1,291 research outputs found
Convective stabilization of a Laplacian moving boundary problem with kinetic undercooling
We study the shape stability of disks moving in an external Laplacian field
in two dimensions. The problem is motivated by the motion of ionization fronts
in streamer-type electric breakdown. It is mathematically equivalent to the
motion of a small bubble in a Hele-Shaw cell with a regularization of kinetic
undercooling type, namely a mixed Dirichlet-Neumann boundary condition for the
Laplacian field on the moving boundary. Using conformal mapping techniques,
linear stability analysis of the uniformly translating disk is recast into a
single PDE which is exactly solvable for certain values of the regularization
parameter. We concentrate on the physically most interesting exactly solvable
and non-trivial case. We show that the circular solutions are linearly stable
against smooth initial perturbations. In the transformation of the PDE to its
normal hyperbolic form, a semigroup of automorphisms of the unit disk plays a
central role. It mediates the convection of perturbations to the back of the
circle where they decay. Exponential convergence to the unperturbed circle
occurs along a unique slow manifold as time . Smooth temporal
eigenfunctions cannot be constructed, but excluding the far back part of the
circle, a discrete set of eigenfunctions does span the function space of
perturbations. We believe that the observed behaviour of a convectively
stabilized circle for a certain value of the regularization parameter is
generic for other shapes and parameter values. Our analytical results are
illustrated by figures of some typical solutions.Comment: 19 pages, 7 figures, accepted for SIAM J. Appl. Mat
Saffman-Taylor fingers with kinetic undercooling
The mathematical model of a steadily propagating Saffman-Taylor finger in a
Hele-Shaw channel has applications to two-dimensional interacting streamer
discharges which are aligned in a periodic array. In the streamer context, the
relevant regularisation on the interface is not provided by surface tension,
but instead has been postulated to involve a mechanism equivalent to kinetic
undercooling, which acts to penalise high velocities and prevent blow-up of the
unregularised solution. Previous asymptotic results for the Hele-Shaw finger
problem with kinetic undercooling suggest that for a given value of the kinetic
undercooling parameter, there is a discrete set of possible finger shapes, each
analytic at the nose and occupying a different fraction of the channel width.
In the limit in which the kinetic undercooling parameter vanishes, the fraction
for each family approaches 1/2, suggesting that this 'selection' of 1/2 by
kinetic undercooling is qualitatively similar to the well-known analogue with
surface tension. We treat the numerical problem of computing these
Saffman-Taylor fingers with kinetic undercooling, which turns out to be more
subtle than the analogue with surface tension, since kinetic undercooling
permits finger shapes which are corner-free but not analytic. We provide
numerical evidence for the selection mechanism by setting up a problem with
both kinetic undercooling and surface tension, and numerically taking the limit
that the surface tension vanishes.Comment: 10 pages, 6 figures, accepted for publication by Physical Review
Multifractality of Brownian motion near absorbing polymers
We characterize the multifractal behavior of Brownian motion in the vicinity
of an absorbing star polymer. We map the problem to an O(M)-symmetric
phi^4-field theory relating higher moments of the Laplacian field of Brownian
motion to corresponding composite operators. The resulting spectra of scaling
dimensions of these operators display the convexity properties which are
necessarily found for multifractal scaling but unusual for power of field
operators in field theory. Using a field-theoretic renormalization group
approach we obtain the multifractal spectrum for absorbtion at the core of a
polymer star as an asymptotic series. We evaluate these series using
resummation techniques.Comment: 18 pages, revtex, 6 ps-figure
Two-dimensional Copolymers and Multifractality: Comparing Perturbative Expansions, MC Simulations, and Exact Results
We analyze the scaling laws for a set of two different species of long
flexible polymer chains joined together at one of their extremities (copolymer
stars) in space dimension D=2. We use a formerly constructed field-theoretic
description and compare our perturbative results for the scaling exponents with
recent conjectures for exact conformal scaling dimensions derived by a
conformal invariance technique in the context of D=2 quantum gravity. A simple
MC simulation brings about reasonable agreement with both approaches. We
analyse the remarkable multifractal properties of the spectrum of scaling
exponents.Comment: 5 page
Interaction of a point charge with the surface of a uniaxial dielectric
We analyze the force on a point charge moving at relativistic speeds parallel
to the surface of a uniaxial dielectric. Two cases are examined: a lossless
dielectric with no dispersion and a dielectric with a plasma type response. The
treatment focuses on the peculiarities of the strength and direction of the
interaction force as compared to the isotropic case. We show that a plasma type
dielectric can, under specific conditions, repel the point charge.Comment: 7 pages, 5 figure
- …