateral migration of particles and droplets suspended ina liquid medium has been extensively studied in the past [e.g. 7, 8, 34, 47].A solid spherical particle in a Stokes flow cannot migrate because the linearsystem is time-reversible. A droplet deforms under shear, and this introducesa nonlinearity into the problem and makes lateral migration possible. It hasbeen shown that in low-Reynolds-number Couette flows, droplets move awayfrom the walls toward the center of the gap [8, 31, 38]. This is commonlyinterpreted as a wall repulsion; the rigid wall produces an asymmetry in thevelocity and pressure fields around the drop. Hence arises the lateral migra-tion force. Naturally one wonders if the same repulsion operates in shearedfoam. This chapter describes an experimental study of lateral migration ofbubbles in a 2D foam sheared steadily in a narrow-gap Couette device. Intoa monodisperse bubble raft we introduce a single bubble of different size andinvestigate its migration. By correlating the migration speed with the shearrate and the bubble size ratio, we propose a hydrodynamic explanation for274.2. ResultsFigure 4.1: Migration trajectories of a single bubble (R = 0.7 mm) at ? = 7 rpm.The curves represent the Chan-Leal formula (Eq. 4.1). The top inset illustratesthe migration schematically (not to scale) and the bottom one depicts the liquidmeniscus above the bubble calculated from the model of Ref. [54].the migration based on bubble deformation.4.2 Results4.2.1 Migration of bubble in a Newtonian fluidAs a baseline, we first study the migration of a single bubble floating on thefree surface. It migrates to the center of the gap from all initial positions.Typical trajectories are shown in Fig. 4.1. The dimensionless drop positions is scaled by the gap width d, with s = 0 at the inner cylinder and 1 atthe outer cylinder. The symmetry between inward and outward trajectoriesconfirms the uniformity of the shear rate across the gap. This migrationis reminiscent of that of neutrally buoyant droplets suspended in a liquidmedium [8, 31]. Thus we have compared the measured trajectories withthose predicted by the theory of Chan and Leal [8]. Chan and Leal [8]284.2. Resultsconsidered the migration of a Newtonian drop in a Newtonian matrix shearedin a Couette device, under the condition of vanishing capillary number andsmall drop deformation. For a bubble of radius R in a matrix of viscosity ?,the dimensional migration velocity can be written as:vm(S) =?? Ca2(RiRo)4{81560R2d2 [1+R2o(Ri + Sd)2]2f(S)? 17R4oR(Ri + Sd)5}, (4.1)where Ca = ???R/? is defined using the shear rate at the inner cylinder,f(S) = S?1 ? (1 ? S)?2 + 2 ? 4S, and we have put the bubble viscosityto zero. The first term in the bracket represents wall repulsion that pushesthe bubble to the center of the gap (S = 0.5), while the second term is dueto the curvature of the streamlines and drives the bubble toward the innercylinder. Thus, the Chan-Leal formula predicts an equilibrium positionbetween the center and the inner cylinder. Note that the prerequisites forthe perturbation theory, Re ? 0, Ca  1 and R  d, are all satisfied bythe experiment. With Re, Ca, R and d being Reynolds number, capillarynumber, bubble radius and gap size respectively. Integrating the aboveusing the experimental parameters produces the trajectories of Fig. 4.1. Theagreement between the measured and predicted trajectories is very close.The formula was derived for a neutrally buoyant drop inside a 3D fluidwhile our bubble ?floats? on the liquid surface. In reality, surface tensionkeeps 99% of the bubble volume below the undisturbed free surface, whichis consistent with theoretical calculations [54] (Fig. 4.1 inset). The viscousfriction in the thin meniscus atop the bubble may be larger than in a fully3D geometry. But apparently the left-right asymmetry dominates and thevertical dimension seems to matter little. Thus, the Chan-Leal formulapredicts the migration in our geometry with no fitting parameter.4.2.2 Migration in two-dimensional monodisperse foamThe main result of the experiment is the migration of a larger bubble of ra-dius R in an otherwise monodisperse bubble raft of radius r. Generally thelarge bubble migrates toward the center of the gap, and the migration speeddepends on the size ratio ? = R/r as well as rotation rate ?. Figure 4.2shows migration trajectories for several ? and ? values. During the migra-tion, the large bubble shifts from one row of bubbles to the next, spending afinite time in each. This is indicated by the horizontal bars on some trajec-tories, forming a staircase pattern. For clarity, the bars are omitted on theother trajectories with only data points plotted at the center of each step.The following observations can be made. (i) There are a threshold ?0for a fixed ? and a threshold ?0for a fixed ?, below which no migrationoccurs. For the conditions of Fig. 4.2(a), ?0lies between 1.43 and 1.54.In particular, a bubble smaller than its neighbors, i.e. with ? < 1, does294.2. Results0 5 10 15 20 25 30 3500.10.20.30.40.5t, Time (sec)s, Dimensionelss position  ? = 1.43, Cae= 5.4 ? 10?3? = 1.54, Cae= 6.5 ? 10?3? = 1.61, Cae= 7.6 ? 10?3? = 2.86, Cae= 1.1 ? 10?2(a)0 5 10 15 20 25 30 35 4000.10.20.30.40.5t, Time (sec)s, Dimensionless position  ? = 2 rpm, Cae= 3.4 ? 10?3? = 2.5 rpm, Cae= 4.3 ? 10?3? = 3 rpm, Cae= 5.2 ? 10?3? = 7 rpm, Cae= 1.2 ? 10?2(b)Figure 4.2: Bubble migration in a 2D foam. (a) Effect of the bubble size ratio ?at ? = 4 rpm. The bubble radii are (in mm): (r,R) = (0.35, 0.5) for ? = 1.43;(0.39, 0.6) for ? = 1.54; (0.435, 0.7) for ? = 1.61 and (0.35, 1) for ? = 2.86. Thesolid and dashed curves are predictions of Eq. (4.3) for ? = 1.61 and 2.86. (b)Effect of the rotational rate ? for fixed bubble sizes (r,R) = (0.35, 0.7) mm. Thecurves are predictions of Eq. (4.3) for ? = 3 and 7 rpm.304.3. A hydrodynamic model to explain the migration in foam	Figure 4.3: Bubble deformation in different environments at ? = 7 rpm. Thelarge bubble (R = 0.7 mm) in (a) deforms much more in a foam of smaller bubbles(r = 0.35 mm) than alone in (b). (c) A smaller bubble (R = 0.4 mm) is shieldedby its neighbors (r = 0.58 mm).not migrate at all. In Fig. 4.2(b), ?0is between 2 and 2.5 rpm. (ii) Forsufficiently large ? and ?, a large bubble migrates all the way to the center(s = 0.5). Below these, the bubble may migrate to an intermediate positionbetween the wall and the center. (iii) The migration speed increases with? and ?. (iv) The migration is much faster than if a bubble of radius Rmigrates on a free surface, without the bubble raft. This can be seen bycomparing Fig. 4.2 with Fig. 4.1; the migration time differs by a factor ofO(102).4.3 A hydrodynamic model to explain themigration in foam4.3.1 Model developmentAll the above observations can be explained by a model based on the de-formation of the migrating bubble. Chan and Leal [8] showed that the wallrepulsion stems from the left-right asymmetry in the flow around the bubbleand the concomitant asymmetric bubble shape. In our experiment, a largerbubble protrudes outside its own row and forces the surrounding bubbles torearrange as they pass around it (Fig. 4.3a). Compared to fluid particles ina continuum, the surrounding bubbles have a finite radius r and a capillarypressure inside, and thus are much harder to displace and deform. Theycontinuously rub and bump into the sides of the large bubble, impartinga force Fb on it. This force is the counterpart of the liquid pressure andviscous force in the single-bubble scenario, but is much larger. A visibleconsequence of Fb is the pronounced deformation of the large bubble, muchmore than a single bubble of the same size subject to the same shear rate(Fig. 4.3b). A less visible one, we surmise, is a strong wall repulsion arising314.3. A hydrodynamic model to explain the migration in foam1 1.5 2 2.5 356789101112?D/CaFigure 4.4: Deformation parameter of a larger bubble in a 2D foam as a functionof the bubble size ratio ?. The error bars indicate the variation among 7 shear ratestested, and the curve is a quadratic fit to the data.from the asymmetry in Fb from the two sides.This idea can be made more precise by plotting the bubble deformationas a function of the size ratio ? (Fig. 4.4). We define a bubble deformationparameter D = (l ? a)/(l + a), l and a being the long and short axes ofthe roughly elliptical deformed bubble. According to Taylor?s celebratedformula [73], a single bubble of negligible internal viscosity in a shearedfluid should have D = Ca. In the bubble raft, we represent the data byD/Ca = g(?) = 2.5?2 ? 7? + 11. Now we equate the larger deformationin a bubble raft to that of a single bubble at a higher ?effective capillarynumber? Cae:Cae = D = Ca ? g(?). (4.2)Plugging this into the Chan-Leal formula (Eq. 4.1) gives us a modified Chan-Leal formulavm(s, ?) =81140R2d2?Ca2e? f(s) g2(?). (4.3)After time integration, this formula predicts well all the migration trajecto-ries recorded in our 2D foam, over the entire range of r, R and ? values.324.3. A hydrodynamic model to explain the migration in foamFor clarity, only a few representative curves are plotted in Fig. 4.2. Notethat the O(10) deformation enhancement in Fig. 4.4 translates to the O(102)increase in the migration velocity. The success of Eq. (4.3) confirms our hy-pothesis in the preceding paragraph. As a corollary, a bubble of the samesize as its neighbors or smaller (? ? 1) does not migrate because it does notjut out of its own row (Fig. 4.3c). Thus, it is not subject to the ?bumpingforce? Fb.4.3.2 Thresholds for migrationFinally, we examine the thresholds ?0and ?0for lateral migration. When amonodisperse 2D foam is sheared, the bubbles typically move in streamwiserows past one another. For a larger bubble (radius R) to migrate laterally,it must squeeze into the next row of bubbles (radius r). The wall repulsionforce Fw driving the migration, therefore, must exceed a threshold in order todeform the bubbles of the next row to create the gap. Because these bubblesin turn interact with multiple moving and changing neighbors on the otherside, it is difficult to posit a precise force balance from which to calculatethe threshold. As an estimation, we take the resistance to migration to beon the same order of magnitude as the capillary force between bubbles inthe row: Fc = (?/r) ? ?a2, where a is the radius of the thin film betweenneighboring bubbles in a 2D foam. For the foam quality used here, a showsa normal distribution among the bubble pairs, with a mean of a = 0.2r [57],which will be used below. On the other hand, the wall repulsion can beestimated from the Stokes formula using the migration velocity of Eq. (4.3):Fw = 6??vmR. We use the Stokes formula as opposed to the Hadamardformula because the bubble surface is immobilized by the high surfactantconcentration [57]. Now the ratio between these two forces is:? =FwFc=24370R3ra2d2 Ca2f g2. (4.4)Note that f(s) gives the wall repulsion at position S. In particular, usingthe largest Fw, for the first row next to the wall, gives us the ratio ?1. Weargue that a ?1value of O(1) gives the threshold for lateral migration ofthe larger bubble.Figure 4.5 plots ?1for all the ? and ? values tested in our experiments.The experimental conditions giving rise to lateral migration are indicatedby filled and half-filled symbols, the latter for partial migration to positionsbetween the wall and the center. The non-migrating conditions are shown byhollow symbols. These two groups are almost perfectly separated by ?1=0.4, thus validating Eq. (4.4) as an approximation for the threshold. Forlower ?, three data points fall on the wrong side of the line; the experimentmay have been more susceptible to external disturbances in these cases.334.3. A hydrodynamic model to explain the migration in foamFigure 4.5: Threshold for bubble migration. ?1, computed from Eq. (4.4) forthe first row of the bubbles next to the wall, is plotted as functions of ? for var-ious ? values. Hollow, half-filled and filled symbols indicate no migration, partialmigration and complete migration to the center. The dashed line is ?1= 0.4.344.4. ConclusionNote that the thresholds reflect the graininess of the bubble raft, and haveno counterpart in the Chan-Leal theory.4.4 ConclusionIn this chapter, we studied lateral migration of a single bubble whose size isdifferent from the neighboring bubbles and showed that the size-differentiatedlateral migration in sheared 2D foam can be achieved under some conditions.The key findings can be summarized as the following:(a) We introduced a single bubbles whose is different than the neighboringbubbles in a monodisperse foam and found out it migrates laterally aslong as the bubble size ratio ? and ? are both above some thresholds.