Conformal mapping methods for interfacial dynamics

TL;DR

This paper reviews conformal mapping techniques for modeling interfacial dynamics across various physical phenomena, highlighting recent advances and pedagogical insights for graduate students in related fields.

Contribution

It provides a comprehensive pedagogical overview of conformal mapping methods applied to interfacial dynamics, including recent developments and extensions to curved surfaces.

Findings

Unified framework for harmonic, bi-harmonic, and non-harmonic field-driven interfacial problems

Application of iterated conformal maps to stochastic interfacial phenomena

Extension of models to curved surfaces via auxiliary conformal mappings

Abstract

The article provides a pedagogical review aimed at graduate students in materials science, physics, and applied mathematics, focusing on recent developments in the subject. Following a brief summary of concepts from complex analysis, the article begins with an overview of continuous conformal-map dynamics. This includes problems of interfacial motion driven by harmonic fields (such as viscous fingering and void electromigration), bi-harmonic fields (such as viscous sintering and elastic pore evolution), and non-harmonic, conformally invariant fields (such as growth by advection-diffusion and electro-deposition). The second part of the article is devoted to iterated conformal maps for analogous problems in stochastic interfacial dynamics (such as diffusion-limited aggregation, dielectric breakdown, brittle fracture, and advection-diffusion-limited aggregation). The third part notes that all of these models can be extended to curved surfaces by an auxilliary conformal mapping from the complex plane, such as stereographic projection to a sphere. The article concludes with an outlook for further research.