Abstract and Applied Analysis
Volume 2004 (2004), Issue 4, Pages 347-360
doi:10.1155/S1085337504306044
Comparison of differential representations for radially symmetric
Stokes flow
1Division of Applied Mathematics, Department of Chemical Engineering, University of Patras, Patras 26504, Greece
2Institute of Chemical Engineering and High Temperature Chemical
Processes (ICE/HT), Foundation for Research Technology-Hellas (FORTH), University of Patras, Patras 26504, Greece
Received 10 September 2002
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
Papkovich and Neuber (PN), and Palaniappan, Nigam, Amaranath, and
Usha (PNAU) proposed two different representations of the
velocity and the pressure fields in Stokes flow, in terms of
harmonic and biharmonic functions, which form a practical tool
for many important physical applications. One is the
particle-in-cell model for Stokes flow through a swarm of
particles. Most of the analytical models in this realm consider
spherical particles since for many interior and exterior flow
problems involving small particles, spherical geometry provides a
very good approximation. In the interest of producing
ready-to-use basic functions for Stokes flow, we calculate the
PNAU and the PN eigensolutions generated by the appropriate
eigenfunctions, and the full series expansion is provided. We
obtain connection formulae by which we can transform any solution
of the Stokes system from the PN to the PNAU eigenform. This
procedure shows that any PNAU eigenform corresponds to a
combination of PN eigenfunctions, a fact that reflects the
flexibility of the second representation. Hence, the advantage of
the PN representation as it compares to the PNAU solution is
obvious. An application is included, which solves the problem of
the flow in a fluid cell filling the space between two concentric
spherical surfaces with Kuwabara-type boundary conditions.