(b) We modified the Chan-Leal theory to account for the observations.The rubbing of large bubbles by the smaller neighboring bubbles re-sulted into an elevated deformation. After accounting for this fairlyhigh deformation, we were able to predict the migration trajectoriesof bubbles using the Chan-Leal theory.(c) We also explained the presence of thresholds for bubble size ratio androtational velocity by using a force balance between the wall repulsionand capillary attraction between bubbles.(d) And finally lateral migration that was reported in this chapter offersa potential explanation for the size-based segregation in sheared 3Dpolydisperse foam [28].35Chapter 5Size segregation in shearedtwo-dimensional polydispersefoam5.1 IntroductionAs we mentioned in previous chapters, it is widely recognized that complexdynamics of foam is rooted in the foam?s microstructure on the bubble scale;the bubbles may undergo breakup, coalescence, coarsening, and morpholog-ical changes [2, 6, 17, 21, 68, 76]. Prior experiments have indicated thepossibility that bubbles may segregate according to size in a flowing poly-disperse foam. But other experiments suggested evidence to the contrary.Herzhaft [28] sheared three-dimensional polydisperse foams between paral-lel disks, and reported that the large bubbles tend to appear at the middleof the gap while smaller ones are closer to the walls. One explanation isthat the bubbles have segregated according to size during the shear. How-ever, an alternative is bubble breakup [27] and coalescence [57] under shear,which could also have produced the observed patterns. In an experimentdesigned expressly to probe bubble migration, Quilliet et al. [65] produceda monolayer of monodisperse bubbles as a two-dimensional (2D) foam, andinserted a bubble larger than its neighbors. Under oscillatory shear, thelarge bubble is seen to migrate toward one of the boundaries of the cell.This is inconsistent with Herzaft?s report of migration away from walls. Ina Hele-Shaw cell, Cantat et al. [5] reported aggregation of large bubblesamong smaller neighbors. Cox et al. [15, 24] studied planar extension ofbidisperse 2D foams experimentally and numerically, and found no sign ofsize-based bubble segregation. Therefore, the question of size segregation inflowing polydisperse foam remains open.For emulsions and suspensions, on the other hand, the segregation andmargination of drops and particles in confined flows are well documented [44,50, 51, 71]. For example, bidisperse suspensions of particles show mild sizesegregation in 2D channel flow [50, 51]. White blood cells and plateletsare found closer to the walls while the red cells aggregate in the centerof the tube [44, 71]. We should note, of course, that foams are differentfrom suspensions or emulsions in that the bubbles are closely packed, with365.2. Size segregation in bidisperse foamrelatively little suspending fluid in between. Thus they have a much reducedmobility.As we showed in previous chapter, we have taken the first step towardanswering the question of size segregation in sheared foam by studying themigration of a single large bubble in an otherwise monodisperse bubble raft[55]. In a Couette shear cell, we saw migration of the large bubble awayfrom the walls toward the center of the gap, apparently driven by a ?wallrepulsion?. This appears consistent with the observations of Herzhaft [28]but not those of Quilliet et al. [65]. Now in bidisperse and polydispersefoams, a new factor is that the large bubbles interact among themselvesas well. How does this interaction affect the migration of the bubbles ofdifferent sizes? Do bubbles segregate based on size, and if yes, what is therole of the area fraction of different species? These are the questions we setout to answer in this chapter.5.2 Size segregation in bidisperse foamWe have done several experiments for bidisperse as well as polydispersefoams. But, it turns out that the key features of size-based segregation aremostly manifested in bidisperse foams already. For the ease of analysis,therefore, we will focus on bidisperse foams in the following, with a finalsubsection devoted to features specific to polydisperse ones.Samples ?1(%) ?2(%) ?3(%) ?4(%)A 80 20 ? ?B 95 ? 5 ?C 90 ? 10 ?Bidisperse D 80 ? 20 ?E 70 ? 30 ?F 50 ? 50 ?G 80 ? ? 20H 90 5 5 ?I 80 10 10 ?J 60 20 20 ?K 40 30 30 ?Polydisperse L 90 ? 5 5M 80 ? 10 10N 60 ? 20 20O 40 ? 30 30Table 5.1: Composition of the bidisperse and polydisperse foam samples used inthe experiments. ?1, ?2, ?3and ?4are the area fractions of the bubble species withradius a1= 350 ?m, a2= 500 ?m, a3= 700 ?m and a4= 875 ?m, respectively.375.2. Size segregation in bidisperse foam  Figure 5.1: Size segregation in sample D under shear (? = 5 rpm). The upperrow consists of snapshots of the foam at different times: (a) t = 0; (b) t = 2 min;(c) t = 5 min. The lower row shows the corresponding large bubble distributions?3(y)/?3.Figure 5.1 illustrates a typical process of size-based segregation of thetwo bubble species under shear. The distribution of the large bubbles, ofradius a3= 700 ?m in this case, are generated by averaging over severalsnapshots taken in repeated experiments. In each snapshot, we divide thevisible domain of the foam into 9 parallel strips of equal thickness across thegap d, and count the number of large bubbles in each strip. This producesa profile of the area fraction for the large bubbles, ?3(y), normalized bythe average fraction ?3= 20%, y being the dimensionless coordinate acrossthe gap with the origin at the center and y = ?0.5 at the walls. The largebubbles are initially released close to the walls (Fig. 5.1a). Under a rotationalrate ? = 5 rpm, the two species mix at first (Fig. 5.1b). In time, however, thelarge bubbles aggregate in the center of the gap, within |y| < ye ? 0.25 in thiscase, and a quasi-steady state is reach at t = 5 min (Fig. 5.1c). In this state,there is no statistically significant variation along the azimuthal direction.The quasi-steady distribution of the bubbles is independent of the initialconfiguration. Figure 5.2 shows that three different initial distributions atthe same ?3= 20% all lead to the same final distribution. Of course, thetime required to reach the final state differs. For brevity, we will refer to thequasi-steady state after prolonged shearing simply as the ?steady state?.The apparent aggregation of large bubbles at the center of the gap is con-sistent with our earlier observations on the migration of single large bubblesin a 2D foam of smaller bubbles [55]. To sum up those findings, a largebubble off the center of the gap experiences an asymmetric ?bumping force?from the small bubbles that pass along its sides under shear. This producesa ?wall repulsion? toward the center of the gap, much as in the migration385.2. Size segregation in bidisperse foam  Figure 5.2: Three different initial configurations of sample D (top row), withthe large bubbles randomly distributed (a), near the walls (b) and segregated intoazimuthal segments (c), lead to the same quasi-steady distribution in the lowerrow after shearing at ? = 7 rpm for 10 min. The arrow indicates the direction ofshearing.of a single drop submerged in a suspending liquid [8, 47]. Furthermore, themigration speed can be predicted by the Chan-Leal formula [8] if the cap-illary number Ca is replaced by an effective capillary number Cae that ishigher than Ca and accounts for the enhanced deformation of the large bub-ble under the continuous impact of the smaller surrounding bubbles. In thepresent study, the obvious difference is that there are multiple large bubblesthat interact among themselves as well.There are two prerequisites for the migration of the single large bubble inan otherwise monodisperse foam of smaller bubbles [55]: that the shear rate?? and the bubble size ratio ? each be above a certain threshold. These reflectthe discreteness of the foam; it takes a minimum force to push a large bubblefrom one row to the next against the capillary pressure in the neighboringbubbles. Such thresholds have also been observed for the bidisperse foamshere. In fact, the two threshold values of ? and ? are expected to be thesame as for a single large bubble [55]. Insofar as they are critical valuescorresponding to the onset of lateral migration of the large bubbles, theyare unaffected by the interaction among large bubbles, which arises onlyafter the thresholds have been crossed. For example, no segregation occursin Sample A for ? up to 7 rpm, the highest rotational rate possible withoutincurring centripetal effects [55]; the bubble size ratio ? = a2/a1= 1.43 istoo small. In Sample D (? = 2), the threshold is around ? = 3 rpm. ForSample G (? = 2.5), it has come down to around 2 rpm. We have previouslypresented detailed experimental data on the thresholds [55], along with ananalytical expression for the critical condition based on scaling arguments.395.2. Size segregation in bidisperse foamFigure 5.3: Steady state distribution of the large bubbles in Sample C after shear-ing at (a) ? = 3 rpm and (b) ? = 7 rpm for 10 min. The solid lines are predictionsof the migration-diffusion model to be discussed in subsection III.E.5.2.1 Effect of shear rateThe shear rate affects both the final steady-state bubble distribution acrossthe gap and the approach to that steady state. Figure 5.3 shows the steady-state distribution of a3in Sample C after shearing at different rotationalspeeds. Evidently, with increasing shear rate the final distribution of thelarge bubbles becomes more narrowly peaked, and the near-wall regions freeof large bubbles widen. For the two cases shown, the half-width of the large-bubble distribution ye ? 