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Mathematical Problems in Engineering
Volume 2014, Article ID 146521, 14 pages
http://dx.doi.org/10.1155/2014/146521
Research Article

Newtonian and Non-Newtonian Fluids through Permeable Boundaries

Department of Mathematical Sciences, University of South Africa, P.O. Box 392, Pretoria 0003, South Africa

Received 20 August 2014; Accepted 17 September 2014; Published 1 October 2014

Academic Editor: Abdon Atangana

Copyright © 2014 Riëtte Maritz and Emile Franc Doungmo Goufo. 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

We considered the situation where a container with a permeable boundary is immersed in a larger body of fluid of the same kind. In this paper, we found mathematical expressions at the permeable interface of a domain , where . is defined as a smooth two-dimensional (at least class ) manifold in . The Sennet-Frenet formulas for curves without torsion were employed to find the expressions on the interface . We modelled the flow of Newtonian as well as non-Newtonian fluids through permeable boundaries which results in nonhomogeneous dynamic and kinematic boundary conditions. The flow is assumed to flow through the boundary only in the direction of the outer normal n, where the tangential components are assumed to be zero. These conditions take into account certain assumptions made on the curvature of the boundary regarding the surface density and the shape of ; namely, that the curvature is constrained in a certain way. Stability of the rest state and uniqueness are proved for a special case where a “shear flow” is assumed.