272 research outputs found
Iterated Conformal Dynamics and Laplacian Growth
The method of iterated conformal maps for the study of Diffusion Limited
Aggregates (DLA) is generalized to the study of Laplacian Growth Patterns and
related processes. We emphasize the fundamental difference between these
processes: DLA is grown serially with constant size particles, while Laplacian
patterns are grown by advancing each boundary point in parallel, proportionally
to the gradient of the Laplacian field. We introduce a 2-parameter family of
growth patterns that interpolates between DLA and a discrete version of
Laplacian growth. The ultraviolet putative finite-time singularities are
regularized here by a minimal tip size, equivalently for all the models in this
family. With this we stress that the difference between DLA and Laplacian
growth is NOT in the manner of ultraviolet regularization, but rather in their
deeply different growth rules. The fractal dimensions of the asymptotic
patterns depend continuously on the two parameters of the family, giving rise
to a "phase diagram" in which DLA and discretized Laplacian growth are at the
extreme ends. In particular we show that the fractal dimension of Laplacian
growth patterns is much higher than the fractal dimension of DLA, with the
possibility of dimension 2 for the former not excluded.Comment: 13 pages, 12 figures, submitted to Phys. Rev.
Quasi-Static Fractures in Disordered Media and Iterated Conformal Maps
We study the geometrical characteristic of quasi-static fractures in
disordered media, using iterated conformal maps to determine the evolution of
the fracture pattern. This method allows an efficient and accurate solution of
the Lam\'e equations without resorting to lattice models. Typical fracture
patterns exhibit increased ramification due to the increase of the stress at
the tips. We find the roughness exponent of the experimentally relevant
backbone of the fracture pattern; it crosses over from about 0.5 for small
scales to about 0.75 for large scales, in excellent agreement with experiments.
We propose that this cross-over reflects the increased ramification of the
fracture pattern.Comment: submitted to Physical Review Letter
Tip Splittings and Phase Transitions in the Dielectric Breakdown Model: Mapping to the DLA Model
We show that the fractal growth described by the dielectric breakdown model
exhibits a phase transition in the multifractal spectrum of the growth measure.
The transition takes place because the tip-splitting of branches forms a fixed
angle. This angle is eta dependent but it can be rescaled onto an
``effectively'' universal angle of the DLA branching process. We derive an
analytic rescaling relation which is in agreement with numerical simulations.
The dimension of the clusters decreases linearly with the angle and the growth
becomes non-fractal at an angle close to 74 degrees (which corresponds to eta=
4.0 +- 0.3).Comment: 4 pages, REVTex, 3 figure
Fractal to Nonfractal Phase Transition in the Dielectric Breakdown Model
A fast method is presented for simulating the dielectric-breakdown model
using iterated conformal mappings. Numerical results for the dimension and for
corrections to scaling are in good agreement with the recent RG prediction of
an upper critical , at which a transition occurs between branching
fractal clusters and one-dimensional nonfractal clusters.Comment: 5 pages, 7 figures; corrections to scaling include
Spreading of thin films assisted by thermal fluctuations
We study the spreading of viscous drops on a solid substrate, taking into
account the effects of thermal fluctuations in the fluid momentum. A nonlinear
stochastic lubrication equation is derived, and studied using numerical
simulations and scaling analysis. We show that asymptotically spreading drops
admit self-similar shapes, whose average radii can increase at rates much
faster than these predicted by Tanner's law. We discuss the physical
realizability of our results for thin molecular and complex fluid films, and
predict that such phenomenon can in principal be observed in various flow
geometries.Comment: 5 pages, 3 figure
Moving boundary approximation for curved streamer ionization fronts: Solvability analysis
The minimal density model for negative streamer ionization fronts is
investigated. An earlier moving boundary approximation for this model consisted
of a "kinetic undercooling" type boundary condition in a Laplacian growth
problem of Hele-Shaw type. Here we derive a curvature correction to the moving
boundary approximation that resembles surface tension. The calculation is based
on solvability analysis with unconventional features, namely, there are three
relevant zero modes of the adjoint operator, one of them diverging;
furthermore, the inner/outer matching ahead of the front has to be performed on
a line rather than on an extended region; and the whole calculation can be
