Connection formulae for differential representations in Stokes flow

Abstract

Stokes flow is described by a pair of partial differential equations connecting the velocity with the pressure field. Papkovich (1932)–Neuber (1934) and Boussinesq (1885)–Galerkin (1935) proposed two different differential representations of the velocity and the pressure in terms of harmonic and biharmonic functions. On the other hand, spherical geometry provides the most widely used framework for representing small particles and obstacles embedded within a viscous, incompressible fluid characterizing the steady and nonaxisymmetric Stokes flow. In the interest of producing ready-to-use basic functions for Stokes flow in spherical coordinates, we calculate the Papkovich–Neuber and the Boussinesq–Galerkin eigensolutions, generated by the well known spherical harmonic and biharmonic eigenfunctions. Furthermore, connection formulae are obtained, by which we can transform any solution of the Stokes system from the Papkovich–Neuber to the Boussinesq–Galerkin eigenform and vice versa.