0.28 and 0.22 for ? = 3 and 7 rpm, respectively.Furthermore, we compare the speed of segregation at different shear ratesstarting from the same uniform initial configurations. Figure 5.4 plots thetemporal evolution of the half-width of the large-bubble distribution, ye(t).At higher shear rate, the size segregation proceeds at higher speed, andthe steady-state distribution is attained within a shorter time. Intuitively,this trend is reasonable. Faster shearing causes more vigorous and frequentimpingement of the small bubbles onto the large ones, which should enhancethe speed of lateral migration for the latter. A more precise analysis callsfor the introduction of another factor, shear-induced diffusion of the largebubbles, which influences the steady-state distribution as well.5.2.2 Effect of area fraction of large bubblesThe size segregation in polydisperse foams differs from the migration of asingle large bubble studied before [55] in that the large bubbles interactamong themselves. Naturally one expects this interaction to depend on thelarge-bubble area fraction. By shearing Samples B, D and F, with ?3= 5%,20% and 50% for the large bubbles, respectively, we compare the steady-state distributions in Fig. 5.5. By increasing ?3, the distribution becomes405.2. Size segregation in bidisperse foamFigure 5.4: Temporal evolution of the half-width of the large-bubble distribution,ye(t), for Sample D at ? = 3 and 7 rpm. The solid and dashed lines indicatepredictions of the migration-diffusion model.Figure 5.5: Steady-state distribution of the large bubbles in Sample B, D and Fafter 10 min of shearing at 7 rpm. These samples have the same bubble sizes butdifferent area fractions for the larger bubbles (see Table 1). The solid lines arepredictions of the migration-diffusion model.broader and the large bubbles are more spread out in the gap. At evenhigher fractions, the large bubbles become essentially uniformly distributedacross the gap.Moreover, Fig. 5.6 compares the temporal development toward the steadystate at two different ?3values. For Sample E at the higher ?3= 30%, the415.2. Size segregation in bidisperse foamFigure 5.6: Temporal evolution of ye(t) for Samples C (?3= 10%) and E (?3=30%) undergoing shear at 7 rpm. The solid and dashed lines indicate predictionsof the migration-diffusion model.equilibrium distribution is achieved more rapidly. In view of the wider dis-tribution in equilibrium (Fig. 5.5c), or equivalently the larger steady-stateye value, the large bubbles initially near the walls need to travel less distanceto reach their equilibrium position. This seems to provide an easy rational-ization of Fig. 5.6. But a more careful examination will be made below withthe help of a quantitative model.5.2.3 Effect of bubble size ratioFigure 5.7 compares the steady-state distributions and temporal evolution ofye for two bidisperse foam samples with the same large-bubble area fraction? but different bubble size ratio (?). Sample G, with the larger ?, exhibitsa more sharply peaked steady distribution, and reaches it more rapidly thanSample D. This mirrors the effects of the shear rate which was explainedin 5.2.1. In the migration of a single large bubble in an otherwise monodis-perse foam of small bubbles [55], we have found that a larger ? increasesthe migration velocity as if by elevating the shear rate. In fact, an effectivecapillary number Cae can be defined based on ? that quantitatively captures425.3. A migration-diffusion modelFigure 5.7: Effect of the bubble size ratio ? on (a) the steady-state distribution ofthe large bubbles, and (b) the transient to the steady state in terms of ye. SampleD has a bubble size ratio ? = 2 and Sample G has ? = 2.5. Both have 20% averagearea fraction for the large bubbles and are subject to ? = 7 rpm. The solid anddashed lines represent the model predictions.this effect. The model presented below will make a similar connection forthe bidisperse foams.5.3 A migration-diffusion modelThe description above indicates that size segregation in sheared foam isdriven by the migration due to wall repulsion, the same mechanism as op-erates on a single large bubble in a medium of smaller ones [55]. A secondkey player, one that distinguishes the bidisperse foam from the single-large-bubble scenario, is the interaction among the large bubbles themselves. Thisinteraction may be described by the idea of shear-induced diffusion that is fa-miliar from prior studies of suspensions and emulsions [22, 34, 41, 48, 50, 51].The competition between these two factors determines the speed of segre-gation between bubbles of different sizes and their final distribution.King and Leighton [41] and Hudson [34] studied the spatial distributionof drops in sheared dilute monodisperse emulsions and investigated the in-terplay between wall migration and shear-induced diffusion. The evolutionof the drop volume fraction ? in a simple shear obeys a convection-diffusionequation:???t = ???y?(vm??D???y?), (5.1)where vm is the velocity of wall-induced migration, D is a diffusivity, andy? = yd is the dimensional coordinate across the gap. The Chan-Leal for-435.3. A migration-diffusion modelmula [8] is used for vm, in terms of the dimensionless y:vm = ?4???a2d2 Ca2[y + 8y(1? 4y2)2], (5.2)where ? is a mildly varying function of the drop-to-matrix viscosity ratiogiven by Chan and Leal [8], ? is the interfacial tension, ? is the ambientfluid viscosity, a and d are the drop radius and gap size, and Ca = ???a/? isthe capillary number. The diffusivity D is written asD = ???a2?, (5.3)where ? is a dimensionless coefficient. Balancing the drop fluxes due to wallmigration and diffusion, Hudson [34] arrived at the following steady-stateprofile:?(y) = ?0+ Pe(1?y22?11? 4y2), (5.4)where ?0= ?(0) is a constant of integration, and the Peclet numberPe = 4?adCa??, (5.5)? being the average volume fraction. Note that both vm and ? divergetoward the walls (y ? ?0.5). The actual profile comprises the positivecentral part of Eq. (5.4) and drop-free layers next to the walls, whose edges(y = ?ye) are determined by setting ?(ye) = 0 in Eq. (5.4). Conservationof the drop volume? ye?ye?(y) dy = ? specifies the centerline volume fraction?0.To adapt this emulsion model to our bidisperse foam, we make the sameanalogy as was used previously to represent the wall-induced migration ofa single large bubble in a sheared monodisperse foam of smaller bubbles[55]. Essentially, we view the smaller bubbles as constituting an effectivecontinuum that suspends and flows around the large bubbles, playing therole of the continuous-phase liquid in the emulsion. Of course, the foam is2D while the emulsion is 3D, and the smaller bubbles exert a hydrodynamicimpact on the larger ones that differs from that of a continuous, viscous liq-uid. Most importantly, the large bubbles are observed to deform much morethan in a viscous liquid under the same capillary number. We previouslyshowed in Chapter 4 that the enhanced deformation can be described by anempirical equation for an effective capillary numberCae = Ca(2.5?2 ? 7? + 11), (5.6)? being the large-to-small bubble size ratio. Cae is larger than Ca and, whenused in the Taylor formula for drop deformation, predicts the observed bub-ble deformation. With Caa being replaced by Cae, the migration velocity445.3. A migration-diffusion modelvm of a single large bubble can be predicted accurately by the Chan-Lealformula [55]. This vm can be used in the emulsion model (Eq. 5.1) for thebidisperse foam at hand. Then we need only to find the counterpart of thediffusivity of Eq. (5.3).As far as we know, the idea of shear-induced diffusion has never beenused for foams before, and no measured data exist for D or ?. In emulsions,one may consider ? a function of the viscosity ratio, the surface mobility,the capillary number Ca, and the drop fraction ?. For surfactant-stabilizeddilute emulsions, King and Leighton [41] have reported ?(Ca) as a weaklyrising function of the capillary number Ca (cf. their Fig. 8). In surfactant-free emulsions, Hudson [34] obtained ? values that are an order of magnitudelarger, owing to the higher surface mobility. Viewed as an emulsion of thelarge bubbles in an effective liquid medium, our bidisperse foam is similarto King and Leighton?s emulsion in that the surfaces are immobilized bysurfactants, and the drop-to-matrix viscosity ratio is negligibly small. Thus,we borrow their dimensionless diffusivity ?, now as a function of the effectivecapillary number Cae. In fact, all our experiments have used low shear ratessuch that Cae < 0.1, in which range ? = 0.02 ? 0.002 remains essentiallyconstant (see Fig. 8 of King and Leighton [41]). Therefore, we have simplytaken ? = 0.02 in our model calculations. Note that this neglects anydependence of ? on ? and possibly also on the bubble size ratio ? in ourfoam. Both prior experiments [34, 41] used dilute emulsions and neitherexplored the effect of ?. We assume that ? is independent of the areafraction of the large bubbles for our bidisperse foam. This assumption will bevalidated a posteriori by comparing the model prediction with experimentaldata over the whole range of area fraction. With the effective continuumanalogy, increasing ? amounts to increasing the effective capillary numberCae through Eq. (5.6). As long as we operate in the low-Cae regime, the ?effect on ? can be safely neglected.Having ? thus determined and noting that ? = 81/140 for an emul-sion of negligible drop viscosity [8], we calculate the Peclet number for ourbidisperse foam asPe = 8135adCae?? . (5.7)With this Peclet number, we can use Eq. (5.4) to predict the steady-state dis-tribution of the large bubbles in our bidisperse foam, and integrate Eq. (5.1)for the transient toward the steady state. In Eq. (5.4), the centerline concen-tration ?0is determined from the conservation of drop volume? ye?ye?dy = ?.Equation (5.1) is integrated using finite difference with boundary conditions?(?ye) = 0, ye being determined iteratively from the drop volume conser-vation by the shooting method. Both the steady ? profile and the transientcan be compared with measurements. In particular, we will examine theeffects of the shear rate ??, the average area fraction ? and the bubble sizeratio ?.455.3. A migration-diffusion modelFigure 5.3 compares the model predictions with the measured steady-state distributions for the bidisperse Sample C at two shear rates, andFig. 5.4 compares the temporal development of the distribution for Sam-ple D. In both cases, the rotational speed of 3 and 7 rpm correspond tocapillary numbers Ca = 5.8 ? 10?3 and 1.4 ? 10?2, which in turn corre-spond respectively to effective capillary numbers Cae = 4.0 ? 10?2 and9.5 ? 10?2. Based on these parameters, the predicted steady-state profileand its temporal development are both in reasonably good agreement withexperimental measurements. With increasing shear rate, the large bubblesmigrate away from the walls more rapidly, and this aggregation at the cen-ter overpowers the shear-induced diffusion that strives to spread the largebubbles uniformly. Consequently, the size-based segregation occurs morerapidly for higher shear rates, and produces a narrower equilibrium distri-bution centered at the middle of the gap y = 0. Note that Eq. (5.1) doesnot predict a t ? ???1 scaling for the transient. It would if vm and D wereboth proportional to ?? or Ca. In reality, vm ? Ca2, and D also depends onCa nonlinearly thanks to ?(Ca) [41]. Our experimental data do not exhibitsuch a scaling either.As the average area fraction of the large bubbles ?3increases, Fig. 5.5shows that the model correctly predicts the widening of the equilibriumdistribution, and the agreement with measurements is quantitatively accu-rate. The idea underlying this prediction is that higher fraction of the largebubbles increases the frequency of their collision and thereby elevates theeffective diffusivity D (cf. Eq. 5.3). This has been confirmed by the experi-ments. We have also studied the effect of area fraction on the speed of sizesegregation. The model predicts that with increasing ?3, the segregationoccurs more rapidly (Fig. 5.6); it takes less time to reach the equilibriumdistribution. This captures the trend in the experimental data if not theprecise values of the segregation time. Qualitatively, increasing ?3increasesthe diffusivity D, which should counteract the migration and lead to a slowersegregation. On the other hand, a higher ?3corresponds to a wider equilib-rium distribution with a larger ye. This means that large bubbles initiallynear the wall need to travel a shorter distance to get to their steady-stateposition. These two effects oppose each other and the outcome seems to bein favor of the latter. King and Leighton [41] have quantified the competi-tion between the two effects in the limiting case of small y. By linearizingthe migration velocity vm of Eq. (5.2) (i.e., reducing the y terms between thebrackets to 9y), they obtained a self-similar solution in which time t scalesonly with d/vm, and is independent of ?3. Our experiment and analysis arenot restricted to the small-y limit, and thus do not exhibit the similarity.Recall that we have assumed ? to be independent of ? in Eq. (5.7). Theclose agreement for the whole range of area fractions studied here indicatesthat this is a reasonable assumption.Finally, we examine the effect of the bubble size ratio ?, which influ-465.4. Polydisperse foamFigure 5.8: Steady-state bubble distributions in the polydisperse foam samplesH, I, J and K, after 10 min of shearing at 7 rpm. The samples have the samethree bubble sizes, a1= 350 ?m, a2= 500 ?m and a3= 700 ?m, at differentaverage area fractions: (a) Sample H, (?1,?2,?3) = (90%, 5%, 5%); (b) Sample I,(80%, 10%, 10%); (c) Sample J, (60%, 20%, 20%); (d) Sample K, (40%, 30%, 30%).The area fractions are normalized for each species.ences the structural evolution of bidisperse foams through the Cae ? Carelationship in our model (Eq. 5.6). Figure 5.7 shows that the model cor-rectly predicts the effects of ? on the steady-state distribution as well as onthe temporal evolution toward it: higher ? produces a faster approach toa narrower steady-state distribution. Qualitatively and quantitatively (viaEq. 