The Average Shape of Transport-Limited Aggregates

Abstract

We study the relation between stochastic and continuous transport-limited growth models, which generalize conformal-mapping formulations of diffusion-limited aggregation (DLA) and viscous fingering, respectively. We derive a nonlinear integro-differential equation for the asymptotic shape (average conformal map) of stochastic aggregates, whose mean-field approximation is the corresponding continuous equation, where the interface moves at its local expected velocity. Our equation accurately describes advection-diffusion-limited aggregation (ADLA), and, due to nonlinear averaging over fluctuations, the average ADLA cluster is similar, but not identical, to an exact solution of the mean-field dynamics. Similar results should apply to all models in our class, thus explaining the known discrepancies between average DLA clusters and viscous fingers in a channel geometry.

References in corpus (6)

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• Dynamics of Conformal Maps for a Class of Non-Laplacian Growth Phenomena
• Steady advection-diffusion around finite absorbers in two-dimensional potential flows
• Renormalization Theory of Stochastic Growth
• Diffusion-limited aggregation in channel geometry
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Cited by in corpus (4)

• Exact solutions and physical analogies for unidirectional flows
• Asymmetric collapse by dissolution or melting in a uniform flow
• Interfacial dynamics in transport-limited dissolution
• Diffusion-Limited Aggregation on Curved Surfaces