Generalized eigenfunctions and complete semiseparable solutions for Stokes flow in spheroidal coordinates

Abstract

Abstract. The stream function y/ for axisymmetric Stokes flow satisfies the wellknown equation E4y/ = 0. In spheroidal coordinates the equation E2y/ = 0 admits separable solutions in the form of products of Gegenbauer functions of the first and second kind, and the general solution is then represented as a series expansion in terms of these eigenfunctions. Unfortunately, this property of separability is not preserved when one seeks solutions of the equation E4i// = 0. The nonseparability of Ea >// = 0 in spheroidal coordinates has impeded considerably the development of theoretical models involving particle-fluid interactions around spheroidal objects. In the present work the complete solution for ^ in spheroidal coordinates is obtained as follows. First, the generalized 0-eigenspace of the operator E2 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 . A rearrangement of the complete expansion, in such a way that the angular-type dependence enters through the Gegenbauer functions of successive order, leads to some kind of semiseparable solutions, which are given in terms of full series expansions. The proper solution subspace that provides velocity and vorticity fields, which are regular on the axis, 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.

Hydrodynamic permeability of aggregates of porous particles with an impermeable core

“…Dassios et al [41] found a solution of Stokes equation in spheroidal coordinates and used it for a Stokes flow in spheroidal particle-in-cell models [42].…”

Model of adsorption–reaction–desorption in a swarm of spheroidal particles

“…The creeping flow of a Newtonian fluid within a porous medium consisting of spheroidal grains can be described by the spheroid-in-cell model (Dassios et al, 1994), which is analogous to the previously proposed sphere-in-cell Kuwabara model (Kuwabara, 1959). According to the Dassios et al (1994) model, the swarm is represented by a solid kernel that is enveloped by a liquid layer, the thickness of which is adjusted so that the porosity of the model is equal to that of the swarm.…”

“…According to the Dassios et al (1994) model, the swarm is represented by a solid kernel that is enveloped by a liquid layer, the thickness of which is adjusted so that the porosity of the model is equal to that of the swarm. The flow field is given analytically in terms of an infinite-series expansion (Dassios et al, 1994), the leading term of which has been shown to be a satisfactory approximation . Using the leading term, the velocity components for the prolate spheroid-in-cell having long semiaxis a 3 and semifocal distance ␣ ϭ ͌ a 3 2 Ϫ 1, are given in the prolate coordinates system (, ) by…”

“…where D, ⌳ 2 , ⌳ 3 , and ⌳ 4 are -and -dependent coefficients defined by Dassios et al (1994) and G n (x) and H n (x) are the Gegenbauer polynomials of the first and second kind, respectively, of degree Ϫ(1/2) and of order n. By assuming constant diffusivity, the governing equation for the steady state mass transport of the solute A, under creeping flow conditions (see Figure 1), can be written in dimensionless form as…”

“…The fluid flow problem within swarm of spheroidal particles has already been examined either numerically (Ammar and Hsieh, 1991;Epstein and Masliyah, 1972) or analytically (Dassios et al, 1994, producing accurate results . These models are useful in the study of convective mass transport processes in swarms of particles.…”