5.6), therefore, increasing ? has similar effects to elevating the shearrate or capillary number.5.4 Polydisperse foamWe now consider polydisperse foams composed of three bubble sizes, a1,a2and a3for Samples H?K and a1, a3and a4for Samples L?O (see Table1). Note that in these samples the two larger species always have the samearea fraction. Figure 5.8 shows the steady-state distributions for Samples475.5. ConclusionH?K. For Sample H with the lowest ?3, the largest bubbles (of radius a3)exhibit a sharply peaked distribution at the center of the gap while the twosmaller bubble species (a1and a2) are more or less uniformly distributed.If the a3bubbles were absent, the a1and a2bubbles would not exhibit sizesegregation as their size ratio ? = 1.43 is below the threshold for ? = 7 rpm[55]. Therefore, the aggregation of the a3bubbles in Sample H is similar tothat in a bidisperse foam. By increasing the area fraction ?2and ?3to 10%and 20% (Samples I and J), the two smaller bubble species are displacedtoward the walls. This is evidently due to the increasing area occupied bythe a3bubbles at the center, and recalls the marginalization of white bloodcells when the more flexible red cells aggregate in the center [44, 71].However, increasing ?2and ?3further to 30% (Sample K) brings aboutan apparent reversal of the marginalization. Now all three species areroughly uniformly distributed in the gap. This can be rationalized by thestronger shear-induced diffusion of the largest bubbles at the higher ?3,much as in the bidisperse foams of Fig. 5.5. Comparing Fig. 5.8 and Fig. 5.5,however, reveals an interesting role for the a2bubbles. In Fig. 5.8(d), the a3distribution flattens for ?3= 30% in the polydisperse Sample K, whereasin the bidisperse Sample F (Fig. 5.5c), the large bubbles are not quite uni-formly distributed even for ?3= 50%. Thus, the a2bubbles are not inert andmerely passively displaced by the a3bubbles. They actively facilitate thespreading of the largest bubbles. This may have occurred through hinderingtheir migration toward the center (via effectively reducing ?) or enhancingthe diffusion of the largest bubbles, or even both.Now we investigate the size segregation in polydisperse Samples L?O inwhich the two larger species, a3and a4, both tend to migrate away fromthe walls and compete with each other to occupy the center of the gap.Figure 5.9 shows the equilibrium distributions of the three bubble speciessubject to shearing at ? = 7 rpm. As it turns out, the two large bubblespecies behave similarly in this case. For ?3= ?4? 20% (Samples L?N), both a3and a4bubbles aggregate at the center of the gap. The a1bubbles are marginalized as seen above. The largest a4species enjoys anarrower distribution with a higher peak than a3. Thus, the larger bubblesize ? affords the former an advantage. With increasing ?3and ?4, thedistributions broaden until at 30%, both become more or less uniformlydistributed across the gap (Sample O). As in Fig. 5.5(c) and Fig. 5.8(d),this can be ascribed to the dominance of the shear-induced diffusion of thea3and a4bubbles.5.5 ConclusionWe have studied the structural evolution of bidisperse and polydisperse 2Dfoams in a narrow-gap Couette shear cell. Within the parameter ranges485.5. ConclusionFigure 5.9: Steady-state bubble distributions in the polydisperse foam Samples L,M, N and O, after 10 min of shearing at 7 rpm. The samples have the same threebubble sizes, a1= 350 ?m, a3= 700 ?m and a4= 875 ?m, at different area frac-tions: (a) Sample L, (?1,?3,?4) = (90%, 5%, 5%); (b) Sample M, (80%, 10%, 10%);(c) Sample N, (60%, 20%, 20%); (d) Sample O, (40%, 30%, 30%). The area fractionsare normalized for each species.tested, the main experimental findings can be summarized as follows.(a) After shearing for a sufficiently long time, the foam achieves a quasi-steady morphology that is independent of the initial configuration.(b) In this quasi-steady state, the bubble species may be uniformly mixedor segregated by size depending on the physical and flow parameters.Size segregation occurs if the bubble size ratio and shear rate are bothabove certain threshold values, and if the area fraction of the largebubbles is not too high. Otherwise a mixed state obtains.(c) In size-segregating bidisperse foams, the segregation occurs more rapidlyand produces a narrower final distribution for higher shear rates andlarger bubble size ratios. Increasing the area fraction of the large bub-bles, on the other hand, leads to a broader final distribution that isachieved in less time.495.5. Conclusion(d) Polydisperse foams behave similarly in that size segregation occurs atrelatively low area fractions of the largest bubbles while a uniformlymixed morphology prevails at higher large-bubble area fractions. Thebubbles of intermediate size tend to facilitate the broadening of thedistribution of the largest bubbles.These observations are rationalized by adapting a migration-diffusionmodel previously developed for monodisperse emulsions. Viewing the largerbubbles as being suspended in an effective continuum comprising the smallerones, we describe the structural evolution in bidisperse foams by a convection-diffusion equation. The model balances two competing factors, the lateralmigration due to wall repulsion and the shear-induced diffusion due to in-teraction among the large bubbles. For bidisperse 2D foams, the modelpredicts all aspects of the experimental observations, often with quantita-tive accuracy.The success of the emulsion model in predicting bubble segregation in apolydisperse bubble raft is quite remarkable, especially in view of the differ-ences between the two systems. The prevailing thinking of foam dynamicsis that it is determined by the interfacial morphology on the local scale.Then the 2D foam studied here can be viewed as a curious exception whereat least one attribute of the dynamics, the migration and segregation ofbubbles based on size, turns out not to be intimately related to the mor-phology of the smaller bubbles. These small bubbles can be replaced, in asense, by an effective continuum while preserving the same segregation ofthe large bubbles. There are some caveats to this analogy, however. The?replacement? of the surrounding bubbles by an effective continuum is so asto produce the same amount of deformation on the large bubbles. This boilsdown to an effective capillary number. One cannot reduce the analogy fur-ther to something more tangible, say an effective viscosity, which would notproduce the correct migration velocity from the Chan-Leal formula (Eq. 5.2).Thus, the effective capillary number embodies intricate local dynamics hav-ing to do with the discreteness of the surrounding bubbles, which exert a?bumping force? on the large bubbles [55] that cannot be ascribed to anelevated medium viscosity. Moreover, the analogy may be limited to certaintypes of foam. In our experiment, the bubbles are closely packed but notpressed against one another so as to produce polygonal facets. If we tryto pack more bubbles into the raft, they tend to pile on top of others anddestroy the two-dimensionality. Thus, the smaller bubbles are essentiallyundeformed in our experiments. In drier foams that undergo more inten-sive interaction among bubbles, e.g. through T1 events [79], the continuumanalogy may no longer hold.To conclude, let us briefly return to prior experiments that motivated ourstudy. Our findings suggest that in the prior experiment of Herzhaft [28],where 3D polydisperse foams are sheared between parallel plates, shear-505.5. Conclusioninduced migration probably have occurred to produce marginalization ofsmaller bubbles to the plates and a central layer rich in large bubbles. How-ever, three-dimensionality affects how neighboring bubbles interact with oneanother, and our 2D model will need to be upgraded before it can be com-pared quantitatively to 3D foam experiments. In addition, it is important tonote the experimental and numerical results of Cox and coworkers [15, 24]that showed no size segregation in 2D foams undergoing cyclic planar ex-tension and compression. The conditions in their studies differ from ours inat least three aspects. In extensional flows the bubbles do not follow par-allel streamlines. Instead, neighboring rows are compressed into one whilebeing elongated in the orthogonal direction. Thus, the interaction amongbubbles differs markedly from the rubbing and bumping in our shear ex-periments. Moreover, the cyclic straining introduces repeated encountersamong bubbles, a feature absent from steady shearing. Finally, the maxi-mum extensional rate in their experiments is only 0.0455 s?1, much belowour threshold value of 4.17 s?1 for approximately the same bubble-size ratio? = 2. Their simulation employed the Surface Evolver, and is thus quasi-static in nature. It appears, therefore, that size segregation in extensionalflows remains an open question that requires further studies, especially athigh strain rates.51Chapter 6Effect of non-Newtonianrheology on bubblemigration in sheared foam6.1 IntroductionFoams are quintessential soft matter in that they admit both a macroscopic,continuum-based description and a microscopic, bubble-scale one. On theone hand, foam rheology is invariably measured on the bulk. In so doing,one implicitly adopts an effective continuum view, and sometimes explicitlyrepresent the foam rheology by continuum models [39, 46]. On the otherhand, bulk flow and deformation produces changes of the microstructure, i.e.bubble-scale morphology. Shearing is known to induce neighbor-swappingrearrangements known as T1 processes [62, 77]. Additional microstructuralchanges include bubble coalescence, breakup, migration and size-based seg-regation [27, 28, 56, 57]. Since foams can be examined on both levels, andindeed manifest a clear link between their microstructure and bulk flow be-havior, they are excellent model systems for studying the coupling betweenthe microscopic and macroscopic scales.We are just beginning to understand the interaction between the twolength scales, and many questions remain to be answered. Even the shearviscosity of a foam is not well understood. In one simple-shear experiment,Golemanov et al. [27] observed a marked increase of the shear stress in timeafter the start of shear, and attributed it to the breakup of the bubbles. Inanother experiment, Herzhaft [28] reported a shear stress that gradually de-clines in time. In addition, foams show shear-thinning, which can be fittedto the continuum Herschel-Bulkley model [39, 46]. Surprisingly, the bubblevelocity profile under shear differs appreciably from that predicted by thecontinuum model [39]. This has been ascribed to a nonlocal effect arisingfrom the cooperative movement of bubbles within a certain ?cooperativity?length scale. Surface tension is also known to produce normal stress differ-ences on the macroscopic scale [25, 45]. Thus, the micro-macro connectionis subtle for foams, and their dynamics is influenced by continuum rheologyas well as textural granularity.We have been investigating another aspect of this connection, through526.1. Introductionthe cross-streamline migration of bubbles in sheared ?two-dimensional? foam,which is a bubble raft floating on a soapy solution. The two-dimensionalityaffords direct visualization of bubble-scale microstructures that would beimpossible for 3D foams. As we showed in the previous chapter, bubblessegregate according to size in sheared polydisperse foams, and that the seg-regation can be understood based on a simple continuum model