Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

A variational inequality approach to Hele-Shaw flow with a moving boundary

C. M. Elliotta1 and V. Janovskýa2

a1 Imperial College, London

a2 Charles University, Prague

Synopsis

This paper is concerned with the study of a mathematical model of the injection of fluid into a finite Hele–Shaw cell. The mathematical problem is one of solving Laplace's equation in an unknown region whose boundary changes with time. By a transformation of the dependent variable, an elliptic variational inequality formulation of the moving boundary problem is obtained. The variational inequality is shown to have a unique solution up to the time at which the cell is filled. Regularity results for the solution of the inequality are obtained by studying a penalty approximation of the inequality.

(Received January 23 1980)

(Revised July 01 1980)

Footnotes

The work of this author was carried out while visiting Brunel University and Oxford University Computing Laboratory, and was supported by The British Council.