Emulsion
Mixtures of immiscible liquids display a wide spectrum of behaviors, and thereby offer a means of achieving tunable material properties. Often they consist of small fraction of a specialized additive in a less expensive bulk liquid. The liquids phase separate into an emulsion containing discrete droplets of various sizes dispersed in a continuous phase. In industrial processing, the flow continually deforms the suspended drops leading to their coalescence and breakup. The evolving microstructure also results in stresses modifying the flow and the finished product. We investigate the dynamics of microstructure and its effects on the overall response (rheology) of the emulsion through direct numerical simulation and analytical techniques at finite Reynolds number. Tol date, research on drop deformation and rheology has mostly been restricted to inertia-less flows and small deformation. We use Front-tracking method to compute deformation of arbitrary magnitude at finite inertia. Stresses are computed from the computed microstructure. The relation between excess stress and imposed strain rate are investigated varying interfacial tension, inertia and frequency. For steady shear, shear thickening and change of sign of normal stress differences are observed with increased inertia. For oscillating extensional flows, the stress-strain relation is a function of the phase between the drop deformation and the imposed flow. At low Reynolds number, the simulation recovers the linear oscillatory rheology (loss and storage moduli) of Oldroyd and Bousmina. At low surface tension, stress is predominantly elastic, while at high surface tension it is viscous. Increased drop inertia leads to resonance and complex phase in deformation. The resulting excess interfacial stress displays a non-monotonic variation with frequency and obtains a negative elastic modulus at low frequency.
Related Publication
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Sarkar K, Singh R 2013 “Spatial ordering due to hydrodynamic interactions between a pair of colliding drops in a confined shear,” Physics of Fluids, 25, 051702.
Pair-collision between viscous drops in a confined shear is simulated to show that the confinement alters the trajectories of the drops spatially ordering them at a finite separation in the center of the domain. In contrast to free shear where drops eventually adopt free streamlines with a finite cross-stream separation, here they move towards the centerline achieving zero cross-stream separation but a net stream-wise separation. The latter varies as inverse of capillary number and cube of the confinement (distance between the walls). The final stream-wise separation does not depend on the initial positions of the drops when the drops are in the same shear plane. The separation decreases approximately linearly with the initial separation in the vorticity direction. An analytical theory explaining the phenomenon is presented. Effects of the ratio of drop to matrix viscosity are briefly investigated
-
Singh R, Sarkar K 2015 “Hydrodynamic interactions between pairs of capsules and drops in a simple shear: effects of viscosity ratio and heterogeneous collision,” Physical Review E, 92, 063029.
Hydrodynamic interactions between a pair of capsules in simple shear are numerically investigated using afront-tracking finite difference method. The membrane of the capsule is modeled using different hyperelasticconstitutive relations. We also compare the pair interactions between drops to those between capsules. Anincreased viscosity ratio leads to a reduced net cross-stream separation between capsules as well as drops aftercollision. At low viscosity ratios, for the same capillary number drop-pairs show higher cross-stream separationthan those for capsule-pairs, while substantially large viscosity ratios result in almost the same value for bothcases. We investigate pair-collisions between two heterogeneous capsules C1and C2with two different capillarynumbers. The maximum deformation of C1was seen to increase with increasing stiffness (decreasing capillarynumber) of C2, even though the stiffness of C1was kept fixed. The findings are similar for a drop-pair, however,with a smaller maximum deformation for the same combinations of capillary numbers. The final cross-streamdrift of the trajectory of C1decreases with the increasing stiffness of C2, but the relative trajectory betweenthe capsules remains unchanged. The maximum deformation and the cross-stream drift of the trajectory of C1are shown to approximately vary with power-law functions of the ratio of the capillary numbers of C1andC2. An analytical explanation of the dependence on the two capillary numbers is offered. Different membraneconstitutive laws result in similar deformation and drift in trajectory.
-
Singh R, Sarkar K 2015 “Hydrodynamic interactions between pairs of capsules and drops in a simple shear: effects of viscosity ratio and heterogeneous collision,” Physical Review E, 92, 063029.
Hydrodynamic interactions between a pair of capsules in simple shear are numerically investigated using afront-tracking finite difference method. The membrane of the capsule is modeled using different hyperelasticconstitutive relations. We also compare the pair interactions between drops to those between capsules. Anincreased viscosity ratio leads to a reduced net cross-stream separation between capsules as well as drops aftercollision. At low viscosity ratios, for the same capillary number drop-pairs show higher cross-stream separationthan those for capsule-pairs, while substantially large viscosity ratios result in almost the same value for bothcases. We investigate pair-collisions between two heterogeneous capsules C1and C2with two different capillarynumbers. The maximum deformation of C1was seen to increase with increasing stiffness (decreasing capillarynumber) of C2, even though the stiffness of C1was kept fixed. The findings are similar for a drop-pair, however,with a smaller maximum deformation for the same combinations of capillary numbers. The final cross-streamdrift of the trajectory of C1decreases with the increasing stiffness of C2, but the relative trajectory betweenthe capsules remains unchanged. The maximum deformation and the cross-stream drift of the trajectory of C1are shown to approximately vary with power-law functions of the ratio of the capillary numbers of C1andC2. An analytical explanation of the dependence on the two capillary numbers is offered. Different membraneconstitutive laws result in similar deformation and drift in trajectory.
-
Singh R, Sarkar K 2015 “Hydrodynamic interactions between pairs of capsules and drops in a simple shear: effects of viscosity ratio and heterogeneous collision,” Physical Review E, 92, 063029.
