European Journal of Applied Mathematics

Research Article

Persistence of corners in free boundaries in Hele-Shaw flow

J. R. Kinga1, A. A. Laceya2 and J. L. Vazqueza3

a1 Department of Theoretical Mechanics, University of Nottingham, Nottingham NG7 2RD, UK

a2 Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK

a3 Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Abstract

In this paper we investigate the movement of free boundaries in the two-dimensional Hele-Shaw problem. By means of the construction of special solutions of self-similar type we can describe the evolution of free boundary corners in terms of the angle at the corner. In particular, we prove that, in the injection case, while obtuse-angled corners move and smooth out instantaneously, acute-angled corners persist until a (finite) waiting time at which, at least for the special solutions, they suddenly jump into an obtuse angle, precisely the supplement of the original one. The critical values of the angle π and π/2 are also considered.

(Revised March 02 1995)