Stationary solutions for the Saffman-Taylor problem with surface tension

Abstract

We report a one-parameter family of solutions for the problem of the motion of an interface between a viscous and a nonviscous two-dimensional fluid. The solutions have the interface moving uniformly while the viscous fluid has a nontrivial potential flow. In an alternative interpretation, the existence of gravitational forces in the plane allow the interface to be at rest while the fluid is in motion. This family of solutions is a generalization of a solution reported previously [L. P. Kadanoff, Phys. Rev. Lett. 65, 2986 (1990)]. The solutions presented here are found to be related to the traveling-wave solutions of the ``Harry Dym equation,'' which is a completely integrable nonlinear evolution equation.