Hydrodynamic interactions between a pair of capsules in simple shear are numerically investigated using afront-tracking finite difference method. The membrane of the capsule is modeled using different hyperelasticconstitutive relations. We also compare the pair interactions between drops to those between capsules. Anincreased viscosity ratio leads to a reduced net cross-stream separation between capsules as well as drops aftercollision. At low viscosity ratios, for the same capillary number drop-pairs show higher cross-stream separationthan those for capsule-pairs, while substantially large viscosity ratios result in almost the same value for bothcases. We investigate pair-collisions between two heterogeneous capsules C1and C2with two different capillarynumbers. The maximum deformation of C1was seen to increase with increasing stiffness (decreasing capillarynumber) of C2, even though the stiffness of C1was kept fixed. The findings are similar for a drop-pair, however,with a smaller maximum deformation for the same combinations of capillary numbers. The final cross-streamdrift of the trajectory of C1decreases with the increasing stiffness of C2, but the relative trajectory betweenthe capsules remains unchanged. The maximum deformation and the cross-stream drift of the trajectory of C1are shown to approximately vary with power-law functions of the ratio of the capillary numbers of C1andC2. An analytical explanation of the dependence on the two capillary numbers is offered. Different membraneconstitutive laws result in similar deformation and drift in trajectory.
-
Singh R, Li X, Sarkar K 2014 “Lateral migration of an elastic capsule in a wall-bounded shear,” Journal of Fluid Mechanics, 739, 421-443.
The migration of a capsule enclosed by an elastic membrane in a wall-bounded linearshear is investigated using a front-tracking method. A detailed comparison with themigration of a viscous drop is presented varying the capillary number (in the caseof a capsule, the elastic capillary number) and the viscosity ratio. In both cases,the deformation breaks the flow reversal symmetry and makes them migrate awayfrom the wall. They quickly go through a transient evolution to eventually reach aquasi-steady state where the dynamics becomes independent of the initial positionand only depends on the wall distance. Previous analytical theories predicted thatfor a viscous drop, in the quasi-steady state, the migration and slip velocities scaleapproximately with the square of the inverse of the drop–wall separation, whereasthe drop deformation scales as the inverse cube of the separation. These power lawrelations are shown to hold for a capsule as well. The deformation and inclinationangle of the capsule and the drop at the same wall separation show a crossoverin their variation with the capillary number: the capsule shows a steeper variationthan that of the drop for smaller capillary numbers and slower variation than thedrop for larger capillary numbers. Using the Green’s function of Stokes flow, asemi-analytic theory is presented to show that the far-field stresslet that causes themigration has two distinct contributions from the interfacial stresses and the viscosityratio, with competing effects between the two defining the dynamics. It predicts thescaling of the migration velocity with the capsule–wall separation, however, matchingwith the simulated result very well only away from the wall. A phenomenologicalcorrelation for the migration velocity as a function of elastic capillary number, walldistance and viscosity ratio is developed using the simulation results. The effects ofdifferent membrane hyperelastic constitutive equations – neo-Hookean, Evans–Skalak,and Skalak – are briefly investigated to show that the behaviour remains similar fordifferent equations.
-
Mukherjee S, Sarkar K 2014 “Lateral migration of a viscoelastic drop in a Newtonian fluid in a shear flow near a wall,” Physics of Fluids, 26, 103102.
Wall induced lateral migration of a viscoelastic (FENE-MCR) drop in a Newtonianfluid is investigated. Just like a Newtonian drop, a viscoelastic drop reaches a quasi-steady state where the lateral velocity only depends on the instantaneous distancefrom the wall. The drop migration velocity and the deformation scale inversely withthe square and the cube of the distance from the wall, respectively. The migration ve-locity varies non-monotonically with increasing viscoelasticity (increasing Deborahnumber); initially increasing and then decreasing. An analytical explanation has beengiven of the effects by computing the migration velocity as arising from an imagestresslet field due to the drop. The semi-analytical expression matches well with thesimulated migration velocity away from the wall. It contains a viscoelastic stressletcomponent apart from those arising from interfacial tension and viscosity ratio. Themigration dynamics is a result of the competition between the viscous (interfacialtension and viscosity ratio) and the viscoelastic effects. The viscoelastic stressletcontribution towards the migration velocity steadily increases. But the interfacialstresslet—arising purely from the drop shape—first increases and then decreases withrising Deborah number causing the migration velocity to be non-monotonic. The ge-ometric effect of the interfacial stresslet is caused by a corresponding nonmonotonicvariation of the drop inclination. High viscosity ratio is briefly considered to showthat the drop viscoelasticity could stabilize a drop against breakup, and the increase inmigration velocity due to viscoelasticity is larger compared to the viscosity-matchedcase.
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Aliabouzar M, Kumar KN, Sarkar K, 2018 “Acoustic vaporization threshold of lipid coated perfluoropentane droplets,” Journal of the Acoustical Society of America, 143, 2001-2012.
Phase shift droplets vaporizable by acoustic stimulation offer the advantages of producing micro-bubbles as contrast agentsin situas well as higher stability and the possibility of achieving smallersizes. Here, the acoustic droplet vaporization (ADV) threshold of a suspension of droplets with aperfluoropentane (PFP) core (diameter 400–3000 nm) is acoustically measured as a function of theexcitation frequency in a tubeless setup at room temperature. The changes in scattered responses—fundamental, sub-, and second harmonic—are investigated, a quantitative criterion is used to deter-mine the ADV phenomenon, and findings are discussed. The average threshold obtained using threedifferent scattered components increases with frequency—1.0560.28 MPa at 2.25 MHz,1.8960.57 MPa at 5 MHz, and 2.3460.014 MPa at 10 MHz. The scattered response from vapor-ized droplets was also found to qualitatively match with that from an independently prepared lipid-coated microbubble suspension in magnitude as well as trends above the determined ADV thresh-old value.
-
Aliabouzar M, Kumar KN, Sarkar K, 2018 “Acoustic vaporization threshold of lipid coated perfluoropentane droplets,” Journal of the Acoustical Society of America, 143, 2001-2012.
