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.
Proc. Roy. Soc. 451, 425-452, 1995
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