Near-contact electrophoretic motion of a sphere parallel to a planar wall

Abstract

The electrophoretic motion of a sphere resulting from an applied electric field directed parallel to a nearby plane wall is analysed for the case of a small sphere–wall gap width. The thin-Debye-layer approximation is employed. Using matched asymptotic expansions, the fluid domain is separated into an ‘inner’ (gap-scaled) region, wherein the electric field and velocity gradients are large, and an ‘outer’ (sphere-scaled) region, wherein field variations are moderate. Asymptotic expressions for the force and torque acting on the sphere are obtained using a reciprocal theorem, thereby avoiding the need to explicitly solve the pertinent Stokes equations. These expressions, as well as the sphere's electrophoretic mobility, become unbounded for near-contact separations. The present asymptotic solution complements existing ‘exact’ bipolar-coordinate eigenfunction expansions, which are numerically suitable only for $O(1)$ gap thicknesses.