• Mukherjee S, Sarkar K 2010 “Effects of viscoelasticity on the retraction of a sheared drop,” Journal of Non-Newtonian Fluid Mechanics, 165, 340-349.

    Effects of drop and matrix viscoelasticity on the retraction of a sheared drop are numerically investigated.Retraction of an Oldroyd-B drop in a Newtonian matrix is initially faster and later slower with increasingdrop Deborah number. The observed behavior is explained using an ordinary differential equation modelrepresenting the dominant balance between various forces during retraction. The initial faster relaxationof viscoelastic drops is due to viscoelastic stresses pulling the drop interface at the tips inward. The laterslower retraction is due to the slowly-relaxing viscoelastic forces at the equator, where they act againstthe capillary force. The drop inclination decreases substantially during retraction unlike in a Newtoniancase. Matrix viscoelasticity slows the relaxation of a Newtonian drop because of the increasingly slowrelaxation of highly stretched polymers near the drop tip with increasing Deborah number. Increasingthe ratio of polymeric to total viscosity further accentuates the viscoelastic effects in both cases. For anOldroyd-B drop in an Oldroyd-B matrix, a competition between the dispersed and the continuous phaseelasticities, represented by their ratio, determines the dynamics; larger values of the ratio leads again toinitial faster and later slower retraction.

  • Katiyar A, Sarkar K, 2010 “Stability Analysis of an Encapsulated Microbubble against Gas Diffusion,”Journal of Colloid and Interface Science, 343, 42-47.

    Linear stability analysis is performed for a mathematical model of diffusion of gases from an encapsulatedmicrobubble. It is an Epstein–Plesset model modified to account for encapsulation elasticity and finite gaspermeability. Although bubbles, containing gases other than air, are considered, the final stable bubble, ifany, contains only air, and stability is achieved only when the surrounding medium is saturated or over-saturated with air. In absence of encapsulation elasticity, only a neutral stability is achieved for zero sur-face tension, the other solution being unstable. For an elastic encapsulation, different equilibriumsolutions are obtained depending on the saturation level and whether the surface tension is smaller orhigher than the elasticity. For an elastic encapsulation, elasticity can stabilize the bubble. However,imposing a non-negativity condition on the effective surface tension (consisting of reference surface ten-sion and the elastic stress) leads to an equilibrium radius which is only neutrally stable. If the encapsu-lation can support a net compressive stress, it achieves actual stability. The linear stability results areconsistent with our recent numerical findings. Physical mechanisms for the stability or instability of var-ious equilibriums are provided.

  • Paul S, Katiyar A, Sarkar K, Chatterjee D, Shi WT, Forsberg F 2010 “Material Characterization of contrast microbubbles using a nonlinear interfacial elasticity model of the encapsulation,” Journal of the Acoustical Society of America, 127, 3846-3857.

    Two nonlinear interfacial elasticity models—interfacial elasticity decreasing linearly andexponentially with area fraction—are developed for the encapsulation of contrast microbubbles. Thestrain softeningdecreasing elasticityresults from the decreasing association between theconstitutive molecules of the encapsulation. The models are used to find the characteristic propertiessurface tension, interfacial elasticity, interfacial viscosity and nonlinear elasticity parametersfor acommercial contrast agent. Properties are found using the ultrasound attenuation measured througha suspension of contrast agent. Dynamics of the resulting models are simulated, compared withother existing models and discussed. Imposing non-negativity on the effective surface tensiontheencapsulation experiences no net compressive stressshows “compression-only” behavior. Theexponential and the quadraticlinearly varying elasticitymodels result in similar behaviors. Thevalidity of the models is investigated by comparing their predictions of the scattered nonlinearresponse for the contrast agent at higher excitations against experimental measurement. All modelspredict well the scattered fundamental response. The nonlinear strain softening included in theproposed elastic models of the encapsulation improves their ability to predict subharmonic response.They predict the threshold excitation for the initiation of subharmonic response and its subsequentsaturation.

  • Olapade P, Singh R, Sarkar K 2009 “Pair-wise interactions between deformable drops in a shear at finite inertia,” Physics of Fluids, 21, 063302.

