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Cusp development in free boundaries, and two-dimensional slow viscous flows

Published online by Cambridge University Press:  26 September 2008

S. D. Howison  and
S. Richardson
S. D. Howison
Affiliation:
Mathematical Institute, 24–29 St Giles, Oxford 0X1 3LB, UK
S. Richardson
Affiliation:
Department of Mathematics and Statistics, University of Edinburgh, Edinburgh EH9 3JZ, UK
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Abstract

We consider a family of problems involving two-dimensional Stokes flows with a time dependent free boundary for which exact analytic solutions can be found; the fluid initially occupies some bounded, simply-connected domain and is withdrawn from a fixed point within that domain. If we suppose there to be no surface tension acting, we find that cusps develop in the free surface before all the fluid has been extracted, and the mathematical solution ceases to be physically relevant after these have appeared. However, if we include a non-zero surface tension in the theory, no matter how small this may be, the cusp development is inhibited and the solution allows all the fluid to be removed.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 6 , Issue 5 , October 1995 , pp. 441 - 454
Copyright
Copyright © Cambridge University Press 1995

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