Conformal dynamics of fractal growth patterns without randomness

Davidovitch, Benny; Feigenbaum, Mitchell J.; Hentschel, H. G. E.; Procaccia, Itamar

doi:10.1103/physreve.62.1706

Cited by 20 publications

(32 citation statements)

References 16 publications

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“…Recently, Hastings & Levitov (1998) proposed an analogous method to describe DLA using iterated conformal maps, which initiated a flurry of activity applying conformal mapping to Laplacian fractal-growth phenomena (Davidovitch et al 1999(Davidovitch et al , 2000Barra et al 2002aBarra et al , 2002bStepanov & Levitov 2001;Hastings 2001;Somfai et al 1999;Ball & Somfai 2002). One of our motivations here is to extend such powerful analytical methods to fractal growth phenomena limited by non-Laplacian transport processes.…”

Section: Introductionmentioning

confidence: 99%

Conformal mapping of some non-harmonic functions in transport theory

Bazant

2004

Proc. R. Soc. Lond. A

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Conformal mapping has been applied mostly to harmonic functions, i.e. solutions of Laplace's equation. In this paper, it is noted that some other equations are also conformally invariant and thus equally well suited for conformal mapping in two dimensions. In physics, these include steady states of various nonlinear diffusion equations, the advection-diffusion equations for potential flows, and the Nernst-Planck equations for bulk electrochemical transport. Exact solutions for complicated geometries are obtained by conformal mapping to simple geometries in the usual way. Novel examples include nonlinear advection-diffusion layers around absorbing objects and concentration polarizations in electrochemical cells. Although some of these results could be obtained by other methods, such as Boussinesq's streamline coordinates, the present approach is based on a simple unifying principle of more general applicability. It reveals a basic geometrical equivalence of similarity solutions for a broad class of transport processes and paves the way for new applications of conformal mapping, e.g. to non-Laplacian fractal growth.

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Section: Introductionmentioning

confidence: 99%

Conformal mapping of some non-harmonic functions in transport theory

Bazant

2004

Proc. R. Soc. Lond. A

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“…In the context of Laplacian growth, for instance, important progress has been made towards an understanding of the generation of the anomalous dimensions of fractal growth. [30][31][32][33][34][35][36] The understanding of the anomalous exponents characterizing Laplacian growth has escaped a controlled renormalization group handling though, since the problem has an infinite upper critical dimension. This point, and the very concept of universality classes in interface growth, remain as some of the outstanding challenges in nonequilibrium physics, but we will not pursue this line here.…”

Section: Introductionmentioning

confidence: 99%

Viscous fingering as a paradigm of interfacial pattern formation: Recent results and new challenges

Casademunt

2004

Chaos: An Interdisciplinary Journal of Nonlinear Science

149

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We review recent results on dynamical aspects of viscous fingering. The Saffman-Taylor instability is studied beyond linear stability analysis by means of a weakly nonlinear analysis and the exact determination of the subcritical branch. A series of contributions pursuing the idea of a dynamical solvability scenario associated to surface tension in analogy with the traditional selection theory is put in perspective and discussed in the light of the asymptotic theory of Tanveer and co-workers. The inherently dynamical singular effects of surface tension are clarified. The dynamical role of viscosity contrast is explored numerically. We find that the basin of attraction of the Saffman-Taylor finger depends on viscosity contrast, and that the sensitivity to this parameter is maximal in the usual limit of high viscosity contrast. The competing attractors are identified as closed bubble solutions. We briefly report on recent results and work in progress concerning rotating Hele-Shaw flows, topological singularities and wetting effects, and also discuss future directions in the context of viscous fingering. © 2004 American Institute of Physics. ͓DOI: 10.1063/1.1784931͔The study of viscous fingering in a Hele-Shaw cell is a longstanding problem which has become an archetype of interfacial pattern formation, but continues to bring up new surprises which challenge our understanding of nonlocal, nonlinear pattern dynamics of interfaces. The problem refers to the dynamics of the interface between two immiscible viscous fluids confined in a quasi-twodimensional geometry, the Hele-Shaw cell, leading to pattern formation through a morphological instability. In this article we briefly review some recent developments on the dynamics of fingering patterns. We discuss the effects of surface tension as a singular perturbation showing that the problem with and without surface tension are essentially different. Within a dynamical systems approach, we describe how the introduction of surface tension dramatically modifies the global "topological… structure of the phase space flow of the system. We also address, in more detail, the effect of varying the parameter viscosity contrast. We show that the dynamics of fingered structures is highly sensitive to this parameter, and that the long time asymptotics is dominated by the competition between the usual Saffman-Taylor single-finger stationary solution and other attractors defined by closed bubbles. In this context and also taking into account recent results on rotating Hele-Shaw flows, we discuss future perspectives in the field concerning the existence of topological singularities in the form of interface pinchoff, wetting effects and applications to other problems such interface roughening in fluid invasion of porous media.

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“…We have presented an alternative new approach for finding the wanted conformal transformation, given in terms of a functional iteration of fundamental conformal maps. The use of iterated conformal maps was pioneered by Hastings and Levitov [38]; it was subsequently turned into a powerful tool for the study of fractal and fracture growth patterns [39][40][41][42][43][44][45]. In the next subsection we describe how, given a crack shape, to construct a conformal map from the complex ω-plane to the physical z-plane such that the conformal map z = Φ(ω) maps the exterior of the unit circle in the ω-plane to the exterior of the crack in the physical z-plane, after n directed growth steps.…”

Section: A Modelmentioning

confidence: 99%

Statistical Physics of Fracture Surfaces Morphology

2006

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Experiments on fracture surface morphologies offer increasing amounts of data that can be analyzed using methods of statistical physics. One finds scaling exponents associated with correlation and structure functions, indicating a rich phenomenology of anomalous scaling. We argue that traditional models of fracture fail to reproduce this rich phenomenology and new ideas and concepts are called for. We present some recent models that introduce the effects of deviations from homogeneous linear elasticity theory on the morphology of fracture surfaces, succeeding to reproduce the multiscaling phenomenology at least in 1+1 dimensions. For surfaces in 2+1 dimensions we introduce novel methods of analysis based on projecting the data on the irreducible representations of the SO(2) symmetry group. It appears that this approach organizes effectively the rich scaling properties. We end up with the proposition of new experiments in which the rotational symmetry is not broken, such that the scaling properties should be particularly simple.It is a privilege to dedicate this paper to Pierre Hohenberg and Jim Langer who contributed, respectively, decisive ideas to scaling concepts in statistical physics and to the study of fracture. It is our hope that the marriage of these two issues in the present paper may give them some degree of pleasure.

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Conformal dynamics of fractal growth patterns without randomness

Cited by 20 publications

References 16 publications

Conformal mapping of some non-harmonic functions in transport theory

Conformal mapping of some non-harmonic functions in transport theory

Viscous fingering as a paradigm of interfacial pattern formation: Recent results and new challenges

Statistical Physics of Fracture Surfaces Morphology

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