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

TL;DR

This paper develops a conformal mapping framework to model non-Laplacian growth phenomena involving complex transport processes, revealing a universal crossover in fractal morphology unaffected by advection.

Contribution

It generalizes conformal-mapping techniques to non-Laplacian growth and introduces a new notion of time in stochastic growth, applied to advection-diffusion-limited aggregation.

Findings

Universal crossover from diffusion-limited to advection-limited fractal patterns.

Fractal dimension remains unchanged despite increased anisotropy and growth rate.

Effective Peclet number governs the crossover in growth morphology.

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.