Phase shift droplets vaporizable by acoustic stimulation offer the advantages of producing micro-bubbles as contrast agentsin situas well as higher stability and the possibility of achieving smallersizes. Here, the acoustic droplet vaporization (ADV) threshold of a suspension of droplets with aperfluoropentane (PFP) core (diameter 400–3000 nm) is acoustically measured as a function of theexcitation frequency in a tubeless setup at room temperature. The changes in scattered responses—fundamental, sub-, and second harmonic—are investigated, a quantitative criterion is used to deter-mine the ADV phenomenon, and findings are discussed. The average threshold obtained using threedifferent scattered components increases with frequency—1.0560.28 MPa at 2.25 MHz,1.8960.57 MPa at 5 MHz, and 2.3460.014 MPa at 10 MHz. The scattered response from vapor-ized droplets was also found to qualitatively match with that from an independently prepared lipid-coated microbubble suspension in magnitude as well as trends above the determined ADV thresh-old value.
-
Singh R, Li X, Sarkar K 2014 “Lateral migration of an elastic capsule in a wall-bounded shear,” Journal of Fluid Mechanics, 739, 421-443.
The migration of a capsule enclosed by an elastic membrane in a wall-bounded linearshear is investigated using a front-tracking method. A detailed comparison with themigration of a viscous drop is presented varying the capillary number (in the caseof a capsule, the elastic capillary number) and the viscosity ratio. In both cases,the deformation breaks the flow reversal symmetry and makes them migrate awayfrom the wall. They quickly go through a transient evolution to eventually reach aquasi-steady state where the dynamics becomes independent of the initial positionand only depends on the wall distance. Previous analytical theories predicted thatfor a viscous drop, in the quasi-steady state, the migration and slip velocities scaleapproximately with the square of the inverse of the drop–wall separation, whereasthe drop deformation scales as the inverse cube of the separation. These power lawrelations are shown to hold for a capsule as well. The deformation and inclinationangle of the capsule and the drop at the same wall separation show a crossoverin their variation with the capillary number: the capsule shows a steeper variationthan that of the drop for smaller capillary numbers and slower variation than thedrop for larger capillary numbers. Using the Green’s function of Stokes flow, asemi-analytic theory is presented to show that the far-field stresslet that causes themigration has two distinct contributions from the interfacial stresses and the viscosityratio, with competing effects between the two defining the dynamics. It predicts thescaling of the migration velocity with the capsule–wall separation, however, matchingwith the simulated result very well only away from the wall. A phenomenologicalcorrelation for the migration velocity as a function of elastic capillary number, walldistance and viscosity ratio is developed using the simulation results. The effects ofdifferent membrane hyperelastic constitutive equations – neo-Hookean, Evans–Skalak,and Skalak – are briefly investigated to show that the behaviour remains similar fordifferent equations.
-
Sarkar K, Singh R 2013 “Spatial ordering due to hydrodynamic interactions between a pair of colliding drops in a confined shear,” Physics of Fluids, 25, 051702.
Pair-collision between viscous drops in a confined shear is simulated to show that the confinement alters the trajectories of the drops spatially ordering them at a finite separation in the center of the domain. In contrast to free shear where drops eventually adopt free streamlines with a finite cross-stream separation, here they move towards the centerline achieving zero cross-stream separation but a net stream-wise separation. The latter varies as inverse of capillary number and cube of the confinement (distance between the walls). The final stream-wise separation does not depend on the initial positions of the drops when the drops are in the same shear plane. The separation decreases approximately linearly with the initial separation in the vorticity direction. An analytical theory explaining the phenomenon is presented. Effects of the ratio of drop to matrix viscosity are briefly investigated
-
Sarkar K, Singh R 2013 “Spatial ordering due to hydrodynamic interactions between a pair of colliding drops in a confined shear,” Physics of Fluids, 25, 051702.
Pair-collision between viscous drops in a confined shear is simulated to show that the confinement alters the trajectories of the drops spatially ordering them at a finite separation in the center of the domain. In contrast to free shear where drops eventually adopt free streamlines with a finite cross-stream separation, here they move towards the centerline achieving zero cross-stream separation but a net stream-wise separation. The latter varies as inverse of capillary number and cube of the confinement (distance between the walls). The final stream-wise separation does not depend on the initial positions of the drops when the drops are in the same shear plane. The separation decreases approximately linearly with the initial separation in the vorticity direction. An analytical theory explaining the phenomenon is presented. Effects of the ratio of drop to matrix viscosity are briefly investigated
-
Singh R, Li X, Sarkar K 2014 “Lateral migration of an elastic capsule in a wall-bounded shear,” Journal of Fluid Mechanics, 739, 421-443.
The migration of a capsule enclosed by an elastic membrane in a wall-bounded linearshear is investigated using a front-tracking method. A detailed comparison with themigration of a viscous drop is presented varying the capillary number (in the caseof a capsule, the elastic capillary number) and the viscosity ratio. In both cases,the deformation breaks the flow reversal symmetry and makes them migrate awayfrom the wall. They quickly go through a transient evolution to eventually reach aquasi-steady state where the dynamics becomes independent of the initial positionand only depends on the wall distance. Previous analytical theories predicted thatfor a viscous drop, in the quasi-steady state, the migration and slip velocities scaleapproximately with the square of the inverse of the drop–wall separation, whereasthe drop deformation scales as the inverse cube of the separation. These power lawrelations are shown to hold for a capsule as well. The deformation and inclinationangle of the capsule and the drop at the same wall separation show a crossoverin their variation with the capillary number: the capsule shows a steeper variationthan that of the drop for smaller capillary numbers and slower variation than thedrop for larger capillary numbers. Using the Green’s function of Stokes flow, asemi-analytic theory is presented to show that the far-field stresslet that causes themigration has two distinct contributions from the interfacial stresses and the viscosityratio, with competing effects between the two defining the dynamics. It predicts thescaling of the migration velocity with the capsule–wall separation, however, matchingwith the simulated result very well only away from the wall. A phenomenologicalcorrelation for the migration velocity as a function of elastic capillary number, walldistance and viscosity ratio is developed using the simulation results. The effects ofdifferent membrane hyperelastic constitutive equations – neo-Hookean, Evans–Skalak,and Skalak – are briefly investigated to show that the behaviour remains similar fordifferent equations.
