Acoustic microstreaming due to an oscillating microbubble, either coated or free, is analytically investigated. The detailed flow field is obtained and the closed streamlines of the ring vortex generated by microstreaming are plotted in both Eulerian and Lagrangian descriptions. Analytical expressions are found for the ring vortex showing that its length depends only on the separation of the microbubble from the wall and the dependence is linear. The circulation as a scalar measure of the vortex is computed quantitatively identifying its spatial location. The functional dependence of circulation on bubble separation and coating parameters is shown to be similar to that of the shear stress.

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.

The shear rheology of an emulsion of viscous drops in the presence of finite inertia is investigated using direct numerical simulation. In the absence of inertia, emulsions display a non-Newtonian rheology with positive first and negative second normal stress differences. However, recently it was discovered that a small amount of drop-level inertia alters their signs – the first normal stress difference becomes negative and the second one becomes positive, each in a small range of capillary numbers (Li & Sarkar, J. Rheol., vol. 49, 2005, pp. 1377–1394). Sign reversal was shown numerically and analytically, but only in the limit of a dilute emulsion where drop–drop interactions were neglected. Here, we compute the rheology of a density- and viscosity-matched emulsion, accounting for the interactions in the volume fraction range of 5 %–27 % and Reynolds number range of 0.1–10. The computed rheological properties (effective shear viscosity and first and second normal stress differences) in the Stokes limit match well with previous theoretical (Choi–Schowalter in the dilute limit) and simulated results (for concentrated systems) using the boundary element method. The two distinct components of the rheology arising from the interfacial stresses at the drop surface and the perturbative Reynolds stresses are investigated as functions of the drop Reynolds number, capillary number and volume fraction. The sign change is caused by the increasing drop inclination in the presence of inertia, which in turn directly affects the interfacial stresses. Increase of the volume fraction or capillary number increases the critical Reynolds number for sign reversals due to enhanced alignment of the drops with the flow directions. The effect of increasing the volume fraction on the rheology is explained by relating it to interactions and specifically to the contact pair-distribution function computed from the simulation. The excess stresses are seen to show an approximately linear behaviour with the Reynolds number in the range of 0.1–5, while with the capillary number and volume fraction, the variation is weakly quadratic.

The migration of a capsule enclosed by an elastic membrane in a wall-bounded linear shear is investigated using a front-tracking method. A detailed comparison with the migration of a viscous drop is presented varying the capillary number (in the case of a capsule, the elastic capillary number) and the viscosity ratio. In both cases, the deformation breaks the flow reversal symmetry and makes them migrate away from the wall. They quickly go through a transient evolution to eventually reach a quasi-steady state where the dynamics becomes independent of the initial position and only depends on the wall distance. Previous analytical theories predicted that for a viscous drop, in the quasi-steady state, the migration and slip velocities scale approximately with the square of the inverse of the drop–wall separation, whereas the drop deformation scales as the inverse cube of the separation. These power law relations are shown to hold for a capsule as well. The deformation and inclination angle of the capsule and the drop at the same wall separation show a crossover in their variation with the capillary number: the capsule shows a steeper variation than that of the drop for smaller capillary numbers and slower variation than the drop for larger capillary numbers. Using the Green’s function of Stokes flow, a semi-analytic theory is presented to show that the far-field stresslet that causes the migration has two distinct contributions from the interfacial stresses and the viscosity ratio, with competing effects between the two defining the dynamics. It predicts the scaling of the migration velocity with the capsule–wall separation, however, matching with the simulated result very well only away from the wall. A phenomenological correlation for the migration velocity as a function of elastic capillary number, wall distance and viscosity ratio is developed using the simulation results. The effects of different membrane hyperelastic constitutive equations – neo-Hookean, Evans–Skalak, and Skalak – are briefly investigated to show that the behaviour remains similar for different equations.

