Hentschel HGE, Popescu MN, Family F. Conformal map modeling of the pinning transition in Laplacian growth.
PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002;
65:036141. [PMID:
11909199 DOI:
10.1103/physreve.65.036141]
[Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/14/2001] [Indexed: 05/23/2023]
Abstract
In Laplacian growth processes pinning may be expected due to a nonlinear response of a material during dielectric breakdown, or due to stick-slip boundary conditions in two-fluid flow in a porous medium, while thermal noise will lead to depinning. Using a method recently proposed by Hastings and Levitov, the size R(max) approximately E(-alpha)(c) of the pinned pattern is shown to scale with the critical field E(c) (electric field for dielectric breakdown, pressure gradient for fluid flow). These pinned patterns have a lower effective fractal dimension d(f) than diffusion-limited aggregation due to the enhancement of growth at the hot tips of the developing pattern. At finite temperature, thermal noise leads to depinning and growth of patterns with a shape and dimensionality dependent on both E(c) and the thermal noise. Using multifractal analysis, scaling expressions are established for this dependency.
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