International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 46, Pages 2429-2441
doi:10.1155/S0161171204312445
The 3D Happel model for complete isotropic Stokes flow
1Division of Applied Mathematics, Department of Chemical Engineering, University of Patras, Patras 265 00, Greece
2Institute of Chemical Engineering and High Temperature Chemical Processes, Foundation for Research and Technology-Hellas (FORTH), 265 04, Patras, Greece
Received 4 December 2003
Copyright © 2004 George Dassios and Panayiotis Vafeas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The creeping flow through a swarm of spherical particles that
move with constant velocity in an arbitrary direction and rotate
with an arbitrary constant angular velocity in a quiescent
Newtonian fluid is analyzed with a 3D sphere-in-cell model. The
mathematical treatment is based on the two-concentric-spheres
model. The inner sphere comprises one of the particles in the
swarm and the outer sphere consists of a fluid envelope. The
appropriate boundary conditions of this non-axisymmetric
formulation are similar to those of the 2D sphere-in-cell Happel
model, namely, nonslip flow condition on the surface of the solid
sphere and nil normal velocity component and shear stress on the
external spherical surface. The boundary value problem is solved
with the aim of the complete Papkovich-Neuber differential
representation of the solutions for Stokes flow, which is valid
in non-axisymmetric geometries and provides us with the velocity
and total pressure fields in terms of harmonic spherical
eigenfunctions. The solution of this 3D model, which is
self-sufficient in mechanical energy, is obtained in closed form
and analytical expressions for the velocity, the total pressure,
the angular velocity, and the stress tensor fields are provided.