-
Sarkar K, Singh R 2013 “Spatial ordering due to hydrodynamic interactions between a pair of colliding drops in a confined shear,” Physics of Fluids, 25, 051702.
Pair-collision between viscous drops in a confined shear is simulated to show that the confinement alters the trajectories of the drops spatially ordering them at a finite separation in the center of the domain. In contrast to free shear where drops eventually adopt free streamlines with a finite cross-stream separation, here they move towards the centerline achieving zero cross-stream separation but a net stream-wise separation. The latter varies as inverse of capillary number and cube of the confinement (distance between the walls). The final stream-wise separation does not depend on the initial positions of the drops when the drops are in the same shear plane. The separation decreases approximately linearly with the initial separation in the vorticity direction. An analytical theory explaining the phenomenon is presented. Effects of the ratio of drop to matrix viscosity are briefly investigated
-
Sarkar K, Singh R 2013 “Spatial ordering due to hydrodynamic interactions between a pair of colliding drops in a confined shear,” Physics of Fluids, 25, 051702.
Pair-collision between viscous drops in a confined shear is simulated to show that the confinement alters the trajectories of the drops spatially ordering them at a finite separation in the center of the domain. In contrast to free shear where drops eventually adopt free streamlines with a finite cross-stream separation, here they move towards the centerline achieving zero cross-stream separation but a net stream-wise separation. The latter varies as inverse of capillary number and cube of the confinement (distance between the walls). The final stream-wise separation does not depend on the initial positions of the drops when the drops are in the same shear plane. The separation decreases approximately linearly with the initial separation in the vorticity direction. An analytical theory explaining the phenomenon is presented. Effects of the ratio of drop to matrix viscosity are briefly investigated
-
Singh R, Li X, Sarkar K 2014 “Lateral migration of an elastic capsule in a wall-bounded shear,” Journal of Fluid Mechanics, 739, 421-443.
The migration of a capsule enclosed by an elastic membrane in a wall-bounded linearshear is investigated using a front-tracking method. A detailed comparison with themigration of a viscous drop is presented varying the capillary number (in the caseof a capsule, the elastic capillary number) and the viscosity ratio. In both cases,the deformation breaks the flow reversal symmetry and makes them migrate awayfrom the wall. They quickly go through a transient evolution to eventually reach aquasi-steady state where the dynamics becomes independent of the initial positionand only depends on the wall distance. Previous analytical theories predicted thatfor a viscous drop, in the quasi-steady state, the migration and slip velocities scaleapproximately with the square of the inverse of the drop–wall separation, whereasthe drop deformation scales as the inverse cube of the separation. These power lawrelations are shown to hold for a capsule as well. The deformation and inclinationangle of the capsule and the drop at the same wall separation show a crossoverin their variation with the capillary number: the capsule shows a steeper variationthan that of the drop for smaller capillary numbers and slower variation than thedrop for larger capillary numbers. Using the Green’s function of Stokes flow, asemi-analytic theory is presented to show that the far-field stresslet that causes themigration has two distinct contributions from the interfacial stresses and the viscosityratio, with competing effects between the two defining the dynamics. It predicts thescaling of the migration velocity with the capsule–wall separation, however, matchingwith the simulated result very well only away from the wall. A phenomenologicalcorrelation for the migration velocity as a function of elastic capillary number, walldistance and viscosity ratio is developed using the simulation results. The effects ofdifferent membrane hyperelastic constitutive equations – neo-Hookean, Evans–Skalak,and Skalak – are briefly investigated to show that the behaviour remains similar fordifferent equations.
-
Singh R, Sarkar K 2015 “Hydrodynamic interactions between pairs of capsules and drops in a simple shear: effects of viscosity ratio and heterogeneous collision,” Physical Review E, 92, 063029.
Hydrodynamic interactions between a pair of capsules in simple shear are numerically investigated using afront-tracking finite difference method. The membrane of the capsule is modeled using different hyperelasticconstitutive relations. We also compare the pair interactions between drops to those between capsules. Anincreased viscosity ratio leads to a reduced net cross-stream separation between capsules as well as drops aftercollision. At low viscosity ratios, for the same capillary number drop-pairs show higher cross-stream separationthan those for capsule-pairs, while substantially large viscosity ratios result in almost the same value for bothcases. We investigate pair-collisions between two heterogeneous capsules C1and C2with two different capillarynumbers. The maximum deformation of C1was seen to increase with increasing stiffness (decreasing capillarynumber) of C2, even though the stiffness of C1was kept fixed. The findings are similar for a drop-pair, however,with a smaller maximum deformation for the same combinations of capillary numbers. The final cross-streamdrift of the trajectory of C1decreases with the increasing stiffness of C2, but the relative trajectory betweenthe capsules remains unchanged. The maximum deformation and the cross-stream drift of the trajectory of C1are shown to approximately vary with power-law functions of the ratio of the capillary numbers of C1andC2. An analytical explanation of the dependence on the two capillary numbers is offered. Different membraneconstitutive laws result in similar deformation and drift in trajectory.
-
Mukherjee S, Sarkar K 2014 “Lateral migration of a viscoelastic drop in a Newtonian fluid in a shear flow near a wall,” Physics of Fluids, 26, 103102.
