Dynamics of Finger Formation in Laplacian Growth without Surface Tension

Abstract

We study the dynamics of "finger" formation in Laplacian growth without surface tension in a channel geometry (the Saffman-Taylor problem). Carefully determining the role of boundary geometry, we construct field equations of motion, these central to the analytic power we can here exercise. We consider an explicit analytic class of maps to the physical space, a basis of solutions for infinite fluid in an infinitely long channel, characterized by meromorphic derivatives. We verify that these maps never lose analyticity in the course of temporal evolution, thus justifying the underlying machinery. However, the great bulk of these solutions can lose conformality in time, this the circumstance of finite-time singularities. By considerations of the nature of the analyticity of all these solutions, we show that those free of such singularities inevitably result in a {\em single} asymptotic "finger". This is purely nonlinear behavior: the very early "finger" actually already has a waist, this having signalled the end of any linear regime. The single "finger" has nevertheless an arbitrary width determined by initial conditions. This is in contradiction with the experimental results that indicate selection of a finger of width 1/2. In the last part of this paper we motivate that such a solution can be determined by the boundary conditions when the fluid is finite. This is a strong signal that {\em finiteness} is determinative of pattern selection.