565 research outputs found
Laplacian transfer across a rough interface: Numerical resolution in the conformal plane
We use a conformal mapping technique to study the Laplacian transfer across a
rough interface. Natural Dirichlet or Von Neumann boundary condition are simply
read by the conformal map. Mixed boundary condition, albeit being more complex
can be efficiently treated in the conformal plane. We show in particular that
an expansion of the potential on a basis of evanescent waves in the conformal
plane allows to write a well-conditioned 1D linear system. These general
principle are illustrated by numerical results on rough interfaces
Chemical etching of a disordered solid: from experiments to field theory
We present a two-dimensional theoretical model for the slow chemical
corrosion of a thin film of a disordered solid by suitable etching solutions.
This model explain different experimental results showing that the corrosion
stops spontaneously in a situation in which the concentration of the etchant is
still finite while the corrosion surface develops clear fractal features. We
show that these properties are strictly related to the percolation theory, and
in particular to its behavior around the critical point. This task is
accomplished both by a direct analysis in terms of a self-organized version of
the Gradient Percolation model and by field theoretical arguments.Comment: 7 pages, 3 figure
Transfer across Random versus Deterministic Fractal Interfaces
A numerical study of the transfer across random fractal surfaces shows that
their responses are very close to the response of deterministic model
geometries with the same fractal dimension. The simulations of several
interfaces with prefractal geometries show that, within very good
approximation, the flux depends only on a few characteristic features of the
interface geometry: the lower and higher cut-offs and the fractal dimension.
Although the active zones are different for different geometries, the electrode
reponses are very nearly the same. In that sense, the fractal dimension is the
essential "universal" exponent which determines the net transfer.Comment: 4 pages, 6 figure
Power laws statistics of cliff failures, scaling and percolation
The size of large cliff failures may be described in several ways, for
instance considering the horizontal eroded area at the cliff top and the
maximum local retreat of the coastline. Field studies suggest that, for large
failures, the frequencies of these two quantities decrease as power laws of the
respective magnitudes, defining two different decay exponents. Moreover, the
horizontal area increases as a power law of the maximum local retreat,
identifying a third exponent. Such observation suggests that the geometry of
cliff failures are statistically similar for different magnitudes. Power laws
are familiar in the physics of critical systems. The corresponding exponents
satisfy precise relations and are proven to be universal features, common to
very different systems. Following the approach typical of statistical physics,
we propose a "scaling hypothesis" resulting in a relation between the three
above exponents: there is a precise, mathematical relation between the
distributions of magnitudes of erosion events and their geometry. Beyond its
theoretical value, such relation could be useful for the validation of field
catalogs analysis. Pushing the statistical physics approach further, we develop
a numerical model of marine erosion that reproduces the observed failure
statistics. Despite the minimality of the model, the exponents resulting from
extensive numerical simulations fairly agree with those measured on the field.
These results suggest that the mathematical theory of percolation, which lies
behind our simple model, can possibly be used as a guide to decipher the
physics of rocky coast erosion and could provide precise predictions to the
statistics of cliff collapses.Comment: 20 pages, 13 figures, 1 table. To appear in Earth Surface Processes
and Lanforms (Rocky Coast special issue
Chemical fracture and distribution of extreme values
When a corrosive solution reaches the limits of a solid sample, a chemical
fracture occurs. An analytical theory for the probability of this chemical
fracture is proposed and confirmed by extensive numerical experiments on a two
dimensional model. This theory follows from the general probability theory of
extreme events given by Gumbel. The analytic law differs from the Weibull law
commonly used to describe mechanical failures for brittle materials. However a
three parameters fit with the Weibull law gives good results, confirming the
empirical value of this kind of analysis.Comment: 7 pages, 5 figures, to appear in Europhysics Letter
Explicit Construction of the Brownian Self-Transport Operator
Applying the technique of characteristic functions developped for
one-dimensional regular surfaces (curves) with compact support, we obtain the
distribution of hitting probabilities for a wide class of finite membranes on
square lattice. Then we generalize it to multi-dimensional finite membranes on
hypercubic lattice. Basing on these distributions, we explicitly construct the
Brownian self-transport operator which governs the Laplacian transfer. In order
to verify the accuracy of the distribution of hitting probabilities, numerical
analysis is carried out for some particular membranes.Comment: 30 pages, 9 figures, 1 tabl
Heat Transport through Rough Channels
We investigate the two-dimensional transport of heat through viscous flow
between two parallel rough interfaces with a given fractal geometry. The flow
and heat transport equations are solved through direct numerical simulations,
and for different conduction-convection conditions. Compared with the behavior
of a channel with smooth interfaces, the results for the rough channel at low
and moderate values of the Peclet number indicate that the effect of roughness
is almost negligible on the efficiency of the heat transport system. This is
explained here in terms of the Makarov's theorem, using the notion of active
zone in Laplacian transport. At sufficiently high Peclet numbers, where
convection becomes the dominant mechanism of heat transport, the role of the
interface roughness is to generally increase both the heat flux across the wall
as well as the active length of heat exchange, when compared with the smooth
channel. Finally, we show that this last behavior is closely related with the
presence of recirculation zones in the reentrant regions of the fractal
geometry.Comment: 12 pages, 8 figure
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