    Interactions between a pair of equal-size viscous drops in shear are numerically investigated at finiteReynolds numberRe=0.1–10. At low Reynolds number the simulation compares well with aprevious experimental observation. Apart from the usual pairwise motion where drops driven byshear pass over each othertype I trajectory, finite inertia introduces a new typetype IIofreversed trajectory where drops approaching each other reverse their initial trajectories. The newtrajectory is explained by a reversed streamline pattern observed around a single drop in an imposedshear, and is similar to what is also observed for rigid spheres at finite inertia. However, dropdeformability introduces a nonuniform transition from one to the other type of trajectory—dropsdisplay type I trajectory for high and low capillary numbers and type II for intermediate capillarynumbers. The phenomenon is explained by noting that increasing capillary number gives rise tocompeting effects—while it increases drop deformation and therefore increases resistance to slidingmotion, it also increases drop flexibility, decreases inclination angle, and overall effect of the drop’spresence is reduced, all helping them to slide by. The nonuniform behavior—type II trajectory foran intermediate range of capillary numbers—occurs only at Reynolds number above a critical value.Further increase in Reynolds number increases the range of capillary numbers for type II trajectory.For type I trajectory, terminal cross-stream separation increases linearly with increasing inertiaindicating an enhanced shear induced diffusion. Increasing initial streamwise separation aids inreversedtype IItrajectory due to increased overlap with the reversed streamline zone. Increasingcross-stream distance expectedly facilitatestype Isliding motion. For passing dropstype Itrajectory, terminal cross-stream separation is not appreciably affected by capillary number andinitial drop separation.

  • Mukherjee S, Sarkar K 2009 “Effects of viscosity ratio on deformation of a viscoelastic drop in a Newtonian matrix under steady shear,” Journal of Non-Newtonian Fluid Mechanics, 160, 104-112.

    Deformation of an Oldroyd B drop in a Newtonian matrix under steady shear is simulated using a fronttracking finite difference method for varying viscosity ratio. For drop viscosity lower than that of thematrix, the long-time steady deformation behavior is similar to that of the viscosity matched system—thedrop shows reduced deformation with increasing Deborah number due to the increased inhibiting vis-coelastic normal stress inside the drop. However for higher viscosity ratio systems, the drop response isnon-monotonic—the steady drop deformation first decreases with increasing Deborah number but abovea critical Deborah number, it increases with further increase in Deborah number, reaching higher thanthe viscous case value for some viscosity ratios. We explain the increase in deformation with Deborahnumber by noting that at higher viscosity ratios, strain rate inside the drop is reduced, thereby reducingthe inhibiting viscoelastic stress. Furthermore, similar to the viscosity matched system, the drop inclina-tion angle increases with increasing Deborah number. A drop aligned more with the maximum stretchingaxis at 45 degree of the imposed shear, experiences increased viscous stretching. With increased ratioof polymeric viscosity to total drop viscosity, the drop deformation decreases and the inclination angleincreases. Our simulation results compare favorably with a number of experimental and computationalresults from other researchers.

  • Sarkar K, Katiyar A, Jain P 2009 “Growth and dissolution of an encapsulated contrast microbubble,”Ultrasound in Medicine and Biology, 35, 1385-1396.

    Gas diffusion from an encapsulated microbubble is modeled using an explicit linear relation for gaspermeation through the encapsulation. Both the cases of single gas (air) and multiple gases (perfluorocarbon insidethe bubble and air dissolved in surrounding liquid) are considered. An analytical expression for the dissolutiontime for an encapsulated air bubble is obtained; it showed that for small permeability the dissolution time increaseslinearly with decreasing permeability. A perfluorocarbon-filled contrast microbubble such as DefinityÒwas pre-dicted to experience a transient growth because of air infusion before it dissolves in conformity with previousexperimental findings. The growth phase occurs only for bubbles with a critical value of initial mole fraction ofperfluorocarbon relative to air. With empirically obtained property values, the dissolution time of a 2.5-microndiameter (same as that of Definity), lipid-coated octafluoropropane bubble, with surface tension 25 mN/m, is pre-dicted to be 42 min in an air-saturated medium. The properties such as shell permeability, surface tension andrelative mole fraction of octafluoropropane are varied to investigate their effects on the time scales of bubblegrowth and dissolution, including their asymptotic scalings where appropriate. The dissolution dynamics scaleswith permeability, in that when the time is nondimensioanlized with permeability, curves for different permeabil-ities collapse on a single curve. Investigation of bubbles filled with other gases (nonoctafluoropropane perfluoro-carbon and sulfur hexafluoride) indicates longer dissolution time because of lower solubility and lowerdiffusivity for larger gas molecules. For such micron-size encapsulated bubbles, lifetime of hours is possibleonly at extremely low surface tension (,1 mN/m) or at extreme oversaturation.(E-mail:sarkar@udel.edu)Ó2009 World Federation for Ultrasound in Medicine & Biology.