in whichthe smallest bubbles are viewed as an effective Newtonian fluid that sus-pends the larger bubbles. In this model, combining shear-induced migrationof individual bubbles and an effective diffusion due to collision among largebubbles can account for the segregation data very well. This adds to thecollection of foam behavior that can be described as continuum. In themean time, the discreteness of the bubbles manifests itself as well, in termsof ?quantized? steps of migration and thresholds in shear-rate and bubblesize ratio under which no migration takes place. Note that the above hasbeen observed in simple shear in a narrow-gap Couette device.The experiments to be presented in this chapter extends the above studyto nonuniform shear in a wide gap Couette device. This seemingly naivechange of geometry, as it turned out, brings out the non-Newtonian rhe-ology of the foam to bear on the migration of bubbles. Therefore, thismay be viewed as an interesting example of the bulk rheology affecting themicrostructural evolution of the foam. Bubble migration in foam can becontrasted with drop migration in non-Newtonian fluids, a subject that hasreceived long-standing attention [8, 9, 26]. As will be demonstrated, bubblemigration in a non-Newtonian liquid holds the key to understanding themigration in foam. In fact, the continuum analogy can be maintained if weview the smaller bubbles as constituting an effective non-Newtonian fluidthat shows shear-thinning and normal stress difference under shear.In quasi-static state liquid foam shows topological changes known as T1and T2 events[21, 46, 62]. Furthermore, bubbles in liquid foam may undergobreakup, coalesce or even segregate according to the size when subject tohigh shear rates [27, 28, 56, 57]. In chapter 4 and chapter 5 in narrow gap,we explored the structural change due to size-based bubble segregation, andfound that the surrounding bubbles can be viewed as an effective Newtoniancontinuum in some sense[55, 56].In this chapter, we show how the continuum idea can be extended to asituation where the foam must be seen as a non-Newtonian fluid. Therefore,the main objective of this work is to study the non-Newtonian behaviorof liquid foam and its effect on structural evolution. As will be demon-strated, bubble migration in a non-Newtonian liquid holds the key to un-derstanding its migration in foam. Studies on migration of a single dropletin non-Newtonian fluids show that the migration differs from the Newtonianfluid [8, 9, 26]. The main objective in this chapter is to understand the mi-gration in foam through the window of non-Newtonian effects. Therefore,we make some well characterized viscous and elastic fluids that represent the536.2. Experimental resultsshear-thinning and normal stresses in sheared foam and study the migrationprocess in them.6.2 Experimental resultsWe have performed two series of experiments?bubble migration and rheo-logical measurements?on three types of mediums: two-dimensional foams,xanthan gum solutions and a Boger fluid. In the following, we present theresult of each experiment in turn.6.2.1 Bubble migration in foamThe experimental protocol for recording bubble migration across streamlinesis similar to that used in narrow-gap Couette cells in chapter 3. We makea monodisperse foam consisting of bubbles of radius r = 0.36 ? 0.02 mm,which covers the entire wide gap of the Couette device in a more or lessregular hexagonal lattice. The foam quality, defined as the area fraction ofthe bubbles, is maintained at 85% for all the experiments to be presented.We then insert a single large bubble of radius R into the foam at differentinitial positions, and shear the foam by rotating the inner cylinder at aconstant angular velocity ?. The two control parameters are the nominalshear rate and the bubble size ratio ? = R/r. Shear rate at the innerwall can be estimated from from the velocity gradient that the first rowof bubble experiences at each rotational speed. Therefore, shear rate ??varies from 1.5 to 8.62 s?1; the upper bound is chosen such that centripetalforce remains negligible in all experiments. Foams in our wide-gap devicemay yield partially, and that introduces unnecessary complication to thediscussion of migration. Thus, from here onward, we will only considershear rates above that required for full yielding: 3.5 s?1 < ?? < 8.62 s?1.For the large bubble we have tested five sizes: R = 0.5, 0.6, 1, 1.4 and 1.8mm, corresponding to ? = 1.39, 1.67, 2.79, 3.91 and 5.03.Similar to what has been reported for the narrow-gap Couette device inchapter 4, the large bubble migrates across the flow direction if the shear rate?? and bubble size ratio ? are each above a threshold value. The migrationis driven by a hydrodynamic force that arises from the asymmetric flow andpressure fields surrounding the deformed bubbles [55, 56]. The thresholdsreflect the discreteness of the foam; the hydrodynamic force has to overcomethe capillary pressure in neighboring bubbles in order to move the largebubble to the next row. The migration is generally away from the walls, andthe hydrodynamic driving force is greatest at the wall and diminishes towardthe center [8]. Thus, a large bubble may migrate across one or several rowsif released near the wall, but not at all if released further away from the wall.For simplicity, we will exclude such partial migration from further discussion,and define the thresholds of ?? and ? according to complete migration, i.e.546.2. Experimental resultsmigration to an equilibrium position regardless of initial positions. As in thenarrow-gap Couette cell, we find the ?? threshold to decrease with increasing?, and the ? threshold to decrease with increasing ?? [55]. The thresholdvalues are comparable to those in the narrow gap. In the following we willconcern ourselves only with the dynamics above these thresholds.0 50 100 150 200 25000.20.40.60.8t, Time (sec)s, Dimensionelss position  ? = 2.79, ?? = 8.62sec?1? = 3.91, ?? = 5.71sec?1? = 3.91, ?? = 6.72sec?1? = 3.91, ?? = 6.72sec?1Figure 6.1: Migration trajectories of bubbles of two size R = 1 mm (? = 2.79)and R = 1.4 mm (? = 3.91), released from different positions in the foam shearedat different shear rates. The bubble center is given by s, its distance from theinner cylinder scaled by the gap width d = Ro?Ri. The curve shows Chan-Leal?sprediction for the bubble size R = 1.4 mm undergoing shear rate of ?? = 5.71 s?1.Figure 6.1 shows the migration trajectories of bubbles of two sizes (? =2.79 and ? = 3.91) at different shear rates. The threshold shear rate isaround 5.71 for the smaller bubble, and around 3.5 for the larger one. Forshear rates above this threshold, the final equilibrium position is reachedfrom all initial positions. This equilibrium position seems to be independentof the shear rate and the bubble size ratio, although the speed of migration556.2. Experimental resultsincreases with ?? and ?. The features described so far are similar to priorobservations in the narrow-gap Couette device [55].The key difference is that the equilibrium position for large ?? is notat the center of the gap, as is the case in the narrow-gap geometry [55].Rather it is some distance inward from the center of the gap, closer to theinner cylinder; in the particular case shown in Fig. 6.1, this position is ats ? 0.36. One naturally seeks a geometrical explanation for the difference.After all, the wide-gap Couette device should produce a nonuniform shearrate profile across the gap, with higher local shear rate in the inner halfthe gap than the outer half. This asymmetry should bias the equilibriumposition of the migrating bubble. This effect can be quantified with the helpof the Chan-Leal theory for lateral migration of droplets in Couette flows[8, 9].Although the Chan-Leal formula was developed for the migration of asingle drop in a continuum suspending fluid, we have demonstrated that itcan accurately describe the migration of a single large bubble in a sea ofmonodisperse bubbles if the enhanced bubble deformation is accounted forthrough an elevated effective capillary number [55, 56]. Since the Chan-Lealtheory was developed for a Couette device, it accounts for the curvaturein the streamlines and the variation of shear rate across the gap. Thus wehave used the modified Chan-Leal formula, containing the effective capillarynumber, to predict the migration of a single bubble corresponding to theconditions of one of the experimental runs of Fig. 6.1, and the result isplotted as a solid curve. It predicts only a slight inward shift of the finalequilibrium position, s = 0.47, which cannot account for the much largershift observed experimentally. Besides, the migration speed is also over-predicted by a wide margin. Therefore, the observations in the wide-gapexperiment cannot be accounted for by the geometry alone.A factor that has not been taken into account in the above comparisonis the non-Newtonian features of the liquid foam. The Chan-Leal formulaused in Fig. 6.1 is for a Newtonian suspending fluid. It has successfullyrepresented the migration observed in our previous experiments [55, 56],which implies that the small-bubble foam can be viewed effectively as aNewtonian suspending fluid. Can it be that the nonuniform shearing in thewide-gap device brings out non-Newtonian rheology that is not manifest inthe narrow-gap Couette cell? Shear-thinning will accentuate the nonlinear-ity of the velocity profile, and a large bubble would thus experience unequalviscosities upon its two sides. Moreover, the first normal stress difference N1would also exhibit an asymmetry between the two sides. To ascertain thesepotential effects on bubble migration in the foam, we need to characterizethe bulk rheology of the foam first. As will become clear in the next sub-section, this has in turn motivated us to make polymer solutions possessingshear thinning and elasticity separately, in which bubble migration may beinvestigated as benchmarks for gauging the bubble migration in foam.566.2. Experimental results10?3 10?2 10?1 100 10110?1100101?? (sec?1)? (Pa)  Foam exp 1Foam exp 2? = 0.32 + 0.77 ??0.47Figure 6.2: Shear flow curve of the two-dimensional foam measured in a rheometerwith a bob-cup fixture. Two data sets are plotted along with a best-fitting curveto the Herschel-Bulkley equation (Eq. 6.4).6.2.2 Bulk rheology of 2D foam and polymer solutionsFoams are known to have a yield stress, and in the fully yielded state exhibitshear-thinning and normal stress differences [25, 39, 42, 60]. To probe theshear-thinning of our 2D foam, we have measured its shear rheology on arotational rheometer using the bob-cup fixture. To accommodate a largenumber of bubbles, we used a wide-gap setup, with the radius of the innercylinder being 22 mm and that of the stationary outer cylinder being 35mm. The local shear rate at the inner cylinder is obtained from the following[23, 43]:?? = 2 ? d(ln?)d(lnM) . If (?o ? ?y ? ?i) (6.1)?? = 2 d(ln?)/d(lnM)1?R2i /R2o???M(d?/dM)ln(Ri/Ro). If (?o > ?y) (6.2)576.2. Experimental results10?3 10?2 10?1 100 101 10210?210?1100101102?? (sec?1)? (Pa.s)  1000 ppm1500 ppm2000 ppm2500 ppm3000 ppm4000 ppmFoamFigure 6.3: Shear viscosity of xanthan gum solutions of various concentrations.The line indicates the foam viscosity.?o, ?i, ?y and M being the shear stress at the outer cylinder, shear stressat the inner cylinder, the yield stress and torque at the inner cylinder re-spectively. The shear rate can be estimated from the maximum of equa-tions(6.1, 6.2). According to Estelle et al. [23] the appropriate shear rate isthe one that maximizes the dissipation in flowing material.Figure 6.2 shows the shear stress as a function of the shear rate for our2D foam. Following prior experiments on 2D and 3D foams [39, 60], we fitthe data by a Herschel-Bulkley model:? = ?y + K??n, (6.3)with a yield stress ?y = 0.32 Pa, consistency K = 0.77 Pa?sn and a power-law index n = 0.47. Thus, our foam shows similar shear-thinning behaviorto previous experiments [39, 60]. Both 2D and 3D foams are known toexhibit a first normal stress difference N1[25, 42, 45, 59]. Labiausse et al.