Dynamics and selection of fingering patterns. Recent developments in the Saffman–Taylor problem

Abstract

We study singular effects of surface tension in the dynamics of the finger competition in the Saffman–Taylor problem with channel geometry. First, we study in detail some relevant classes of exact solutions in the absence of surface tension and compare them to finite surface tension. We conclude that (nonsingular) zero-surface tension solutions are generically unphysical. We show that the elementary two-finger competition process in the absence of surface tension is structurally unstable and this fact is ultimately responsible for the lack of genuine finger competition. Second, we generalize solvability theory to study selection of multifinger configurations with finite surface tension. We find that a discrete set of nontrivial multifinger solutions, with stationary coexistence of unequal fingers, are selected by surface tension out of a continuum of solutions. We discuss the implications of these results on a possible dynamic solvability scenario of selection.