  • Singh R, Sarkar K 2009 “Effects of viscosity ratio and three dimensional positioning on hydrodynamic interactions between two viscous drops in a shear flow at Finite Inertia,” Physics of Fluids, 21, 103303.

    Drops driven toward each other by shear at finite inertia follow two distinct types of trajectories.Type I trajectory is similar to the one in Stokes flow where drops slide past each other. However, atfinite inertia, drops display a new type II trajectory, where they reverse their paths. Increasingviscosity ratio results in a transition from type II to type I trajectory. The transition is caused bydecreased drop deformation and increased alignment with the flow at higher drop viscosity; bothdecrease the zone of reversed streamlines that accompanies a drop at finite inertia. The transition isdelineated in a phase diagram of Reynolds number and viscosity ratio for different capillarynumbers. The critical viscosity ratio, where a type II transitions into type I, increases with Reynoldsnumber except at higher capillary numbers, where the critical viscosity ratio shows a slightnonmonotonic variation with Reynolds number. Also, it is nonmonotonic with capillary numbers inthat for a fixed Reynolds number, the critical viscosity ratio first increases with increasing capillarynumber and then decreases. Similar to the Stokes regime, increased viscosity ratio leads to adecreased postcollision cross-stream separation effectively decreasing the shear induced diffusion.Higher viscosity ratio results in an increased separation between drops during encounter, whichresults in a smaller interaction time. With drops placed initially at different shear planes, drops comeunder the influence of the reversed flow zone around a single drop that broadens off the central shearplane. Consequently, the trajectory changes from type I to type II as the offset in the vorticitydirection increases. The change depends on the initial offset in the shear direction as well. The finaldisplacement in the shear direction varies linearly with the initial offset. The net relativedisplacement in the shear direction shows a gradual decrease with increasing offset. The net relativedisplacement in the vorticity direction with increasing offset first increases from a zero value whendrops are placed at the same shear plane to a maximum and then decreases. For certain cases, itreaches a negative value.

  • Katiyar A, Sarkar K, Jain P, 2009 “Effects of encapsulation elasticity on the stability of an encapsulated microbubble,” Journal of Colloid and Interface Science, 336, 519-525.

    A model for gas transport from an encapsulated microbubble into the surrounding medium is developedand investigated incorporating the effects of encapsulation elasticity. Encapsulation elasticity stabilizesmicrobubbles against dissolution and explains the long shelf life of microbubble contrast agent. We con-sider air bubbles as well as bubbles containing perfluorocarbon gas. Analytical conditions between satu-ration level, surface tension and interfacial dilatational elasticity are determined for attaining non-zeroequilibrium radius for these microbubbles. Numerical solution of the equation verifies the stability ofthe equilibrium radii. In an undersaturated medium all encapsulated bubbles dissolve. In a saturatedmedium, an encapsulated bubble is found to achieve a long-time stable radius when interfacial dilata-tional elasticity is larger than equilibrium surface tension. For bubbles with interfacial dilatational elas-ticity smaller than the equilibrium surface tension, stable bubble of non-zero radius can be achieved onlywhen the saturation level is greater than a critical value. Even if they initially contain a gas other than air,bubbles that reach a stable radius finally become air bubbles. The model is applied to an octafluoropro-pane filled lipid-coated 2.5lm bubble, which displayed a transient swelling due to air intake beforereaching an equilibrium size. Effects of elasticity, shell permeability, initial mole fraction, initial radiusand saturation level are investigated and discussed. Shell permeability and mole fraction do not affectthe final equilibrium radius of the microbubble but affect the time scale and the transient dynamics. Sim-ilarly, the ratio of equilibrium radius to initial radius remains unaffected by the variation in initial radius.

  • Aggarwal N, Sarkar K 2008 “Effects of matrix viscoelasticity on viscous and viscoelastic drop deformation in a shear flow,” Journal of Fluid Mechanics, 601, 63-84.