[45] measured N1for a 3D foam in the pre-yielding regime. Kraynik et al.[42] further determined that N1is on the same order of magnitude as theshear stress for 3D foam before yielding. In simulations of a random 2D foam586.2. Experimental results1000 1500 2000 2500 3000 3500 40000.20.30.40.50.60.7c, Concentration (ppm)n, Powerlaw index1000 1500 2000 2500 3000 3500 400000.511.522.5c, Concentration (ppm)K, Consistency factorFigure 6.4: Comparison of the power-law viscosity of xanthan solutions of differentconcentrations (symbols) with that of the foam (horizontal lines). (a) The power-law index n; (b) the consistency factor K.undergoing simple shear in the yielded regime, Okuzono et al. [59] recordedN1values roughly twice as large as the shear stress over a range of shearrates. For our 2D foam, we have not been able to measure N1directly. Inview of the limited data in the literature, we have decided to use the resultsof Okuzono et al. [59] as a guideline, and assume that for our 2D foam N1ison the same order of magnitude as the shear stress, which we have measuredwith confidence.596.2. Experimental resultsAs shear-thinning and normal stress act simultaneously on bubble mi-gration in our foam, it is impossible to identify and analyze their individualcontributions. Therefore, we have sought to probe the two effects separatelyby using shear-thinning and viscoelastic polymer solutions that representeach aspect of the foam?s rheology.Figure 6.5: Shear rheology of the Boger fluid, with open circles for the shearviscosity and filled diamonds for the first normal stress difference N1. The straightline is a power-law fitting for N1with a slope close to 2. The filled squares showthe shear stress of the foam, which is comparable in magnitude to N1of the Bogerfluid, especially near the upper bound of the shear rate.Aqueous solution of xanthan gum are known to exhibit shear thinningbut negligible elasticity [1, 3]. We have tested a series of xanthan solutionsand chosen the closest one to the foam rheology. Figure 6.3 shows the shearviscosity of xanthan solutions of 6 concentrations. For comparison, we haveplotted the viscosity of our foam in the range of shear rates encountered inthe bubble-migration experiments. Furthermore, we fit a power-law to thexanthan viscosities in the same range, and plot in Fig. 6.4 the consistencyK and the power-law index n for the xanthan solutions together with thevalues measured for our foam. The closest one to the foam appears to be the606.2. Experimental resultssolution at a concentration of 2500 ppm. Therefore, we choose this solutionas the representative for the shear-thinning behavior of foam.Similarly, Fig. 6.5 compares the shear rheology of the Boger fluid withthat of the foam. Within the range of shear rates tested, the Boger fluidexhibits an essentially constant shear viscosity, and an N1that scales ap-proximately with ??2. Ideally, we would have liked N1of the Boger fluid tomatch the foam shear stress in the ?? range of interest, up to 6 s?1. Thisturns out to be difficult to realize experimentally. For one, increasing thepolymer concentration in the Boger fluid brings forth appreciable shear thin-ning. Thus, we have accepted this Boger fluid as roughly representing theorder of magnitude of the normal-stress in the foam.6.2.3 Bubble migration in shear-thinning and Boger fluidsWe have conducted bubble migration experiments in the shear-thinning xan-than solution and the Boger fluid, using the same wide-gap Couette device,bubble sizes and operating conditions as in the foam experiments. Figure 6.6depicts migration of large bubbles of two sizes in the xanthan solution atdifferent shear rates and initial positions. Our results show that the sameequilibrium position, s ? 0.25, is reached from different initial positions.This position is roughly midway between the center of the gap and theinner cylinder, and is much more inward than that in a Newtonian fluid(s = 0.47 according to the modified Chan-Leal formula; see Fig. 6.1). Be-sides, the equilibrium position does not depend on the bubble size, nor onthe shear rate. But the speed of migration does increase with the bubble sizeand the shear rate. Thus, shear-thinning tends to shift the bubble furthertoward the inner cylinder. This conclusion is consistent with the previousexperimental results of Gauthier et al. [26]. In a wide gap Couette device,Gauthier et al. studied migration of a deformable droplet in a shear-thinningfluid with power-law index n = 0.71. Droplets of different sizes starting fromdifferent initial positions all end up at an equilibrium position s ? 0.4.Our xanthan solution has stronger shear-thinning (n = 0.43) than theirfluid, and it is reasonable that the bubbles assume a position farther inwardthan in their case. Regarding the hydrodynamic origin of the effect, one mayimagine that the bubble experiences reduced viscosity on the side closer tothe inner wall, where the shear rate is higher. This may have biased thelateral force in favor of inward migration. However, such a naive argumentfails to anticipate the apparent insensitivity of the equilibrium position to?? and ?. In the next subsection, we will use numerical simulations of dropmigration in shear-thinning fluids to explore these questions.The opposite trend is observed in the Boger fluid. Figure 6.7 showstypical migration trajectories of bubbles of three sizes, released at differentinitial positions, at two shear rates. In all cases, the bubble migrates to anequilibrium position close to s = 0.55 in the outer half of the gap. This616.2. Experimental results0 500 1000 1500 200000.20.40.60.8t, Time (sec)s, Dimensionless position  R = 0.6 mm, ?? = 8.0 sec?1R = 1 mm, ?? = 8.0 sec?1R = 1 mm, ?? = 3.15 sec?1R = 1 mm, ?? = 8.0 sec?1R = 0.6 mm, ?? = 8.0 sec?1Figure 6.6: Migration trajectories of bubbles in the xanthan solution starting fromdifferent initial positions. The bubbles are of two sizes R = 0.6 mm and R = 1mm, and the shear rate is varied from ?? = 3.5 s?1 to ?? = 8.62 s?1.suggests that the normal stress N1tends to force the bubble outward. Fur-thermore, the equilibrium position shows no dependence on the shear rate ??and little dependence on the bubble size ?, although the migration speedincreases with both.Chan and Leal [8, 9] have carried out experimental and theoretical stud-ies of the migration of a single suspended drop in a Boger fluid sheared ina Couette device. The predictions of the Chan-Leal formula, for the experi-mental conditions used here, are plotted as solid and dashed lines in Fig. 6.7.First, the formula correctly predicts the outward shift of the bubble?s equi-librium position in all cases. Second, for a fixed bubble size, the formulapredicts a final equilibrium position that is independent of the shear rate,in agreement with our observations. In Fig. 6.7(a), the predicted s ? 0.6differs from our measurement by 8%. This difference in the destination af-fects the prediction of the migration speed, but it still falls within reasonableagreement with experimental data. Third, the Chan-Leal formula predictsthe equilibrium position s to shift away from the outer wall as the bubble626.2. Experimental results0 1000 2000 300000.20.40.60.81t, Time (sec)s, Dimensionless position  (a)R = 1 mm, ?? = 2.64 sec?1R = 1 mm, ?? = 4.77 sec?1R = 1 mm, ?? = 4.77 sec?10 1000 2000 300000.20.40.60.81t, Time (sec)s, Dimensionless position  (b)R = 0.7 mm, ?? = 4.77sec?1R = 1 mm, ?? = 4.77sec?1R = 1.4 mm, ?? = 4.77sec?1Figure 6.7: Migration trajectories of a single bubble in the Boger fluid. (a) Abubble of radius R = 1 mm released from two initial positions at two shear rates.(b) The effect of bubble size at a fixed shear rate (?? = 4.77 s?1). The curves showthe predictions of the Chan-Leal formula.size R increases relative to the gap d = Ro ? Ri. This effect is stronger forsmaller R/d and saturates for larger R/d. In comparison, our experimentaldata shows a much weaker effect of bubble size. As R increases from 0.7mm to 1 mm and then to 1.4 mm, s seems to shift slightly inward towardthe centerline, but the magnitude is much below the roughly 10% change ins predicted by the Chan-Leal formula.636.3. Discussion6.3 DiscussionIn this section we will try to understand the underlying physics behind theexperimental results. To do so, we will strive to answer three importantquestions separately: (a)- Why does elasticity (N1) shift the equilibriumposition outward? (b)- Why does shear thinning have an opposite effect?(c)- What is the effect of combination of these two factors?6.3.1 Effect of elasticityWe start with the first question: how does elasticity (N1) shift the equilib-rium position outward? Karnis and Mason [37] suggested that a particlein a Boger fluid experiences normal forces due to N1on both of its sides.In the nonuniform shear of a wide-gap Couette device, the normal force islarger on the inner side as the shear rate is higher there. This asymmetrypushes the droplet toward regions of lower shear rates, i.e. toward the outerwall. Ho and Leal [29] introduced the idea of ?hoop thrust? to rationalizethe N1effect on particle migration. They suggested that the presence of theparticle disturbs the flow field around it and generates ?bowed streamlines?around the particle. The tension along these streamlines thus produce ahoop stress that tends to drive the particle toward regions of lower shearrates. Both explanations are similar in essence; the gradient in shear rateleads to a gradient in normal stress, which then pushes the particle towardthe outer wall.Another intriguing feature of Fig. 6.7(a) is the independence of the equi-librium bubble position to the shear rate ??. There are two mechanismsgoverning migration of a deformable particle in a Boger fluid, the normalstress as discussed above, and the deformation of the bubble or drop. Asdemonstrated by Chan and Leal [8] in Newtonian as well as second-orderfluids, the bubble deformation creates an asymmetry in the flow and stressfields in the vicinities, which tends to push the bubble away from solid walls.Thus, this effect is opposite to that of N1in the outer half of the Couettedevice. When the shear rate ?? is elevated, N1increases and so does thebubble deformation. The Chan-Leal theory shows that these two effectscancel out such that the final equilibrium position is independent of ??. Thishas been born out by our experimental data as well (Fig. 6.7a). Our exper-imental conditions also satisfy the constraints under which the Chan-Lealasymptotic theory holds. These conditions are Ca  1, R/d  1, De  1and De << Ca.The effect of the bubble size R can be considered in a similar way. Alarger bubble will experience larger deformation on the one hand, and alarger N1-based normal force on the other. Again the two factors tend tooppose each other. The Chan-Leal calculation shows, however, that theydo not exactly cancel each other. Smaller bubbles tend to favor the N1646.3. Discussion0 0.2 0.4 0.6 0.8 1?1?0.500.51 x 10?6sForce (N)  Newtonian forceElastic forceFigure 6.8: Calculated force for Boger fluid and Newtonian fluid vs radius on abubble with radius of 1 mm in wide-gap Couette geometry using Chan-Leal formula.0 0.05 0.1 0.150.50.60.70.8R/ds e,DimensionlessEquilibriumPosition  Chan-Leal modelExperimentsFigure 6.9: Dimensionless equilibrium position vs dimensionless radius of the bub-ble in Boger fluid sheared at ?? = 4.77 sec?1. Curve shows the prediction of Chan-Leal?s model and the data are experimental results.656.3. Discussioneffect, and thus attain an equilibrium position closer to the outer wall. Withincreasing bubble size, the equilibrium position shifts inward but levels offfor R/d ? 