Multi-Cuts Solutions of Laplacian Growth

TL;DR

This paper introduces a new class of time-dependent conformal map solutions for Laplacian growth with zero surface tension, encompassing all known solutions and modeling oil fjords with specific geometries in viscous fingering experiments.

Contribution

It presents a novel class of solutions governed by a nonlinear integral equation, including a linear subclass, and connects these solutions to physical phenomena in Hele-Shaw cell experiments.

Findings

Solutions include all known Laplacian growth solutions.

Identifies integrals of motion via Schwarz function singularities.

Models oil fjords with constant opening angles in experiments.

Abstract

A new class of solutions to Laplacian growth with zero surface tension is presented and shown to contain all other known solutions as special or limiting cases. These solutions, which are time-dependent conformal maps with branch cuts inside the unit circle, are governed by a nonlinear integral equation and describe oil fjords with non-parallel walls in viscous fingering experiments in Hele-Shaw cells. Integrals of motion for the multi-cut Laplacian growth solutions in terms of singularities of the Schwarz function are found, and the dynamics of densities (jumps) on the cuts are derived. The subclass of these solutions with linear Cauchy densities on the cuts of the Schwarz function is of particular interest, because in this case the integral equation for the conformal map becomes linear. These solutions can also be of physical importance by representing oil/air interfaces, which form oil fjords with a constant opening angle, in accordance with recent experiments in a Hele-shaw cell.