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Complex variable methods in Hele–Shaw moving boundary problems

Published online by Cambridge University Press:  16 July 2009

S. D. Howison
S. D. Howison
Affiliation:
Mathematical Institute, 24–29 St. Giles, Oxford OX1 3LB, UK
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Abstract

We discuss the one-phase Hele–Shaw problem in two space dimensions. We review exact solutions in the zero-surface-tension case, giving a unified account of the Schwarz function and conformal mapping approaches. We discuss the extension of the former method to the cases in which surface tension or ‘kinetic undercooling’ terms apply on the moving boundary, and we give some conjectures on the resulting singularity structure. Finally, we give a new interpretation of the linear stability analysis of the zero-surface-tension problem, and we suggest a possible regularization of ill-posed problems by the imposition of a unilateral constraint on the moving boundary.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 3 , Issue 3 , September 1992 , pp. 209 - 224
Copyright
Copyright © Cambridge University Press 1992

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