Fingered growth in channel geometry: A Loewner equation approach

TL;DR

This paper extends the Loewner equation model of Laplacian finger growth to channel geometries, revealing how side walls influence finger evolution and screening effects, with implications for understanding pattern formation.

Contribution

It introduces a novel extension of the Loewner equation approach to channel geometries, highlighting the impact of side walls on finger growth dynamics.

Findings

Side walls significantly affect finger evolution.

Screening process is influenced by channel boundaries.

Longer fingers suppress the growth of shorter ones.

Abstract

A simple model of Laplacian growth is considered, in which the growth takes place only at the tips of long, thin fingers. In a recent paper, Carleson and Makarov used the deterministic Loewner equation to describe the evolution of such a system. We extend their approach to a channel geometry and show that the presence of the side walls has a significant influence on the evolution of the fingers and the dynamics of the screening process, in which longer fingers suppress the growth of the shorter ones.