performed analytically. The analysis reveals a relation between the fields
ahead and behind a slowly evolving curved front, the curvature and the
generated conductivity. This relation forces us to give up the ideal
conductivity approximation, and we suggest to replace it by a constant
conductivity approximation. This implies that the electric potential in the
streamer interior is no longer constant but solves a Laplace equation; this
leads to a Muskat-type problem.Comment: 22 pages, 6 figure
A prototypical model for tensional wrinkling in thin sheets
The buckling and wrinkling of thin films has recently seen a surge of interest among physicists, biologists, mathematicians and engineers. This has been triggered by the growing interest in developing technologies at ever decreasing scales and the resulting necessity to control the mechanics of tiny structures, as well as by the realization that morphogenetic processes, such as the tissue-shaping instabilities occurring in animal epithelia or plant leaves, often emerge from mechanical instabilities of cell sheets. While the most basic buckling instability of uniaxially compressed plates was understood by Euler more than 200 years ago, recent experiments on nanometrically thin (ultrathin) films have shown significant deviations from predictions of standard buckling theory. Motivated by this puzzle, we introduce here a theoretical model that allows for a systematic analysis of wrinkling in sheets far from their instability threshold. We focus on the simplest extension of Euler buckling that exhibits wrinkles of finite length - a sheet under axisymmetric tensile loads. This geometry, whose first study is attributed to Lam´e, allows us to construct\ud
a phase diagram that demonstrates the dramatic variation of wrinkling patterns from near-threshold to far-from-threshold conditions. Theoretical arguments and comparison to experiments show that for thin sheets the far-from-threshold regime is expected to emerge under extremely small compressive loads, emphasizing the relevance of our analysis for nanomechanics applications
Capillary deformations of bendable films
We address the partial wetting of liquid drops on ultrathin solid sheets resting on a deformable foundation. Considering the membrane limit of sheets that can relax compression through wrinkling at negligible energetic cost, we revisit the classical theory for the contact of liquid drops on solids. Our calculations and experiments show that the liquid-solid-vapor contact angle is modified from the Young angle, even though the elastic bulk modulus (E) of the sheet is so large that the ratio between the surface tension γ and E is of molecular size. This finding establishes a new type of “soft capillarity” that stems from the bendability of thin elastic bodies rather than from material softness. We also show that the size of the wrinkle pattern that emerges in the sheet is fully predictable, thus resolving a puzzle noticed in several previous attempts to model “drop-on-a-floating-sheet” experiments, and enabling a reliable usage of this setup for the metrology of ultrathin films
Bi-Laplacian Growth Patterns in Disordered Media
Experiments in quasi 2-dimensional geometry (Hele Shaw cells) in which a
fluid is injected into a visco-elastic medium (foam, clay or
associating-polymers) show patterns akin to fracture in brittle materials, very
different from standard Laplacian growth patterns of viscous fingering. An
analytic theory is lacking since a pre-requisite to describing the fracture of
elastic material is the solution of the bi-Laplace rather than the Laplace
equation. In this Letter we close this gap, offering a theory of bi-Laplacian
growth patterns based on the method of iterated conformal maps.Comment: Submitted to PRL. For further information see
http://www.weizmann.ac.il/chemphys/ander
Quasi-Static Brittle Fracture in Inhomogeneous Media and Iterated Conformal Maps: Modes I, II and III
The method of iterated conformal maps is developed for quasi-static fracture
of brittle materials, for all modes of fracture. Previous theory, that was
relevant for mode III only, is extended here to mode I and II. The latter
require solution of the bi-Laplace rather than the Laplace equation. For all
cases we can consider quenched randomness in the brittle material itself, as
well as randomness in the succession of fracture events. While mode III calls
for the advance (in time) of one analytic function, mode I and II call for the
advance of two analytic functions. This fundamental difference creates
different stress distribution around the cracks. As a result the geometric
characteristics of the cracks differ, putting mode III in a different class
compared to modes I and II.Comment: submitted to PRE For a version with qualitatively better figures see:
http://www.weizmann.ac.il/chemphys/ander
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