The dynamics of a drop deforming, orienting and moving in a shear flow of a viscoelastic liquid near a wall is numerically investigated using a front-tracking finite-difference method and a semi-analytic theory. The viscoelasticity is modelled using the modified FENE-CR constitutive equation. In a Newtonian system, deformation in a drop breaks the reversal symmetry of the system resulting in a migration away from the wall. This study shows that the matrix elasticity reduces the migration velocity, the reduction scaling approximately linearly with viscoelasticity (product of the Deborah number De and the ratio of polymer viscosity to total viscosity $\beta $). Similar to a Newtonian system, for small Deborah numbers, the dynamics quickly reaches a quasi-steady state where deformation, inclination, as well as migration and slip velocities become independent of the initial drop–wall separation. They all approximately scale inversely with the square of the instantaneous separation except for deformation which scales inversely with the cube of separation. The deformation shows a non-monotonic variation with increasing viscoelasticity similar to the case of a drop in an unbounded shear and is found to influence little the change in migration. Two competing effects due to matrix viscoelasticity on drop migration are identified. The first stems from the reduced inclination angle of the drop with increasing viscoelasticity that tries to enhance migration velocity. However, it is overcome by the second effect inhibiting migration that results from the normal stress differences from the curved streamlines around the drop; they are more curved on the side away from the wall compared with those in the gap between the wall and the drop, an effect that is also present for a rigid particle. A perturbative theory of migration is developed for small ratio of the drop size to its separation from the wall that clearly shows the migration to be caused by the image stresslet field due to the drop in presence of the wall. The theory delineates the two competing viscoelastic effects, their relative magnitudes, and predicts migration that matches well with the simulation. Using the simulation results and the stresslet theory, we develop an algebraic expression for the quasi-steady migration velocity as a function of Ca, De and $\beta $. The transient dynamics of the migrating drop is seen to be governed by the finite time needed for development of the viscoelastic stresses. For larger capillary numbers, in both Newtonian and viscoelastic matrices, a viscous drop fails to reach a quasi-steady state independent of initial drop–wall separation. Matrix viscoelasticity tends to prevent drop breakup. Drops that break up in a Newtonian matrix are stabilized in a viscoelastic matrix if it is initially far away from the wall. Initial proximity to the wall enhances deformation and aids in drop breakup.

Deformation of a viscous drop in shear at finite inertia and the streamlines around it are numerically investigated. Inertia destroys the closed streamlines found in Stokes flow. It creates reversed streamlines and streamlines spiralling around the vorticity axis. Spiralling streamlines spiral either towards the central shear plane or away from it depending on the viscosity ratio and the inertia. The zones of open or reversed streamlines as well as streamlines spiralling towards or away from the central shear plane are delineated for varying viscosity ratio and Reynolds number. In contrast to the infinite extent of the closed Stokes streamlines around a rigid sphere in shear, the region of the spiralling streamlines in the vorticity direction both for a rigid sphere and a drop shrinks with inertia. Inertia increases deformation, and introduces oscillations in drop shape. An approximate analysis explains the scaling of oscillation frequency and damping with Reynolds and capillary numbers. The steady-state drop inclination angle with the flow axis increases with increasing Reynolds number for small Reynolds number. But it decreases at higher Reynolds number, especially for larger capillary numbers. For smaller capillary numbers, drop inclination reaches higher than (the direction of maximum extension), critically affecting the interfacial stresses due to the drop. It changes the sign of first and second normal interfacial stress differences (and thereby these components of the effective stresses of an emulsion of such drops). Increasing viscosity ratio orients the drop towards the flow axis, which increases the critical Reynolds number above which the drop inclination reaches more than .

The deformation of a Newtonian/viscoelastic drop suspended in a viscoelastic fluid is investigated using a three-dimensional front-tracking finite-difference method. The viscoelasticity is modelled using the Oldroyd-B constitutive equation. Matrix viscoelasticity affects the drop deformation and the inclination angle with the flow direction. Numerical predictions of these quantities are compared with previous experimental measurements using Boger fluids. The elastic and viscous stresses at the interface, polymer orientation, and the elastic and viscous forces in the domain are carefully investigated as they affect the drop response. Significant change in the drop inclination with increasing viscoelasticity is observed; this is explained in terms of the first normal stress difference. A non-monotonic change – a decrease followed by an increase – in the steady-state drop deformation is observed with increasing Deborah number (De) and explained in terms of the competition between increased localized polymer stretching at the drop tips and the decreased viscous stretching due to change in drop orientation angle. The transient drop orientation angle is found to evolve on the polymer relaxation time scale for high De. The breakup of a viscous drop in a viscoelastic matrix is inhibited for small De, and promoted at higher De. Polymeric to total viscosity ratio β was seen to affect the result through the combined parameter βDe indicating a dominant role of the first normal stress difference. A viscoelastic drop in a viscoelastic matrix with matched relaxation time experiences 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 decreased deformation.

The deformation of a viscoelastic drop suspended in a Newtonian fluid subjected to a steady shear is investigated using a front-tracking finite-difference method. The viscoelasticity is modelled using the Oldroyd-B constitutive equation. The drop response with increasing relaxation time λ and varying polymeric to the total drop viscosity ratio β is studied and explained by examining the elastic and viscous stresses at the interface. Steady-state drop deformation was seen to decrease from its Newtonian value with increasing viscoelasticity. A slight non-monotonicity in steady-state deformation with increasing Deborah number is observed at high Capillary numbers. Transient drop deformation displays an overshoot before settling down to a lower value of deformation. The overshoot increases with increasing β. The drop shows slightly decreased alignment with the flow with increasing viscoelasticity. A simple ordinary differential equation model is developed to explain the various behaviours and the scalings observed numerically. The critical Capillary number for drop breakup is observed to increase with Deborah number owing to the inhibitive effects of viscoelasticity, the increase being linear for small Deborah number.

