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Hele–Shaw flows with a free boundary produced by multipoles

Published online by Cambridge University Press:  26 September 2008

Vladimir M. Entov ,
Pavel I. Etingof  and
Dmitry Ya. Kleinbock
Vladimir M. Entov
Affiliation:
Institute for Problems in Mechanics, Russian Academy of Sciences, prosp. Vernadskogo, 101 Moscow, Russia
Pavel I. Etingof
Affiliation:
Yale University, Department of Mathematics, 2155 Yale Station, New Haven, CT 06520USA
Dmitry Ya. Kleinbock
Affiliation:
Yale University, Department of Mathematics, 2155 Yale Station, New Haven, CT 06520USA
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Abstract

We study Hele–Shaw flows with a moving boundary and multipole singularities. We find that such flows can be defined only on a finite time interval. Using a complex variable approach, we construct a family of explicit solutions for a single multipole. These solutions turn out to have the maximal possible lifetime in a certain class of solutions.

We also discuss the generalized Hele-Shaw model in which surface tension at the moving boundary is considered, and develop a method of finding steady shapes. This method yields new one-parameter families of stationary solutions. In the Appendix we discuss a connection between these solutions and a variational problem of potential theory.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 4 , Issue 2 , June 1993 , pp. 97 - 120
Copyright
Copyright © Cambridge University Press 1993

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