The symmetry of mobility laws for viscous flow along arbitrarily patterned surfaces

Abstract

Generalizations of the no-slip boundary condition to allow for slip at a patterned fluid-solid boundary introduce a surface mobility tensor, which relates the shear traction vector tangent to the mean surface to an apparent surface velocity vector. For steady, low-Reynolds-number fluid motions over planar surfaces perturbed by arbitrary periodic height and Navier slip fluctuations, we prove that the resulting mobility tensor is always symmetric, which had previously been conjectured. We describe generalizations of the results to three other families of geometries, which typically have unsteady flow.