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Some Hele Shaw flows with time-dependent free boundaries

Published online by Cambridge University Press:  20 April 2006

S. Richardson
S. Richardson
Affiliation:
Department of Mathematics, University of Edinburgh, Scotland
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Abstract

We consider a blob of Newtonian fluid sandwiched in the narrow gap between two plane parallel surfaces. At some initial instant, its plan-view occupies a given, simply connected domain, and its growth as further fluid is injected at a number of injection points in its interior is to be determined. It is shown that certain functionals of the domain of a purely geometric character, infinite in number, evolve in a predictable manner, and that these may be exploited in some cases of interest to yield a complete description of the motion.

By invoking images, these results may be used to solve certain problems involving the growth of a blob in a gap containing barriers. Injection at a point in a half-plane bounded by a straight line, with an initially empty gap, is shown to lead to a blob whose outline is part of an elliptic lemniscate of Booth for which there is a simple geometrical construction. Injection into a quarter-plane is also considered in some detail when conditions are such that the image domain involved is simply connected.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 102 , January 1981 , pp. 263 - 278
Copyright
© 1981 Cambridge University Press

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