Multipole expansions in Stokes flow

Abstract

Maxwell's theory of multipoles is extended from potential theory to Stokes flow field, and from spherical to spheroidal geometry. The expansion is based on an exterior integral representation of the velocity and the pressure field of Stokes flow as well as the appropriate fundamental solution. It is shown that the velocity field is expandable in terms of five different multipoles, four of which are weighted multipoles. On the other hand, the pressure as well as the vorticity field, have multipole expansions that involve only the non-weighted multipoles. In fact, a more general result is demonstrated according to which the pressure and the vorticity are given as the scalar and the vector invariants of the same harmonic dyadic field. The importance of the multipole expansion for the velocity and the pressure field is well known, and it refers both to the theoretical understanding of the flow, as well as to practical applications and numerical implementations.