Loewner equation for Laplacian growth: A Schwarz-Christoffel-transformation approach

TL;DR

This paper introduces a novel Schwarz-Christoffel transformation-based method to derive the Loewner equation for various Laplacian growth problems, including slit-like fingers and bubble expansion in the half-plane.

Contribution

A new approach using Schwarz-Christoffel transformations to derive Loewner equations for a broad class of Laplacian growth phenomena.

Findings

Reproduces known Loewner evolution for slit-like fingers

Extends to derive Loewner equation for bubble growth

Provides a unified framework for different growth geometries

Abstract

The problem of Laplacian growth is considered within the Loewner-equation framework. A new method of deriving the Loewner equation for a large class of growth problems in the half-plane is presented. The method is based on the Schwarz-Christoffel transformation between the so-called `mathematical planes' at two infinitesimally separated times. Our method not only reproduces the correct Loewner evolution for the case of slit-like fingers but also can be extended to treat more general growth problems. In particular, the Loewner equation for the case of a bubble growing into the half-plane is presented.