On a Saffman–Taylor problem in an infinite wedge

Abstract

We consider a zero-surface-tension two-dimensional Hele–Shaw flow in an infinite wedge. There exists a self-similar interface evolution in this wedge, an analogue of the famous Saffman–Taylor finger in a channel, exact shape of which has been given by Kadanoff. One of the main features of this evolution is its infinite time of existence and stability for the Hadamard ill-posed problem. We derive several exact solutions existing infinitely by generalizing and perturbing the one given by Kadanoff.

Keywords:

• Hele–Shaw problem
• Saffman–Taylor finger
• Conformal map

Keywords:

• 2000 Mathematics Subject Classifications: Primary: 76D27
• Secondary: 30C35

Acknowledgements

The authors Irina Markina and Alexander Vasil’ev were supported by the grant of the University of Bergen (Norway) and the grant of the Norwegian Research Council #177355; the author Rodrigo Meneses was partially supported by Project Fondecyt #1040333, and UTFSM #12.05.23 (Chile).