Interface growth in the channel geometry and tripolar Loewner evolutions

TL;DR

This paper explores Laplacian growth in channel geometries using tripolar Loewner evolutions, presenting exact solutions for fingered growth and a generalized model demonstrating complex interface dynamics like tip competition.

Contribution

It introduces a generalized tripolar Loewner framework for interface growth, extending previous models with new exact solutions and complex dynamical behaviors.

Findings

Exact solutions for stationary finger growth

Modeling of multiple tips and interfaces

Observation of tip and finger competition phenomena

Abstract

A class of Laplacian growth models in the channel geometry is studied using the formalism of tripolar Loewner evolutions, in which three points, namely, the channel corners and infinity, are kept fixed. Initially, the problem of fingered growth, where growth takes place only at the tips of slit-like fingers, is revisited and a class of exact exact solutions of the corresponding Loewner equation is presented for the case of stationary driving functions. A model for interface growth is then formulated in terms of a generalized tripolar Loewner equation and several examples are presented, including interfaces with multiple tips as well as multiple growing interfaces. The model exhibits interesting dynamical features, such as tip and finger competition.