    The deformation of a Newtonian/viscoelastic drop suspended in a viscoelastic fluidis investigated using a three-dimensional front-tracking finite-difference method.The viscoelasticity is modelled using the Oldroyd-B constitutive equation. Matrixviscoelasticity affects the drop deformation and the inclination angle with the flowdirection. Numerical predictions of these quantities are compared with previousexperimental measurements using Boger fluids. The elastic and viscous stresses atthe interface, polymer orientation, and the elastic and viscous forces in the domainare carefully investigated as they affect the drop response. Significant change in thedrop inclination with increasing viscoelasticity is observed; this is explained in termsof the first normal stress difference. A non-monotonic change – a decrease followedby an increase – in the steady-state drop deformation is observed with increasingDeborah number (De) and explained in terms of the competition between increasedlocalized polymer stretching at the drop tips and the decreased viscous stretchingdue to change in drop orientation angle. The transient drop orientation angle isfound to evolve on the polymer relaxation time scale for highDe. The breakup ofa viscous drop in a viscoelastic matrix is inhibited for smallDe, and promoted athigherDe. Polymeric to total viscosity ratioβwas seen to affect the result throughthe combined parameterβDeindicating a dominant role of the first normal stressdifference. A viscoelastic drop in a viscoelastic matrix with matched relaxation timeexperiences less deformation compared to the case when one of the phases is viscous;but the inclination angle assumes an intermediate value between two extreme cases.Increased drop phase viscoelasticity compared to matrix phase leads to decreaseddeformation.

  • Li X, Sarkar K 2008 “Front-tracking simulation of a liquid capsule enclosed by an elastic membrane,”Journal of Computational Physics, 227, 4998-5018.

    The dynamics of a liquid capsule enclosed by an elastic membrane in a shear flow is investigated using a front trackingfinite difference method. We compute deformation, orientation and tank-treading of the capsule, as functions of the forcing(capillary number) and the viscosity ratio for two different membrane constitutive equations – Neo-Hookean and Skalak.The computed results compare very well with those obtained by high-order boundary element methods as well as the smalldeformation perturbation analysis. The simulation shows that a drop and a capsule, even under those circumstances thatresult in the same Taylor deformation criterion for both, attain very different shapes. The tank-treading period even fordifferent capillary numbers as well as capsules with different constitutive laws, is primarily determined by the deformationand the viscosity ratio. At low capillary numbers the simulation predicts buckling due to large compressive stresses on themembrane. However, we show that in shear, unlike in extension, the tank-treading motion can inhibit the buckling insta-bility and gives rise to a stable evolution even in presence of membrane compressive stresses. At large capillary numbers thecapsule experiences large bounded shape followed by tip buckling indicating possible membrane breakup.Ó2008 Elsevier Inc. All rights reserved.

  • Aggarwal, N, Sarkar K 2008 “Rheology of an emulsion of viscoelastic drops in steady shear,” Journal of Non-Newtonian Fluid Mechanics, 150, 19-31.

    Steady shear rheology of a dilute emulsion with viscoelastic inclusions is numerically investigated using direct numerical simulations. Batchelor’sformulation for rheology of a viscous emulsion is extended for a viscoelastic system. Viscoelasticity is modeled using the Oldroyd-B constitutiveequation. A front-tracking finite difference code is used to numerically determine the drop shape, and solve for the velocity and stress fields. Theeffective stress of the viscoelastic emulsion has three different components due to interfacial tension, viscosity difference (not considered here) andthe drop phase viscoelasticity. The interfacial contributions – first and second normal stress differences and shear stresses – vary with Capillarynumber in a manner similar to those of a Newtonian system. However the shear viscosity decreases with viscoelasticity at low Capillary numbers,and increases at high Capillary numbers. The first normal stress difference due to interfacial contribution decreases with increasing drop phaseviscoelasticity. The first normal stress difference due to the drop phase viscoelasticity is found to have a complex dependence on Capillary andDeborah numbers, in contrast with the linear mixing rule. Drop phase viscoelasticity does not contribute significantly to effective shear viscosityof the emulsion. The total first normal stress difference shows an increase with drop phase viscoelasticity at high Capillary numbers. However atlow Capillary numbers, a non-monotonic behavior is observed. The results are explained by examining the stress field and the drop shape.© 2007 Elsevier B.V. All rights reserved.