Document Type Doctoral Thesis Author Maritz, Riette URN etd-02222006-143343 Document Title Flow of second-grade fluids in regions with permeable boundaries Degree PhD (Applied Mathematics) Department Mathematics and Applied Mathematics Supervisor

Advisor Name Title Prof N Sauer Committee Chair Keywords

- Navier-Stokes equations
- applied mathematics
- equations of motion Newtonian fluids
Date 2001-09-01 Availability unrestricted AbstractThe equation of motion for the flows of incompressible Newtonian fluids (Navier Stokes equations) under no-slip boundary conditions have been studied deeply from many perspectives. The questions of existence and uniqueness of both classical and weak solutions have received more than a fair share of attention. In this study the same problem for non-Newtonian fluids of second grade has been studied from the point of view of weak solutions and classical solutions for non-homogeneous boundary data, i.e., dynamical boundary conditions in regions with permeable boundaries. We consider the situation where a container is immersed in a larger fluid body and the boundary admits fluid particles moving across it in the direction of the normal.In this study we give alternative approaches through formulations of' dynamics at the boundary', the idea being that the normal component of velocity at the boundary is viewed as an unknown function which satisfies a differential equation intricately coupled to the flow in the region 'enclosed' by the boundary.

We describe two mathematical models denoted by

ProblemP_{I}andProblemP_{2}. These models lead to dynamics at a permeable boundary, and a kinematical boundary condition for normal flow through the boundary. These conditions take into account the curvature of the boundary which enforces certain stresses. We then show with the help of the energy method that for fluids of second grade, the dynamics at the boundary and the boundary condition lead to conditional stability of the rest state forProblemP_{1}andProblemP_{2}. We also prove uniqueness of classical solutions for the two models. The existence of a weak solution for this system of evolution equations is proved only forProblemP_{2}with the help of the Faedo-Galerkin method with a special basis. In this case the special basis is formed by eigenfunctions.The existence proof of at least one classical solution, local in time is established by means of a version of the Fixed-point Theorem of Bohnenblust and Karlin, and the Ascoli-Arzela Theorem.

© 2001, University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria.

Please cite as follows:Maritz, R 2001,

Flow of second-grade fluids in regions with permeable boundaries, PhD thesis, University of Pretoria, Pretoria, viewedyymmdd< http://upetd.up.ac.za/thesis/available/etd-02222006-143343/ >H752/ag

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