0.11 (c.f. Fig. 6.9). The experimental data, covering a modestrange of R/d, do show a clear downward trend, although the slope is notas steep, and the equilibrium positions are generally farther from the outercylinder than the theoretical prediction. The largest discrepancy is for thesmallest drop (R = 0.7 mm). At present we have no explanation.We can rationalize the migration results of a bubble in Boger fluid byestimating the lateral force on the bubble during the migration process.The wall repulsive force can be computed from the Stokes formula using themigration velocity of the Chan-Leal theory for both Newtonian and Bogerfluids. Fig 6.9 shows calculated force vs dimensionless position in differentfluids. For a Newtonian fluid this force goes to zero at s ? 0.48 which indi-cates the equilibrium position of a migrating bubble and for similar bubblein Boger fluid equilibrium position shifts toward the outer cylinder s ? 0.6.This confirms our experimental results on migrating bubbles in Boger fluid.6.3.2 Effect of shear thinningShear thinning behavior of foam is an important non-Newtonian featurethat might have affected the migration of bubble under shear. To the bestof our knowledge, there is no theory that can explain the effect of shearthinning on lateral migration of a deformable object. Therefore, we havecarried out several 2D simulations using finite element method. Simulationswould allow us to analyze the bubble trajectory and force on a bubble in-side a shear thinning fluid. The shear thinning parameters were chosen torepresent the xanthan gum solution with following parameters: n = 0.43,K = 1 (Pa.s0.43), 0.025 ? R/d ? 0.1 and 1.37s?1 ? ?? ? 11s?1 in a Couetteco-axial cylinders geometry with Ri = 2 cm and Ro = 3 cm. The size ofcomputational domain is different from the one in experiments( Ri = 8.1 cmand Ri = 9.9 cm ). This is mainly due to the technical difficulties in simu-lating of very large domains.Fig.6.10 shows the trajectory of a migrating bubble in shear thinningfluid obtained by simulations. For a Newtonian fluid Chan-Leal?s theorygives an equilibrium position of s ? 0.49. It indicates that the equilib-rium position of bubble shifts further towards the inner cylinder when shearthinning behavior is introduced (s ? 0.377 < 0.49). This result is essen-tially in agreement with our experimental observations for shear thinningfluid (c.f. Fig.6.6). The predicted equilibrium position is different from theone in experiments and this is due to the different geometry chosen in sim-ulations. Moreover, fig. 6.10 shows that the final equilibrium position doesnot depend on the shear rate. This is consistent with our experimental ob-servations reported in figure 6.6 and also prediction of Chan-Leal?s theory.Furthermore, Fig. 6.11 shows the bubble trajectories for different bubble666.3. Discussion0 500 1000 1500 2000 25000.10.20.30.40.50.60.70.8t, Time (sec)s, Dimensionless position  ?? = 1.37s?1?? = 1.37s?1?? = 2.74s?1?? = 10.96s?1?? = 10.96s?1Figure 6.10: Bubble migration trajectories in shear thinning fluid for R/d = 0.05at different shear rates.sizes and fixed shear rate. The final equilibrium position clearly moves awayfrom the inner cylinder when bubble size is increased. On the other hand,the final equilibrium position for different bubble size tested in experimentis s = 0.25 ? 0.01. Therefore, we do not see the effect of bubble size inexperiments. Mason et al [26] have also measured the trajectory of dropletsin shear thinning fluid with different size and did not see a considerablechange in equilibrium position. They reported equilibrium position of s ?0.43? 0.05 for R/d = 0.073 and s ? 0.453 for R/d = 0.11. At this momentwe do not have an explanation for the disagreement between experimentsand simulations.To rationalize the effect of shear-thinning on the final equilibrium po-sition of the bubble, we consider two lateral forces acting on it. The firstarises from wall repulsion, and is the same as in a Newtonian fluid thatcan be estimated from the Chan-Leal formula. The second is due to shear-thinning. Specifically, we expect the inner side of the bubble to experiencea higher shear rate and thus lower viscosity than the outer side. This radialasymmetry in the viscosity is probably the direct cause of an inward lateralforce on the bubble, which shifts the bubble closer to the inner wall againstthe wall repulsion.676.3. Discussion0 1000 2000 30000.20.30.40.50.6t, Time (sec)s, Dimensionless position  R/d = 0.025R/d = 0.025R/d = 0.05R/d = 0.05R/d = 0.1Figure 6.11: Bubble migration trajectories in shear thinning fluid for ?? = 2.74s?1at different shear rates.We can estimate the net lateral force on the bubble from the computedbubble trajectory. To convert the lateral velocity of the bubble to a force,we borrow the following formula from a cylinder moving in a fluid boundedby a wall [36]:Fl =4??Vmig[log(g+aR )?ag ], (6.4)where a2 = g2 ? R2. R and g are the bubble size and its distance from thewall. Fig. 6.12 compares this force experienced by a bubble sheared in shearthinning fluid with the one in Newtonian fluid. It appears that increasingof the shear rate does not change this force balance. But, increasing ofthe bubble size tends to empower the wall repulsive force. Consequently,the equilibrium position of the bubble shifts further away from the innercylinder for larger bubbles.Now we turn to the foam that has both shear-thinning and elasticity.We know that elasticity generates a net outward motion of bubble whichopposes the inward motion of the bubble. This force is on the same or-der of magnitude as the shear thinning force but in the opposite direction.Therefore, if we combine these two effects, the net outcome is that the final686.3. Discussion0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9?4?2024 x 10?7s, Dimensionless positionForce (N)  Newtonian FluidR/d = 0.025, ?? = 2.74 s?1R/d = 0.1, ?? = 2.74 s?1R/d = 0.1, ?? = 5.48 s?1Figure 6.12: Calculated force for shear thinning fluid and Newtonian fluid vsdimensionless position on a bubble with in wide-gap Couette geometry. Newtonianforce was evaluated using Chan-Leal formula.equilibrium position should be in the range of equilibrium positions boundedby those for the purely shear-thinning and purely elastic fluids.For migration experiments in foam the final position is between thoseexpected from the two ?rheologically pure? liquids. Although this forcebalance is a simplified version of what happens in real case, it is capable ofrationalizing the experimental observations nonetheless.In our discussion so far, we have tacitly taken the 2D foam of smallerbubbles as an effective continuum, a non-Newtonian fluid exhibiting shear-thinning and normal stress difference. The conclusion of the above discussionis that as far as the bubble migration is concerned, the analogy seems to hold.This amazing fact is reminiscent of the use of the falling ball rheometer formeasuring the viscosity of a suspension, made of particles comparable in sizewith that of the falling ball [4]. Nevertheless, the foam is a heterogeneousmedium and its granularity manifests itself in certain ways. An example ofthis is the so-called nonlocal effect. Figure 6.2 shows the bulk shear rheologyof the foam, measured from the torque on the inner cylinder. Alternativelywe can measure the velocity profile of the foam using PIV [57], with typicalprofiles shown in Fig. 6.13. In the same figure we have plotted the velocity696.4. Conclusion0 0.2 0.4 0.6 0.8 100.20.40.60.81(r?Ri)(Ro?Ri)v/v i  Newtonian Fluid?? = 1.5 sec?1?? = 3.5 sec?1Figure 6.13: Normalized velocity versus normalized radius. Symbols show thelocal measurements by using PIV and curves show the corresponding predictionsusing global measurements.profile computed from the Herschel-Bulkley viscosity of the foam, and thereis a disagreement between the global measurement (of viscosity) and thelocal measurement (of the v(r) profile). Katgert et al. [39] rationalized thisdiscrepancy by a nonlocal effect in flowing foam, with clusters of bubblesmoving cooperatively over a certain correlation length. This serves as areminder of the subtle dynamics of sheared foam and of the limitations ofthe continuum analogy.6.4 ConclusionTo conclude, we studied lateral migration of a bubble in sheared two-dimensionalfoam in a wide-gap Couette geometry. We reported two thresholds for shearrate and bubble size ratio. The final equilibrium position of a migrating bub-ble in wide-gap geometry differs from the narrow gap device. We showedthat for a wide-gap geometry equilibrium positions shifts further towards theinner cylinder compared to a narrow-gap device. This shift was attributedto the non-Newtonian rheology of foam. It appears that the viscoelasticity706.4. Conclusionof foam alters the migration behavior. We then understood the effect ofshear thinning and elasticity on the migration of a bubble by conductingexperiments on xanthan gum and Boger fluids separately. It is of interestthat we could rationalize the migration results in foam by using the migra-tion experiments in those aforementioned fluids. Foam also shows non-localeffects in terms of rheology [39]. In this work we used the result of bulkrheology to justify the migration experiments, how about using local mea-surements? i.e. Can local measurements of rheology be used to understandmigration in foam? This remains an open question that can be addressedin future studies.71Chapter 7Conclusion andrecommendationsThe overarching theme of the research conducted in this thesis is to use theexperiments to explore the structural evolution of two-dimensional foam un-dergoing flow. This is a necessary and significant step in studying the flowof foams in microscopic as well as macroscopic level. Up to now, most exper-iments on foams have focused on the structural changes in static or quasi-static states, and less attention has been given to the changes in structurein dynamical processes. There have been a limited number of experimentswhich investigated the dynamical change in foam structure. Those exper-iments are not in line with each other when they are put in one picture.For example, some experiments show that size based segregation happensin three dimensional foams in which larger bubbles tend to move away fromthe walls, but some other experiments in two-dimensional foams showedthat large bubbles rather to move towards the wall. Therefore, knowledgein this area is limited and has lacked a firm scientific foundation. A welldesigned experiment with a simplified geometry fills a much needed role inthis context.Developing a new shear device with Couette co-axial cylinder geometryhave allowed us to study the dynamics of two-dimensional foam thorough insimple shear flow. This device is coupled with three different cameras thatallowed us to visualize the microstructure of the foam as foam undergoesshear simultaneously. The use of two-dimensional foam is preferred to three-dimensional foam, since the latter is opaque and its microstructure is difficultto visualize.We also used the PIV method to track and compute the velocity profileof foam across the gap and used this as additional tool to rationalize theexperimental observations. In view of the outcome of the four projects, onecan claim that the experiments have succeeded in identifying and clarifyingthe mechanisms underlying structural changes in foam. In addition, we haveuncovered novel physics in sheared two-dimensional foams.In the following, I will first summarize the key findings, and then reflecton their significance and limitations.727.1. Summary of key findings7.1 Summary of key findings7.1.1 Coalescence of bubbles in sheared two-dimensionalmonodisperse foamTo thoroughly understand the mechanisms responsible for structural evolu-tion in three-dimensional foams, we made the problem tractable by simplify-ing the sample to two-dimensional foam. This allowed us to simply visualizethe foam structure at any point during experiments. We performed experi-ments in simple shear flow of two-dimensional foam in fairly high shear ratesand observed appearance of large bubbles shortly after the start. Smallerbubbles have been coalescing to make larger bubbles only if a minimumthreshold is passed. We did not observe breakup of bubbles in this case.Then, we explored the effect of different parameters including bubble size,viscosity of liquid and shear rate on the coalescence