On the Spheroidal Semiseparation for Stokes Flow

Abstract

Many heat and mass transport problems involve particle-fluid systems, where the assumption of Stokes flow provides a very good approximation for representing small particles embedded within a viscous, incompressible fluid characterizing the steady, creeping flow. The present work is concerned with some interesting practical aspects of the theoretical analysis of Stokes flow in spheroidal domains. The stream function 𝜓, for axisymmetric Stokes flow, satisfies the well-known equation 𝐸4𝜓=0. Despite the fact that in spherical coordinates this equation admits separable solutions, this property is not preserved when one seeks solutions in the spheroidal geometry. Nevertheless, defining some kind of semiseparability, the complete solution for 𝜓 in spheroidal coordinates has been obtained in the form of products combining Gegenbauer functions of different degrees. Thus, the general solution is represented in a full-series expansion in terms of eigenfunctions, which are elements of the space 𝑘𝑒𝑟𝐸2 (separable solutions), and in terms of generalized eigenfunctions, which are elements of the space 𝑘𝑒𝑟𝐸4 (semiseparable solutions). In this work we revisit this aspect by introducing a different and simpler way of representing the aforementioned generalized eigenfunctions. Consequently, additional semiseparable solutions are provided in terms of the Gegenbauer functions, whereas the completeness is preserved and the full-series expansion is rewritten in terms of these functions.