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Hele-Shaw flows with a free boundary produced by multipoles. (English) Zbl 0780.76024

Summary: 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.

MSC:

76D99 Incompressible viscous fluids
76D45 Capillarity (surface tension) for incompressible viscous fluids
30C20 Conformal mappings of special domains
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