process. The minimumthreshold for coalescence of bubbles in foam contradicts the conventionalwisdom on coalescence in which coalescence occurs for gentler collisions.Then, we made an effort to rationalize the experimental observations usingdifferent theories that could possibly explain the results. Though none ofthe theories worked quantitatively, the most promising one was the modelbased on inertia of fluid. This theory can explain the minimum thresholdfor start of coalescence and the dependency of this shear rate to the bubblesize, but overestimate the effect of viscosity.Perhaps the most important finding of this study is that if we sheara two-dimensional foam fast enough, we would observe the coalescence ofbubbles and therefore, formation of new large bubbles in foam. This is thefirst observation of its kind.7.1.2 Cross stream-line migration of a single large bubblein monodisperse foams.One of the potential mechanisms for structural evolution in foam is cross-stream line migration of bubbles away from walls during the flow of foam.This phenomena has been extensively reported in flow of suspensions andemulsions. For foams, on the other hand, there have been relatively fewstudies on structural evolution in the literature, and they tend to contradicteach other. Therefore, it is vital to understand the physics of the problemto resolve the apparent contradictions.In this study we used a simple Couette co-axial cylinder and filled itwith monodisperse foam. Then, we introduced a single bubble whose sizewas different than its surroundings. We reported cross-streamline migrationfor the cases in which bubble size ratio and shear rate were both above somethresholds. For those cases, bubble migrates from any initial position tothe middle of the gap. Then, we modified a model based on perturbation737.1. Summary of key findingstheory to account for the migration in foam. The results of theory andexperiments were in good agreement with each other. In addition to thatwe used a force balance between wall repulsion and capillary attraction toaccount for the presence of thresholds; again this simple force balance was inreasonable agreement with experimental observations. This was a key steptowards understanding the mechanisms responsible for structural evolutionin a confined foam flow.7.1.3 Size-based segregation in sheared two-dimensionalpolydisperse foamFollowing the study of single-bubble migration, the next step for us was tointroduce more than one large bubble into a monodisperse foam and performsimilar experiments as before in Couette co-axial cylinder geometry. Whenwe have several large bubbles in a sea of smaller bubbles, they will interactwith each other resulting in an effective diffusion. This diffusion tends toevenly distribute the large bubbles in the gap while the lateral migrationdoes the opposite. The competition between these two mechanisms lead todistribution of large bubbles across the gap, which is peaked at the middle ofthe gap. There are also regions close to walls where no large bubble can befound. Then, we studied the effects of different parameters such as bubblesize ratio, shear rate, area fraction of large bubbles and initial configura-tion of foam to better understand the process of size-segregation. Here, weagain observed thresholds for migration similar to previous work. Beyondthese thresholds, foam structure evolves to the state where large bubblesare mainly accumulated in the middle of the gap. Initial configuration ofpolydisperse foam does not seem to have any effect on the final equilibriumstate as long as we are above the thresholds. Then, we used a model that ac-counts for both cross stream-line migration and diffusion due to interactionof large bubbles with each other. It turned out that model could predict theexperimental observations to a good degree.7.1.4 Effect of non-Newtonian rheology on bubblemigration in sheared foam.We have done experiments in a narrow gap Couette co-axial cylinder inwhich two-dimensional foam behaved essentially as a Newtonian fluid. There-fore, the non-Newtonian feature of the foam has not been manifested inprevious experiments. One naturally wonders how does the non-Newtonianbehavior of foam impact the structural evolution? To explore this question,we widen the gap of Couette co-axial cylinder to introduce a nonuniformshear rate profile within the gap. Two-dimensional foam in wide gap ex-hibits non-Newtonian behaviors including the shear thinning, yielding andelasticity. For simplicity we have focused on experiments in which foam is747.2. Significance and limitationsfully yielded in the gap. Therefore, the remaining non-Newtonian featuresare shear thinning and elasticity. We made a monodisperse foam and placedit between two cylinders and then introduced one large bubble inside thesea of smaller ones. We again report the lateral migration for shear ratesand bubble size ratio above some thresholds, similar to our previous results.The main difference between wide-gap experimental results and narrow gapis the final equilibrium position that the bubble attains. For narrow gapgeometry this position is at the middle of the gap, while for the wide-gapgeometry it is at the inner half closer to the inner cylinder. We also modifiedthe perturbation theory of Chan-Leal to account for trajectory of bubbles.In this model we included the curvature of stream-lines, shear rate profileand elevated deformation of bubble. It turned out that model fails to explainthe experiments. Therefore, there has to be another factor that leads to thisdeviation. We believe that this is due to the non-Newtonian features of thefoam. To understand the process of migration in a viscoelastic fluid, one hasto study the effect of shear thinning and elasticity separately. For foam thisis impossible. Therefore, we designed some polymer solutions which mimicthe shear thinning behavior and elasticity of foam. In addition to that, weused theoretical as well as simulation results to explain these two effectsseparately. Both experiments and calculations are pointing to the same di-rection that the shear thinning behavior pushes the bubble further towardsthe inner cylinder while the elasticity does the opposite. Therefore, for afoam which is viscoelastic fluid the final equilibrium position has to be inthe middle of values for shear thinning and elastic fluids. This is consistentwith the results of experiments in foam.7.2 Significance and limitationsThe insights gained from this research are potentially useful in two generalways. First, the novel phenomena that we have discovered in foam experi-ments may inspire further, more in-depth research in foam and flow of othermulticomponent fluids. These experiments serve as the first window towardsa better understanding of fundamental dynamics of foam as complex fluids.Second, This study provides potential guidelines for designing bubble struc-tures in engineering processing. This understanding in turn would lead toproducts with better qualities.For example, the final properties of some cosmetic products such asshaving cream is determined by their microstructure. It is shown in thisthesis that the structure of a sheared foam tends to evolve, therefore, thequality of product would change. Hence, one can tune foam?s propertiesby knowing its dynamics through the process of manufacturing. In anotherexample, foam is used in process of enhanced oil recovery. Its efficiencycan be improved by tuning its structure. To do so, one should identify the757.3. Recommendationmechanisms responsible for structural evolution in foam and then try totune them based on the requirements in each application.The limitations of this research can be summarized as follows:1. Two dimensionality. A real liquid foam has a complex 3D structure,but the opacity of three dimensional foam has forced us to study the2D foam. With this simplification we might lose some basics physicswhich emanates from three dimensionality of foam. For instance, abubble is surrounded by more bubbles in 3D foam than in a 2D one.This might change its deformation in the same flow field and therefore,its dynamics.2. Limited range of bubble size. We could not test a very wide range ofbubble size in the experiments with narrow gap Couette device dueto some limitations. Foam structure has to be stable in the absenceof dynamic flow field. In order to increase the stability of foam weneeded to increase the viscosity of soap solution. This would result inbubbles with larger size. Moreover, we are bounded to put at least 10bubbles across the gap of narrow Couette as a rule of thumb to assumethe foam as a continuum fluid. Given the gap size, this gives us anupper bound for the bubble size.3. Limited range of shear rate. For the Couette co-axial geometry we arebound to use a limited range of shear rate due to the presence of somecomplicating forces. If the shear rate is too high, centripetal forcescome into play and contaminate the clear picture of lateral migrationor size segregation in foam. The shear rate is also bounded frombelow by the need to ensure yielding throughout the entire domain.This avoids the unnecessary trouble of dealing with an uncertain yieldsurface in the experiments. Therefore, we had to make sure that wewere operating within the range that the inertia of the fluid and alsothe yielding could not have affected the lateral migration and sizesegregation experiments.4. Different geometry or flow field. Currently, our experimental observa-tions are limited to Couette co-axial cylinder geometry. What aboutsome other geometries such as channel flow or simple shear flow inparallel plate geometry? We can avoid the previous limitation for theshear rate range by investigating a different geometry.7.3 RecommendationWe have taken the fist step to identify the main mechanisms responsible forstructural evolution in polydisperse foam. One can carry out experiments767.3. Recommendationto further detailed understanding of foam dynamics. In the following, I listsome potential works that can be carried out in future.1. Size-based segregation in wide-gap Couette co-axial geometry.Following chapter 5, one can perform experiments on polydispersefoam and investigate the effect of nonuniform shear rate on size-basedsegregation. Then, possibly a model can be developed to describethe distribution of large bubbles among smaller ones similar to resultsreported in ( chapter 5) for narrow gap system.2. Correlation between Rheology and Size-based segregation in Couetteco-axial geometry.Another potential future works on polydisperse foam is to study therheology of two-dimensional polydisperse foam in conjunction with thevisualization techniques. We have studied the size-based segregationin polydisperse foam in chapter 5. Now, one can measure the rheolog-ical properties of two-dimensional polydisperse foam as its structureevolves. Would there be any correlation between size-based segre-gation and rheological properties of foam? And if yes, what is theunderlying physics behind this correlation?3. Complex flow fields and geometries.There are also other factors that can be investigated. For instance,what is the effect of different flow fields on the aforementioned obser-vations in chapter 4 and 5? Since foam processing necessarily involvescomplex flow fields that combine shear and elongation, and are spa-tially heterogeneous, it will be highly valuable to test foam flows insuch complex geometries.This summary makes it clear that there are myriad new problems thatremain to be investigated in flows of foam. It is our hope that the findingsof this thesis will inspire other researchers to contribute to this area of re-search, perhaps by bringing new tools and strategies to the as yet unresolvedquestions.77Bibliography[1] M. Aytouna, J. Paredes, N. Shahidzadeh-Bonn, S. Moulinet, C. Wag-ner, Y. Amarouchene, J. Eggers, , and D. Bonn. Drop formation innon-newtonian fluids. Phys. Rev. Lett., 110:034501, 2013.[2] Anne-Laure Biance, Aline Delbos, and Olivier Pitois. 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