Conformal mapping methods for interfacial dynamics

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.

37 pages, 12 (mostly color) figures

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Cited by in corpus (8)

• 2D growth processes: SLE and Loewner chains
• Hastings-Levitov aggregation in the small-particle limit
• Asymmetric collapse by dissolution or melting in a uniform flow
• Planar aggregation and the coalescing Brownian flow
• Optimal convection cooling flows in general geometries
• Genus-zero Whitham hierarchies in conformal-map dynamics
• Penguin huddling: a continuum model
• Discrete scale invariance and stochastic Loewner evolution