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Bubble growth in porous media and Hele–Shaw cells

Published online by Cambridge University Press:  14 November 2011

S. D. Howison
S. D. Howison
Affiliation:
Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy N.Y. 12181, U.S.A.
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Synopsis

We consider the characterisation of a class of free boundary problems arising in the flow of a viscous liquid in a porous medium (or, in two dimensions, a Hele–Shaw cell). Injected air forms a bubble which grows as time increases; it is shown that three kinds of behaviour can occur. Firstly, the solution may cease to exist in finite time; secondly, the solution may exist for all time and the free boundary may have one or more limit points as t tends to infinity; and thirdly, the bubble may exist for all time and fill the whole space as t tends to infinity. Two-dimensional explicit examples arc given of all three types of behaviour, and it is proved that the only solutions of the third kind are those in which the bubble is always elliptical; the proof uses the theory of null quadrature domains. It is shown that solutions for ellipsoidal bubbles exist in three dimensions and it is conjectured that the only three-dimensional null quadrature domains with finite complement are those whose complement is an ellipsoid.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 102 , Issue 1-2 , 1986 , pp. 141 - 148
Copyright
Copyright © Royal Society of Edinburgh 1986

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