Velocity Selection and the Saffman-Taylor Problem

Abstract

A new approach to the velocity selection problem is presented. It establishes a relation between the existence of propagating steady states as a function of a parameter and a certain inhomogeneous linear problem, the solvability of which determines the set of allowed velocities. The method is illustrated on the example of a geometrical model of solidification. It is used to explain analytically the fact that at large velocity the Saffman-Taylor "fingers" have width close to ½ and to predict the scaling exponent for the dependence of the finger width on velocity.