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Fluid flow in a medium distorted by a quasiconformal map can produce fractal boundaries

Published online by Cambridge University Press:  26 September 2008

Olli Martio  and
Bernt Øksendal
Olli Martio
Affiliation:
Department of Mathematics, P.O. Box 4, FIN-00014University of Helsinki, Finland
Bernt Øksendal
Affiliation:
Department of Mathematics/VISTA, University of Oslo, Box 1053 Blindern, N-0316 Oslo, Norway
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Abstract

Physical experiments indicate that when an expanding fluid flows through a porous rock then the boundary between the wet and the dry region can be very irregular (e.g. see [OMBAFJ] and the references therein). In fact, it has been conjectured that this boundary is a fractal with Hausdorff dimension about 2.5. The (one-phase) fluid flow in a porous medium can be modelled mathematically by a system of partial differential equations, which, under some simplifying assumptions, can be reduced to a family of semi-elliptic boundary value problems involving the (unknown) pressure p(x) of the fluid (at the point x and at t) and the (unknown) wet region Ut at time t. (See equations (1.5)–(1.7) below). This set of equations, called the moving boundary problem involves the permeability matrix K(x) of the medium at x. A question which has been debated is whether this relatively simple mathematical model can explain such a complicated fractal nature of ∂Ut. More precisely, does there exist a symmetric non-negative definite matrix K(x) such that the solution Ut of the corresponding (expanding) moving boundary problem has a fractal boundary? The purpose of this paper is to prove that this is indeed the case. More precisely, we show that a porous medium which produce fractal wet boundaries can be obtained by distorting a completely homogeneous medium by means of a quasiconformal map.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 7 , Issue 1 , February 1996 , pp. 1 - 10
Copyright
Copyright © Cambridge University Press 1996

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