Wall induced lateral migration of a viscoelastic (FENE-MCR) drop in a Newtonianfluid is investigated. Just like a Newtonian drop, a viscoelastic drop reaches a quasi-steady state where the lateral velocity only depends on the instantaneous distancefrom the wall. The drop migration velocity and the deformation scale inversely withthe square and the cube of the distance from the wall, respectively. The migration ve-locity varies non-monotonically with increasing viscoelasticity (increasing Deborahnumber); initially increasing and then decreasing. An analytical explanation has beengiven of the effects by computing the migration velocity as arising from an imagestresslet field due to the drop. The semi-analytical expression matches well with thesimulated migration velocity away from the wall. It contains a viscoelastic stressletcomponent apart from those arising from interfacial tension and viscosity ratio. Themigration dynamics is a result of the competition between the viscous (interfacialtension and viscosity ratio) and the viscoelastic effects. The viscoelastic stressletcontribution towards the migration velocity steadily increases. But the interfacialstresslet—arising purely from the drop shape—first increases and then decreases withrising Deborah number causing the migration velocity to be non-monotonic. The ge-ometric effect of the interfacial stresslet is caused by a corresponding nonmonotonicvariation of the drop inclination. High viscosity ratio is briefly considered to showthat the drop viscoelasticity could stabilize a drop against breakup, and the increase inmigration velocity due to viscoelasticity is larger compared to the viscosity-matchedcase.
-
Aliabouzar M, Kumar KN, Sarkar K, 2018 “Acoustic vaporization threshold of lipid coated perfluoropentane droplets,” Journal of the Acoustical Society of America, 143, 2001-2012.
Phase shift droplets vaporizable by acoustic stimulation offer the advantages of producing micro-bubbles as contrast agentsin situas well as higher stability and the possibility of achieving smallersizes. Here, the acoustic droplet vaporization (ADV) threshold of a suspension of droplets with aperfluoropentane (PFP) core (diameter 400–3000 nm) is acoustically measured as a function of theexcitation frequency in a tubeless setup at room temperature. The changes in scattered responses—fundamental, sub-, and second harmonic—are investigated, a quantitative criterion is used to deter-mine the ADV phenomenon, and findings are discussed. The average threshold obtained using threedifferent scattered components increases with frequency—1.0560.28 MPa at 2.25 MHz,1.8960.57 MPa at 5 MHz, and 2.3460.014 MPa at 10 MHz. The scattered response from vapor-ized droplets was also found to qualitatively match with that from an independently prepared lipid-coated microbubble suspension in magnitude as well as trends above the determined ADV thresh-old value.
-
Aliabouzar M, Kumar KN, Sarkar K, 2018 “Acoustic vaporization threshold of lipid coated perfluoropentane droplets,” Journal of the Acoustical Society of America, 143, 2001-2012.
Phase shift droplets vaporizable by acoustic stimulation offer the advantages of producing micro-bubbles as contrast agentsin situas well as higher stability and the possibility of achieving smallersizes. Here, the acoustic droplet vaporization (ADV) threshold of a suspension of droplets with aperfluoropentane (PFP) core (diameter 400–3000 nm) is acoustically measured as a function of theexcitation frequency in a tubeless setup at room temperature. The changes in scattered responses—fundamental, sub-, and second harmonic—are investigated, a quantitative criterion is used to deter-mine the ADV phenomenon, and findings are discussed. The average threshold obtained using threedifferent scattered components increases with frequency—1.0560.28 MPa at 2.25 MHz,1.8960.57 MPa at 5 MHz, and 2.3460.014 MPa at 10 MHz. The scattered response from vapor-ized droplets was also found to qualitatively match with that from an independently prepared lipid-coated microbubble suspension in magnitude as well as trends above the determined ADV thresh-old value.
-
Singh R, Sarkar K 2015 “Hydrodynamic interactions between pairs of capsules and drops in a simple shear: effects of viscosity ratio and heterogeneous collision,” Physical Review E, 92, 063029.
Hydrodynamic interactions between a pair of capsules in simple shear are numerically investigated using afront-tracking finite difference method. The membrane of the capsule is modeled using different hyperelasticconstitutive relations. We also compare the pair interactions between drops to those between capsules. Anincreased viscosity ratio leads to a reduced net cross-stream separation between capsules as well as drops aftercollision. At low viscosity ratios, for the same capillary number drop-pairs show higher cross-stream separationthan those for capsule-pairs, while substantially large viscosity ratios result in almost the same value for bothcases. We investigate pair-collisions between two heterogeneous capsules C1and C2with two different capillarynumbers. The maximum deformation of C1was seen to increase with increasing stiffness (decreasing capillarynumber) of C2, even though the stiffness of C1was kept fixed. The findings are similar for a drop-pair, however,with a smaller maximum deformation for the same combinations of capillary numbers. The final cross-streamdrift of the trajectory of C1decreases with the increasing stiffness of C2, but the relative trajectory betweenthe capsules remains unchanged. The maximum deformation and the cross-stream drift of the trajectory of C1are shown to approximately vary with power-law functions of the ratio of the capillary numbers of C1andC2. An analytical explanation of the dependence on the two capillary numbers is offered. Different membraneconstitutive laws result in similar deformation and drift in trajectory.
-
Mukherjee S, Sarkar K 2014 “Lateral migration of a viscoelastic drop in a Newtonian fluid in a shear flow near a wall,” Physics of Fluids, 26, 103102.
