159 research outputs found
Onsager approach to the one-dimensional solidification problem and its relation to the phase-field description
We give a general phenomenological description of the steady-state 1D front propagation problem in two cases: the solidification of a pure material and the isothermal solidification of two-component dilute alloys. The solidification of a pure material is controlled by the heat transport in the bulk and the interface kinetics. The isothermal solidification of two-component alloys is controlled by the diffusion in the bulk and the interface kinetics. We find that the condition of positive-definiteness of the symmetric Onsager matrix of interface kinetic coefficients still allows an arbitrary sign of the slope of the velocity-concentration line near the solidus in the alloy problem or of the velocity-temperature line in the case of solidification of a pure material. This result offers a very simple and elegant way to describe the interesting phenomenon of a possible non-single-value behavior of velocity versus concentration that has previously been discussed by different approaches. We also discuss the relation of this Onsager approach to the thin-interface limit of the phase-field description
Anomalous Dynamic Scaling in Locally-Conserved Coarsening of Fractal Clusters
We report two-dimensional phase-field simulations of locally-conserved
coarsening dynamics of random fractal clusters with fractal dimension D=1.7 and
1.5. The correlation function, cluster perimeter and solute mass are measured
as functions of time. Analyzing the correlation function dynamics, we identify
two different time-dependent length scales that exhibit power laws in time. The
exponents of these power laws are independent of D, one of them is apparently
the classic exponent 1/3. The solute mass versus time exhibits dynamic scaling
with a D-dependent exponent, in agreement with a simple scaling theory.Comment: 5 pages, 4 figure
Smectic Liquid Crystals: Materials with One-Dimensional, Periodic Order
Smectic liquid crystals are materials formed by stacking deformable, fluid
layers. Though smectics prefer to have flat, uniformly-spaced layers, boundary
conditions can impose curvature on the layers. Since the layer spacing and
curvature are intertwined, the problem of finding minimal configurations for
the layers becomes highly nontrivial. We discuss various topological and
geometrical aspects of these materials and present recent progress on finding
some exact layer configurations. We also exhibit connections to the study of
certain embedded minimal surfaces and briefly summarize some important open
problems.Comment: 16 page
A model for interacting instabilities and texture dynamics of patterns
A simple model to study interacting instabilities and textures of resulting
patterns for thermal convection is presented. The model consisting of
twelve-mode dynamical system derived for periodic square lattice describes
convective patterns in the form of stripes and patchwork quilt. The interaction
between stationary zig-zag stripes and standing patchwork quilt pattern leads
to spatiotemporal patterns of twisted patchwork quilt. Textures of these
patterns, which depend strongly on Prandtl number, are investigated numerically
using the model. The model also shows an interesting possibility of a
multicritical point, where stability boundaries of four different structures
meet.Comment: 4 pages including 4 figures, page width revise
Viscous fingering in liquid crystals: Anisotropy and morphological transitions
We show that a minimal model for viscous fingering with a nematic liquid
crystal in which anisotropy is considered to enter through two different
viscosities in two perpendicular directions can be mapped to a two-fold
anisotropy in the surface tension. We numerically integrate the dynamics of the
resulting problem with the phase-field approach to find and characterize a
transition between tip-splitting and side-branching as a function of both
anisotropy and dimensionless surface tension. This anisotropy dependence could
explain the experimentally observed (reentrant) transition as temperature and
applied pressure are varied. Our observations are also consistent with previous
experimental evidence in viscous fingering within an etched cell and
simulations of solidification.Comment: 12 pages, 3 figures. Submitted to PR
Dynamical Systems approach to Saffman-Taylor fingering. A Dynamical Solvability Scenario
A dynamical systems approach to competition of Saffman-Taylor fingers in a
channel is developed. This is based on the global study of the phase space
structure of the low-dimensional ODE's defined by the classes of exact
solutions of the problem without surface tension. Some simple examples are
studied in detail, and general proofs concerning properties of fixed points and
existence of finite-time singularities for broad classes of solutions are
given. The existence of a continuum of multifinger fixed points and its
dynamical implications are discussed. The main conclusion is that exact
zero-surface tension solutions taken in a global sense as families of
trajectories in phase space spanning a sufficiently large set of initial
conditions, are unphysical because the multifinger fixed points are
nonhyperbolic, and an unfolding of them does not exist within the same class of
solutions. Hyperbolicity (saddle-point structure) of the multifinger fixed
points is argued to be essential to the physically correct qualitative
description of finger competition. The restoring of hyperbolicity by surface
tension is discussed as the key point for a generic Dynamical Solvability
Scenario which is proposed for a general context of interfacial pattern
selection.Comment: 3 figures added, major rewriting of some sections, submitted to Phys.
Rev.
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