Dassios, G., Hadjinicolaou, M., Peyatakes, A.C.  Generalized eigenfunctions and complete semiseparable solutions for Stokes flow in spheroidal coordinates, Quarterly of Applied Mathematics  52 (1), pp. 157-191, 1994  
 

Abstract

The stream function psi for axisymmetric Stokes flow satisfies the well-known equation E"SUP 4" psi=0. In the present work the complete solution for psi in spheroidal coordinates is obtained as follows. First, the generalized 0-eigenspace of the operator E"SUP 2" is investigated and a complete set of generalized eigenfunctions is given in closed form, in terms of products of Gegenbauer functions with mixed order. The general Stokes stream function is then represented as the sum of two functions: one from the 0-eigenspace and one from the generalized 0-eigenspace of the operator E"SUP 2" . The proper solution subspace that provides velocity and vorticity fields is given explicitly. Finally, it is shown how these simple and generalized eigenfunctions reduce to the corresponding spherical eigenfunctions as the focal distance of the spheroidal system tends to zero, in which case the separability is regained. The usefulness of the method is demonstrated by solving the problem of the flow in a fluid cell contained between two confocal spheroidal surfaces with Kuwabara-type boundary conditions.