Wall induced lateral migration of a viscoelastic (FENE-MCR) drop in a Newtonianfluid is investigated. Just like a Newtonian drop, a viscoelastic drop reaches a quasi-steady state where the lateral velocity only depends on the instantaneous distancefrom the wall. The drop migration velocity and the deformation scale inversely withthe square and the cube of the distance from the wall, respectively. The migration ve-locity varies non-monotonically with increasing viscoelasticity (increasing Deborahnumber); initially increasing and then decreasing. An analytical explanation has beengiven of the effects by computing the migration velocity as arising from an imagestresslet field due to the drop. The semi-analytical expression matches well with thesimulated migration velocity away from the wall. It contains a viscoelastic stressletcomponent apart from those arising from interfacial tension and viscosity ratio. Themigration dynamics is a result of the competition between the viscous (interfacialtension and viscosity ratio) and the viscoelastic effects. The viscoelastic stressletcontribution towards the migration velocity steadily increases. But the interfacialstresslet—arising purely from the drop shape—first increases and then decreases withrising Deborah number causing the migration velocity to be non-monotonic. The ge-ometric effect of the interfacial stresslet is caused by a corresponding nonmonotonicvariation of the drop inclination. High viscosity ratio is briefly considered to showthat the drop viscoelasticity could stabilize a drop against breakup, and the increase inmigration velocity due to viscoelasticity is larger compared to the viscosity-matchedcase.
-
Singh R, Li X, Sarkar K 2014 “Lateral migration of an elastic capsule in a wall-bounded shear,” Journal of Fluid Mechanics, 739, 421-443.
The migration of a capsule enclosed by an elastic membrane in a wall-bounded linearshear is investigated using a front-tracking method. A detailed comparison with themigration of a viscous drop is presented varying the capillary number (in the caseof a capsule, the elastic capillary number) and the viscosity ratio. In both cases,the deformation breaks the flow reversal symmetry and makes them migrate awayfrom the wall. They quickly go through a transient evolution to eventually reach aquasi-steady state where the dynamics becomes independent of the initial positionand only depends on the wall distance. Previous analytical theories predicted thatfor a viscous drop, in the quasi-steady state, the migration and slip velocities scaleapproximately with the square of the inverse of the drop–wall separation, whereasthe drop deformation scales as the inverse cube of the separation. These power lawrelations are shown to hold for a capsule as well. The deformation and inclinationangle of the capsule and the drop at the same wall separation show a crossoverin their variation with the capillary number: the capsule shows a steeper variationthan that of the drop for smaller capillary numbers and slower variation than thedrop for larger capillary numbers. Using the Green’s function of Stokes flow, asemi-analytic theory is presented to show that the far-field stresslet that causes themigration has two distinct contributions from the interfacial stresses and the viscosityratio, with competing effects between the two defining the dynamics. It predicts thescaling of the migration velocity with the capsule–wall separation, however, matchingwith the simulated result very well only away from the wall. A phenomenologicalcorrelation for the migration velocity as a function of elastic capillary number, walldistance and viscosity ratio is developed using the simulation results. The effects ofdifferent membrane hyperelastic constitutive equations – neo-Hookean, Evans–Skalak,and Skalak – are briefly investigated to show that the behaviour remains similar fordifferent equations.
-
Sarkar K, Singh R 2013 “Spatial ordering due to hydrodynamic interactions between a pair of colliding drops in a confined shear,” Physics of Fluids, 25, 051702.
Pair-collision between viscous drops in a confined shear is simulated to show that the confinement alters the trajectories of the drops spatially ordering them at a finite separation in the center of the domain. In contrast to free shear where drops eventually adopt free streamlines with a finite cross-stream separation, here they move towards the centerline achieving zero cross-stream separation but a net stream-wise separation. The latter varies as inverse of capillary number and cube of the confinement (distance between the walls). The final stream-wise separation does not depend on the initial positions of the drops when the drops are in the same shear plane. The separation decreases approximately linearly with the initial separation in the vorticity direction. An analytical theory explaining the phenomenon is presented. Effects of the ratio of drop to matrix viscosity are briefly investigated
-
Aliabouzar M, Kumar KN, Sarkar K, 2018 “Acoustic vaporization threshold of lipid coated perfluoropentane droplets,” Journal of the Acoustical Society of America, 143, 2001-2012.
Phase shift droplets vaporizable by acoustic stimulation offer the advantages of producing micro-bubbles as contrast agentsin situas well as higher stability and the possibility of achieving smallersizes. Here, the acoustic droplet vaporization (ADV) threshold of a suspension of droplets with aperfluoropentane (PFP) core (diameter 400–3000 nm) is acoustically measured as a function of theexcitation frequency in a tubeless setup at room temperature. The changes in scattered responses—fundamental, sub-, and second harmonic—are investigated, a quantitative criterion is used to deter-mine the ADV phenomenon, and findings are discussed. The average threshold obtained using threedifferent scattered components increases with frequency—1.0560.28 MPa at 2.25 MHz,1.8960.57 MPa at 5 MHz, and 2.3460.014 MPa at 10 MHz. The scattered response from vapor-ized droplets was also found to qualitatively match with that from an independently prepared lipid-coated microbubble suspension in magnitude as well as trends above the determined ADV thresh-old value.
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Aliabouzar M, Kumar KN, Sarkar K, 2018 “Acoustic vaporization threshold of lipid coated perfluoropentane droplets,” Journal of the Acoustical Society of America, 143, 2001-2012.
Phase shift droplets vaporizable by acoustic stimulation offer the advantages of producing micro-bubbles as contrast agentsin situas well as higher stability and the possibility of achieving smallersizes. Here, the acoustic droplet vaporization (ADV) threshold of a suspension of droplets with aperfluoropentane (PFP) core (diameter 400–3000 nm) is acoustically measured as a function of theexcitation frequency in a tubeless setup at room temperature. The changes in scattered responses—fundamental, sub-, and second harmonic—are investigated, a quantitative criterion is used to deter-mine the ADV phenomenon, and findings are discussed. The average threshold obtained using threedifferent scattered components increases with frequency—1.0560.28 MPa at 2.25 MHz,1.8960.57 MPa at 5 MHz, and 2.3460.014 MPa at 10 MHz. The scattered response from vapor-ized droplets was also found to qualitatively match with that from an independently prepared lipid-coated microbubble suspension in magnitude as well as trends above the determined ADV thresh-old value.
