1,271 research outputs found
Conformal mapping methods for interfacial dynamics
The article provides a pedagogical review aimed at graduate students in
materials science, physics, and applied mathematics, focusing on recent
developments in the subject. Following a brief summary of concepts from complex
analysis, the article begins with an overview of continuous conformal-map
dynamics. This includes problems of interfacial motion driven by harmonic
fields (such as viscous fingering and void electromigration), bi-harmonic
fields (such as viscous sintering and elastic pore evolution), and
non-harmonic, conformally invariant fields (such as growth by
advection-diffusion and electro-deposition). The second part of the article is
devoted to iterated conformal maps for analogous problems in stochastic
interfacial dynamics (such as diffusion-limited aggregation, dielectric
breakdown, brittle fracture, and advection-diffusion-limited aggregation). The
third part notes that all of these models can be extended to curved surfaces by
an auxilliary conformal mapping from the complex plane, such as stereographic
projection to a sphere. The article concludes with an outlook for further
research.Comment: 37 pages, 12 (mostly color) figure
Dynamics of Conformal Maps for a Class of Non-Laplacian Growth Phenomena
Time-dependent conformal maps are used to model a class of growth phenomena
limited by coupled non-Laplacian transport processes, such as nonlinear
diffusion, advection, and electro-migration. Both continuous and stochastic
dynamics are described by generalizing conformal-mapping techniques for viscous
fingering and diffusion-limited aggregation, respectively. A general notion of
time in stochastic growth is also introduced. The theory is applied to
simulations of advection-diffusion-limited aggregation in a background
potential flow. A universal crossover in morphology is observed from
diffusion-limited to advection-limited fractal patterns with an associated
crossover in the growth rate, controlled by a time-dependent effective Peclet
number. Remarkably, the fractal dimension is not affected by advection, in
spite of dramatic increases in anisotropy and growth rate, due to the
persistence of diffusion limitation at small scales.Comment: 4 pages, 2 figures (six color plates
Influence of pore-scale disorder on viscous fingering during drainage
We study viscous fingering during drainage experiments in linear Hele-Shaw
cells filled with a random porous medium. The central zone of the cell is found
to be statistically more occupied than the average, and to have a lateral width
of 40% of the system width, irrespectively of the capillary number . A
crossover length separates lower scales where the
invader's fractal dimension is identical to capillary fingering,
and larger scales where the dimension is found to be . The lateral
width and the large scale dimension are lower than the results for Diffusion
Limited Aggregation, but can be explained in terms of Dielectric Breakdown
Model. Indeed, we show that when averaging over the quenched disorder in
capillary thresholds, an effective law relates the
average interface growth rate and the local pressure gradient.Comment: 4 pages, 4 figures, submitted to Phys Rev Letter
Interface growth in two dimensions: A Loewner-equation approach
The problem of Laplacian growth in two dimensions is considered within the
Loewner-equation framework. Initially the problem of fingered growth recently
discussed by Gubiec and Szymczak [T. Gubiec and P. Szymczak, Phys. Rev. E 77,
041602 (2008)] is revisited and a new exact solution for a three-finger
configuration is reported. Then a general class of growth models for an
interface growing in the upper-half plane is introduced and the corresponding
Loewner equation for the problem is derived. Several examples are given
including interfaces with one or more tips as well as multiple growing
interfaces. A generalization of our interface growth model in terms of
``Loewner domains,'' where the growth rule is specified by a time evolving
measure, is briefly discussed.Comment: To appear in Physical Review
Modelling the evaporation of thin films of colloidal suspensions using Dynamical Density Functional Theory
Recent experiments have shown that various structures may be formed during
the evaporative dewetting of thin films of colloidal suspensions. Nano-particle
deposits of strongly branched `flower-like', labyrinthine and network
structures are observed. They are caused by the different transport processes
and the rich phase behaviour of the system. We develop a model for the system,
based on a dynamical density functional theory, which reproduces these
structures. The model is employed to determine the influences of the solvent
evaporation and of the diffusion of the colloidal particles and of the liquid
over the surface. Finally, we investigate the conditions needed for
`liquid-particle' phase separation to occur and discuss its effect on the
self-organised nano-structures
Interfacial dynamics in transport-limited dissolution
Various model problems of ``transport-limited dissolution'' in two dimensions
are analyzed using time-dependent conformal maps. For diffusion-limited
dissolution (reverse Laplacian growth), several exact solutions are discussed
for the smoothing of corrugated surfaces, including the continuous analogs of
``internal diffusion-limited aggregation'' and ``diffusion-limited erosion''. A
class of non-Laplacian, transport-limited dissolution processes are also
considered, which raise the general question of when and where a finite solid
will disappear. In a case of dissolution by advection-diffusion, a tilted
ellipse maintains its shape during collapse, as its center of mass drifts
obliquely away from the background fluid flow, but other initial shapes have
more complicated dynamics.Comment: 5 pages, 4 fig
Non-Gaussian buoyancy statistics in fingering convection
We examine the statistics of active scalar fluctuations in high-Rayleigh
number fingering convection with high-resolution three-dimensional numerical
experiments. The one-point distribution of buoyancy fluctuations is found to
present significantly non-Gaussian tails.
A modified theory based on an original approach by Yakhot (1989) is used to
model the active scalar distributions as a function of the conditional
expectation values of scalar dissipation and fluxes in the flow. Simple models
for these two quantities highlight the role of blob-like coherent structures
for scalar statistics in fingering convection
- …