Dynamics of Conformal Maps for a Class of Non-Laplacian Growth Phenomena

Abstract

Time-dependent conformal maps are used to model a class of growth phenomena limited by coupled non-Laplacian transport processes, such as nonlinear diffusion, advection, and electro-migration. Both continuous and stochastic dynamics are described by generalizing conformal-mapping techniques for viscous fingering and diffusion-limited aggregation, respectively. A general notion of time in stochastic growth is also introduced. The theory is applied to simulations of advection-diffusion-limited aggregation in a background potential flow. A universal crossover in morphology is observed from diffusion-limited to advection-limited fractal patterns with an associated crossover in the growth rate, controlled by a time-dependent effective Peclet number. Remarkably, the fractal dimension is not affected by advection, in spite of dramatic increases in anisotropy and growth rate, due to the persistence of diffusion limitation at small scales.

4 pages, 2 figures (six color plates)

References in corpus (3)

• Laplacian growth as one-dimensional turbulence
• Fractal to Nonfractal Phase Transition in the Dielectric Breakdown Model
• Quasi-Static Fractures in Disordered Media and Iterated Conformal Maps

Cited by in corpus (14)

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• Asymmetric collapse by dissolution or melting in a uniform flow
• Interfacial dynamics in transport-limited dissolution
• Electro-osmotic Instability of Concentration Enrichment in Curved Geometries for an Aqueous Electrolyte
• Diffusion-Limited Aggregation on Curved Surfaces
• The Average Shape of Transport-Limited Aggregates
• Growth morphology and symmetry selection of interfacial instabilities in anisotropic environments
• Growth instability due to lattice-induced topological currents in limited mobility epitaxial growth models
• Fractal Structure of Hastings-Levitov Patterns Restricted in a Sector Geometry
• Conformal mapping of some non-harmonic functions in transport theory