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Malipeddy Reddy A, Sarkar K, 2019 “Shear-induced collective diffusivity down a concentration gradient in a viscous emulsion,” Journal of Fluid Mechanics, 868, 5-25.
The shear-induced collective diffusivity down a concentration gradient in a viscous
emulsion is computed using direct numerical simulation. A layer of randomly packed
drops subjected to a shear flow, shows the layer width to increase with the 1=3
power of time, consistent with a semi-dilute theory that assumes a diffusivity linear
with concentration. This characteristic scaling and the underlying theory are used
to compute the collective diffusivity coefficient. This is the first ever computation
of this quantity for a system of deformable particles using fully resolved numerical
simulation. The results match very well with previous experimental observations.
The coefficient of collective diffusivity varies non-monotonically with the capillary
number, due to the competing effects of increasing deformation and drop orientation.
A phenomenological correlation for the collective diffusivity coefficient as a function
of capillary number is presented. We also apply an alternative approach to compute
collective diffusivity, developed originally for a statistically homogeneous rigid sphere
suspension – computing the dynamic structure factor from the simulated droplet
positions and examining its time variation at small wavenumber. We show that
the results from this alternative approach qualitatively agree with our computation
of collective diffusivity including the prediction of the non-monotonic variation of
diffusivity with the capillary number. -
Malipeddy Reddy A, Sarkar K, 2019 “Shear-induced collective diffusivity down a concentration gradient in a viscous emulsion,” Journal of Fluid Mechanics, 868, 5-25.
The shear-induced collective diffusivity down a concentration gradient in a viscous
emulsion is computed using direct numerical simulation. A layer of randomly packed
drops subjected to a shear flow, shows the layer width to increase with the 1=3
power of time, consistent with a semi-dilute theory that assumes a diffusivity linear
with concentration. This characteristic scaling and the underlying theory are used
to compute the collective diffusivity coefficient. This is the first ever computation
of this quantity for a system of deformable particles using fully resolved numerical
simulation. The results match very well with previous experimental observations.
The coefficient of collective diffusivity varies non-monotonically with the capillary
number, due to the competing effects of increasing deformation and drop orientation.
A phenomenological correlation for the collective diffusivity coefficient as a function
of capillary number is presented. We also apply an alternative approach to compute
collective diffusivity, developed originally for a statistically homogeneous rigid sphere
suspension – computing the dynamic structure factor from the simulated droplet
positions and examining its time variation at small wavenumber. We show that
the results from this alternative approach qualitatively agree with our computation
of collective diffusivity including the prediction of the non-monotonic variation of
diffusivity with the capillary number. -
Malipeddy Reddy A, Sarkar K, 2019 “Shear-induced collective diffusivity down a concentration gradient in a viscous emulsion,” Journal of Fluid Mechanics, 868, 5-25.
The shear-induced collective diffusivity down a concentration gradient in a viscous
emulsion is computed using direct numerical simulation. A layer of randomly packed
drops subjected to a shear flow, shows the layer width to increase with the 1=3
power of time, consistent with a semi-dilute theory that assumes a diffusivity linear
with concentration. This characteristic scaling and the underlying theory are used
to compute the collective diffusivity coefficient. This is the first ever computation
of this quantity for a system of deformable particles using fully resolved numerical
simulation. The results match very well with previous experimental observations.
The coefficient of collective diffusivity varies non-monotonically with the capillary
number, due to the competing effects of increasing deformation and drop orientation.
A phenomenological correlation for the collective diffusivity coefficient as a function
of capillary number is presented. We also apply an alternative approach to compute
collective diffusivity, developed originally for a statistically homogeneous rigid sphere
suspension – computing the dynamic structure factor from the simulated droplet
positions and examining its time variation at small wavenumber. We show that
the results from this alternative approach qualitatively agree with our computation
of collective diffusivity including the prediction of the non-monotonic variation of
diffusivity with the capillary number. -
Singh R, Sarkar K 2015 “Hydrodynamic interactions between pairs of capsules and drops in a simple shear: effects of viscosity ratio and heterogeneous collision,” Physical Review E, 92, 063029.
Hydrodynamic interactions between a pair of capsules in simple shear are numerically investigated using afront-tracking finite difference method. The membrane of the capsule is modeled using different hyperelasticconstitutive relations. We also compare the pair interactions between drops to those between capsules. Anincreased viscosity ratio leads to a reduced net cross-stream separation between capsules as well as drops aftercollision. At low viscosity ratios, for the same capillary number drop-pairs show higher cross-stream separationthan those for capsule-pairs, while substantially large viscosity ratios result in almost the same value for bothcases. We investigate pair-collisions between two heterogeneous capsules C1and C2with two different capillarynumbers. The maximum deformation of C1was seen to increase with increasing stiffness (decreasing capillarynumber) of C2, even though the stiffness of C1was kept fixed. The findings are similar for a drop-pair, however,with a smaller maximum deformation for the same combinations of capillary numbers. The final cross-streamdrift of the trajectory of C1decreases with the increasing stiffness of C2, but the relative trajectory betweenthe capsules remains unchanged. The maximum deformation and the cross-stream drift of the trajectory of C1are shown to approximately vary with power-law functions of the ratio of the capillary numbers of C1andC2. An analytical explanation of the dependence on the two capillary numbers is offered. Different membraneconstitutive laws result in similar deformation and drift in trajectory.
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Singh R, Li X, Sarkar K 2014 “Lateral migration of an elastic capsule in a wall-bounded shear,” Journal of Fluid Mechanics, 739, 421-443.
