Effective boundary conditions for the Laplace equation with a rough boundary

Abstract

The problem of replacing Dirichlet or Neumann conditions on a stochastically embossed surface by approximate effective conditions on a smooth surface is studied for potential fields satisfying the Laplace equation. A combination of ensemble averaging and multiple-scattering techniques is used. It is shown that for the Dirichlet case the effective boundary condition becomes mixed and establishes a relation between the averaged field and its normal derivative. For the Neumann problem the normal derivative on the smooth surface equals a suitable combination of first- and second-order derivatives tangent to the surface. Explicit results are given for small boss concentration and illustrated with the examples of spheroidal and spherical bosses. For the Dirichlet case with hemispherical bosses, direct numerical-simulation results are presented up to area coverages of 75%. An application of the results to the calculation of the added mass of a rough sphere in potential flow, of the capacitance of a rough spherical conductor, and of the transmission and reflection of long water waves at a smooth-rough bottom transition aids in their physical interpretation.