Velocity selection (without surface tension) in multi-connected Laplacian growth

Abstract

We predict a novel selection phenomenon in nonlinear interface dynamics out of equilibrium. Using a recently developed formalism based on the Schottky-Klein prime functions, we extended the existing integrable theory from a single interface to multiple moving interfaces. After applying this extended theory to the two-dimensional Laplacian growth, we derive a new rich class of exact (non-singular) solutions for the unsteady dynamics of an arbitrary assembly of air bubbles within a layer of a viscous fluid in a Hele-Shaw cell. These solutions demonstrate that all bubbles reach an asymptotic velocity, $U$, which is {\it precisely twice} greater than the velocity, $V$, of the uniform background flow, i.e., $U=2V$. The result does not depend on the number of bubbles. It is worth to mention that contrary to common belief, the predicted velocity selection does not require surface tension.Comment: 5 pages, 1 figure. Updated versio