The gravity-driven migration of a drop in an unsteady Stokes flow

Abstract

The motion of a spherical droplet through a fluid driven by gravity is revisited theoretically. Analytical expressions for the translational velocity of the droplet as well as its internal and external flows coupled at their interface and described by the unsteady Stokes equations are derived combining two-point Padé and Padé-type approximants with the Laplace transform. These expressions, involving scaled complementary error functions, are fast and globally convergent to the exact solutions of the equations. The migration of the droplet depends on three non-dimensional parameters, two of which determine the flow patterns inside and outside, and the last is related to the timescale. The analytical solutions reveal that there is a diffusive shear layer of thickness O(t) on either side of the interface in the early stage. Several plots are made to compare migration velocities of the droplet for different parameter sets and changes in the flow field with time.