International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 46, Pages 2429-2441

The 3D Happel model for complete isotropic Stokes flow

George Dassios1,2 and Panayiotis Vafeas1,2

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.


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.