The migration of a capsule enclosed by an elastic membrane in a wall-bounded linearshear is investigated using a front-tracking method. A detailed comparison with themigration of a viscous drop is presented varying the capillary number (in the caseof a capsule, the elastic capillary number) and the viscosity ratio. In both cases,the deformation breaks the flow reversal symmetry and makes them migrate awayfrom the wall. They quickly go through a transient evolution to eventually reach aquasi-steady state where the dynamics becomes independent of the initial positionand only depends on the wall distance. Previous analytical theories predicted thatfor a viscous drop, in the quasi-steady state, the migration and slip velocities scaleapproximately with the square of the inverse of the drop–wall separation, whereasthe drop deformation scales as the inverse cube of the separation. These power lawrelations are shown to hold for a capsule as well. The deformation and inclinationangle of the capsule and the drop at the same wall separation show a crossoverin their variation with the capillary number: the capsule shows a steeper variationthan that of the drop for smaller capillary numbers and slower variation than thedrop for larger capillary numbers. Using the Green’s function of Stokes flow, asemi-analytic theory is presented to show that the far-field stresslet that causes themigration has two distinct contributions from the interfacial stresses and the viscosityratio, with competing effects between the two defining the dynamics. It predicts thescaling of the migration velocity with the capsule–wall separation, however, matchingwith the simulated result very well only away from the wall. A phenomenologicalcorrelation for the migration velocity as a function of elastic capillary number, walldistance and viscosity ratio is developed using the simulation results. The effects ofdifferent membrane hyperelastic constitutive equations – neo-Hookean, Evans–Skalak,and Skalak – are briefly investigated to show that the behaviour remains similar fordifferent equations.
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Mukherjee S, Sarkar K 2014 “Lateral migration of a viscoelastic drop in a Newtonian fluid in a shear flow near a wall,” Physics of Fluids, 26, 103102.
Wall induced lateral migration of a viscoelastic (FENE-MCR) drop in a Newtonianfluid is investigated. Just like a Newtonian drop, a viscoelastic drop reaches a quasi-steady state where the lateral velocity only depends on the instantaneous distancefrom the wall. The drop migration velocity and the deformation scale inversely withthe square and the cube of the distance from the wall, respectively. The migration ve-locity varies non-monotonically with increasing viscoelasticity (increasing Deborahnumber); initially increasing and then decreasing. An analytical explanation has beengiven of the effects by computing the migration velocity as arising from an imagestresslet field due to the drop. The semi-analytical expression matches well with thesimulated migration velocity away from the wall. It contains a viscoelastic stressletcomponent apart from those arising from interfacial tension and viscosity ratio. Themigration dynamics is a result of the competition between the viscous (interfacialtension and viscosity ratio) and the viscoelastic effects. The viscoelastic stressletcontribution towards the migration velocity steadily increases. But the interfacialstresslet—arising purely from the drop shape—first increases and then decreases withrising Deborah number causing the migration velocity to be non-monotonic. The ge-ometric effect of the interfacial stresslet is caused by a corresponding nonmonotonicvariation of the drop inclination. High viscosity ratio is briefly considered to showthat the drop viscoelasticity could stabilize a drop against breakup, and the increase inmigration velocity due to viscoelasticity is larger compared to the viscosity-matchedcase.
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Sarkar K, Singh R 2013 “Spatial ordering due to hydrodynamic interactions between a pair of colliding drops in a confined shear,” Physics of Fluids, 25, 051702.
Pair-collision between viscous drops in a confined shear is simulated to show that the confinement alters the trajectories of the drops spatially ordering them at a finite separation in the center of the domain. In contrast to free shear where drops eventually adopt free streamlines with a finite cross-stream separation, here they move towards the centerline achieving zero cross-stream separation but a net stream-wise separation. The latter varies as inverse of capillary number and cube of the confinement (distance between the walls). The final stream-wise separation does not depend on the initial positions of the drops when the drops are in the same shear plane. The separation decreases approximately linearly with the initial separation in the vorticity direction. An analytical theory explaining the phenomenon is presented. Effects of the ratio of drop to matrix viscosity are briefly investigated
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Malipeddy AR, Sarkar K 2019 “Collective diffusivity in a sheared viscous emulsion: effects of viscosity ratio,” Physical Review Fluids, 4, 093603.
The shear-induced collective or gradient diffusivity in an emulsion of viscous drops,
specifically as a function of viscosity ratio, was computed using a fully resolved numerical method. An initially randomly packed layer of viscous drops spreading due to drop-drop interactions in an imposed shear has been simulated. The collective diffusivity coefficient was computed using a self-similar solution of the drop concentration profile. We also obtained the collective diffusivity (the collective diffusivity coefficient multiplied by the average drop volume fraction), computing the dynamic structure factor from the simulated drop positions—an analysis typically applied only to homogeneous systems. The two quantities computed using entirely different methods are in broad agreement, including their predictions of nonmonotonic variations with increasing capillary number and viscosity ratio. The computed values were also found to match with past experimental
measurements. The collective diffusivity coefficient computed here, as expected, is 1 order of magnitude larger than the self-diffusivity coefficient for a dilute emulsion previously computed using pairwise simulation of viscous drops in shear. The collective diffusivity coefficient computed here shows a nonmonotonic variation with viscosity ratio, in contrast to self-diffusivity computed using pairwise computation. The difference might point to an intrinsic difference in physics underlying the two diffusivities. Alternatively, it also might
arise from drops not reaching equilibrium deformation in the period after one interaction and before the next—an effect absent in the pairwise simulation used for the computation of self-diffusivity. We offer a qualitative explanation of the nonmonotonic variation by relating it to average nonmonotonic drop deformation with increasing viscosity ratio. We
also provide empirical correlations of the collective diffusivity as a function of viscosity ratio and capillary number.
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