Hydrodynamic interaction of a spherical particle with a planar boundary I. Free surface

Abstract

We construct the grand resistance and mobility matrices for a hard sphere moving in an incompressible viscous fluid with a planar boundary. Using the result for a single sphere in an unbounded fluid we express the hydrodynamic interaction between the sphere and wall in terms of the Green function for the bounded fluid. We express the resistance and mobility matrices in terms of a set of scalar resistance and mobility functions that depend on the sphere radius and the distance to the boundary. We derive a reflection theorem for a complete set of solutions to the Navier-Stokes equations and use it to compute a series expansion of the scalar transport functions in inverse powers of the distance to the wall. For the simplest case, a boundary which is a free surface, we give numerical tables of the first twenty expansion coefficients for the functions describing translation and rotation of a sphere with stick boundary conditions.