Conducting drops subject to electric fields in 2D Stokes flows

Abstract

This paper presents an analysis of a variety of problems of electrohydrodynamics involving conducting uncharged drops (or bubbles) in 2D Stokes flow. The assumption of an uncharged droplet implies that there is no net force on the bubble and, hence, that a solution to this 2D problem is not disqualified by a 'Stokes paradox' phenomenon. We first study the dynamics of conducting, incompressible bubbles subject to electric fields and ambient straining flows and then extend the scope to include compressible bubbles containing ideal gas. The free boundary problems are first studied using full numerical simulations based on a complex variable formulation coupled with conformal mapping theory. It is found that even under the combined effects of surface tension, electric field effects, compressibility and ambient strain flows, there exist equilibria for the bubble shape which are remarkably close to elliptical. Therefore, in each case, a simple model system of (at most two) non-linear, non-local ordinary differential equations is proposed which is found to approximate the dynamics up to equilibrium to high accuracy.