Motions of a fluid drop near a deformable interface

Abstract

Analytical results are presented for the motion of a viscous Newtonian fluid drop in the presence of a plane, deformable, interface in the velocity range for which inertial effects may be neglected. The zeroth-order approximation for a spherical drop near a flat interface is expressed in terms of fundamental singularity solutions for Stokes flow, and used to evaluate the drag on the fluid drop in translation either perpendicular or parallel to the interface. The present approximate results for drag are in good agreement with exact-solution results where available. The first corrections for the shapes of the plane interface and the drop are then determined by reformulating the small deformation problem in terms of equivalent boundary conditions on a flat interface and a spherical drop surface. We consider the influence of the viscosity ratios, density differences and interfacial tensions (or Bond number and capillary numbers) and the drop position relative to the interface, in determining the degree of distortion of the plane interface and the fluid drop surface, and the hydrodynamic drag force on the drop. Among the most interesting results is the prediction of lateral migration induced by the drop and the interface deformations.