Deformation of a two-dimensional viscous drop in time-periodic extensional flows: Analytical treatment (Q2731220)

In their companion paper [see the foregoing entry], the authors have numerically investigated the shape deformation of a two-dimensional drop in several time-dependent flow fields. Here the same problem is treated analytically. Starting from the full equations of momentum and mass conservation and assuming periodic-in-time dependent variables and small but non-zero Reynolds numbers in comparison to Strouhal number, the authors arrive at two non-dimensional unsteady Stokes equations (expressed in the frequency domain), that govern the flow in the domain of continuous phase (surrounding liquid) and in the suspended drop domain. The other non-dimensional parameters entering into the computation via the boundary conditions at the moving drop interface are the ratios of density and viscosity of the two phases, and the inverse of non-dimensional capillary number. In the far-field the continuous phase flow should agree with the imposed time-periodic potential flow. The authors consider the same three kinds of forcing flows as in the companion paper. Assuming the drop deformation to be small, the problem is solved analytically using a first-order perturbation method. The limiting cases (Stokes flow and inviscid flow) are treated separately. It is found that the analytical results compare satisfactorily with those obtained from the numerical simulation in the companion paper.