Deformation of a two-dimensional viscous drop in time-periodic extensional flows: analytical treatment

Abstract

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.