Deformation and breakup of a viscous drop in a potential vortex are numerically simulated. Capillary number, Reynolds number, and viscosity and density ratios are varied to investigate their effects on the drop dynamics. The vortex locally gives rise to an extensional flow near the drop with the axis of extension rotating at a constant rate, as the drop revolves around the vortex centre. The rotation of the axis plays a critical role in the competing dynamics between the flow-induced stretching and the interfacial tension. The relation between the rotating extensional flow and a shear flow is explored. For low capillary numbers, a periodic state is reached, where the drop deforms into an ellipsoidal shape and undergoes steady rotation with a distinct phase lag behind the imposed flow. For density-and-viscosity-matched drops, increased interfacial tension results in decreased deformation and reduced phase lag. Increased inertia promotes deformation. In the presence of inertia, decreasing capillary number leads to a negative phase lag. The rotation of the extension axis inhibits deformation at low values of the Reynolds number. But at high Reynolds numbers, rotation-induced centrifugal forces promote deformation. At low and high viscosity ratios, an increase in viscosity ratio leads to enhancement and reduction in deformation, respectively. At density ratios larger than unity, the drop deformation displays resonance in that it varies non-monotonically with a distinct peak with variation of interfacial tension and density ratio. The peak corresponds to the natural frequency of the drop deformation matching with the frequency of rotation due to the vortex. A simple physical model is used to explain various observations including asymptotic scalings. We also explore different mechanisms for drop breakup at different Reynolds number, and provide critical capillary numbers as functions of other parameters. In particular, vortex-induced resonance offers an alternative mechanism for size-selective drop breakup. Details of flow fields and transients are also presented and discussed.

Ensemble averaging combined with multiple scattering ideas is applied to the Stokes flow over a stochastic rough surface. The surface roughness is modelled by compact protrusions on an underlying smooth surface. It is established that the effect of the roughness on the flow far from the boundary may be represented by replacing the no-slip condition on the exact boundary by a partial slip condition on the smooth surface. An approximate analysis is presented for a sparse distribution of arbitrarily shaped protrusions and explicit numerical results are given for hemispheres. Analogous conclusions for the two-dimensional case are obtained. It is shown that in certain cases a traction force develops on the surface at an angle with the direction of the flow.

The shape of a two-dimensional viscous drop deforming in several time-dependent flow fields, including that due to a potential vortex, has been studied. Vortex flow was approximated by linearizing the induced velocity field at the drop centre, giving rise to an extensional flow with rotating axes of stretching. A generalization of the potential vortex, a flow we have called rotating extensional flow, occurs when the frequency of revolution of the flow is varied independently of the shear rate. Drops subjected to this forcing flow exhibit an interesting resonance phenomenon. Finally we have studied drop deformation in an oscillatory extensional flow.

Calculations were performed at small but non-zero Reynolds numbers using an ADI front-tracking/finite difference method. We investigate the effects of interfacial tension, periodicity, viscosity ratio, and Reynolds number on the drop dynamics. The simulation reveals interesting behaviour for steady stretching flows, as well as time-dependent flows. For a steady extensional flow, the drop deformation is found to be non-monotonic with time in its approach to an equilibrium value. At sufficiently high Reynolds numbers, the drop experiences multiple growth–collapse cycles, with possible axes reversal, before reaching a final shape. For a vortex flow, the long-time deformation reaches a steady value, and the drop attains a revolving steady elliptic shape. For rotating extensional flows as well as oscillatory extensional flows, the maximum value of deformation displays resonance with variation in parameters, first increasing and then decreasing with increasing interfacial tension or forcing frequency. A simple ODE model with proper forcing is offered to explain the observed phenomena.

In Sarkar & Schowalter (2001), we reported results from numerical simulations of drop deformation in various classes of time-periodic straining flows at non-zero Reynolds number. As often occurs, analytical solutions provide more effective understanding of the structure and significance of a phenomenon. Here we describe drop deformation predicted from analytical solutions to linear time-periodic straining flows. Three different limiting cases are considered: an unsteady Stokes flow that retains all but the nonlinear advection terms, a Stokes flow that neglects inertia altogether, and an inviscid potential flow. The first limit is in clear contrast to the common approach in emulsion literature that resorts almost always to the Stokes flow assumption. The analysis clearly shows the forced–damped mass–spring system underlying the physical phenomena, which distinguishes it from the inertialess Stokes flow. The potential flow also depicts resonance, albeit of an undamped system, and provides an important limit of the problem. The drop deformation is assumed to be small, and a perturbative approach has been employed. The first-order problem has been solved to arrive at either an evolution equation (in Stokes and potential flow limits) or the long-time periodic drop response (for unsteady Stokes analysis). The analytical results compare satisfactorily with those obtained from the numerical simulation in Sarkar & Schowalter (2001), and the resonance characteristics are quantitatively explained. The three different solutions are compared with each other, and the results are presented for different parameters such as frequency, interfacial tension, viscosity ratio, density ratio and Reynolds number. Furthermore, the simple ODE model presented in the Appendix of Sarkar & Schowalter (2001) is shown to explain the asymptotic limits of the present solution.