6 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
Streamers, sprites, leaders, lightning: from micro- to macroscales
"Streamers, sprites, leaders, lightning: from micro- to macroscales" was the
theme of a workshop in October 2007 in Leiden, The Netherlands; it brought
researchers from plasma physics, electrical engineering and industry,
geophysics and space physics, computational science and nonlinear dynamics
together around the common topic of generation, structure and products of
streamer-like electric breakdown. The present cluster issue collects relevant
articles within this area; most of them were presented during the workshop. We
here briefly discuss the research questions and very shortly review the papers
in the cluster issue, and we also refer to a few recent papers in other
journals.Comment: Editorial introduction for the cluster issue on "Streamers, sprites
and lightning" in J. Phys. D, 13 pages, 74 reference
Streamers, sprites, leaders, lightning: from micro- to macroscales
‘Streamers, sprites, leaders, lightning: from micro- to macroscales’ was the theme of a workshop in October 2007 in Leiden, The Netherlands; it brought together researchers from plasma physics, electrical engineering and industry, geophysics and space physics, computational science and nonlinear dynamics around the common topic of generation, structure and products of streamer-like electric breakdown. The present cluster issue collects relevant papers within this area; most of them were presented during the workshop. We here briefly discuss the research questions and very shortly review the papers in the cluster issue, and we also refer to a few recent papers in this and other journals
Corner and finger formation in Hele--Shaw flow with kinetic undercooling regularisation
We examine the effect of a kinetic undercooling condition on the evolution of
a free boundary in Hele--Shaw flow, in both bubble and channel geometries. We
present analytical and numerical evidence that the bubble boundary is unstable
and may develop one or more corners in finite time, for both expansion and
contraction cases. This loss of regularity is interesting because it occurs
regardless of whether the less viscous fluid is displacing the more viscous
fluid, or vice versa. We show that small contracting bubbles are described to
leading order by a well-studied geometric flow rule. Exact solutions to this
asymptotic problem continue past the corner formation until the bubble
contracts to a point as a slit in the limit. Lastly, we consider the evolving
boundary with kinetic undercooling in a Saffman--Taylor channel geometry. The
boundary may either form corners in finite time, or evolve to a single long
finger travelling at constant speed, depending on the strength of kinetic
undercooling. We demonstrate these two different behaviours numerically. For
the travelling finger, we present results of a numerical solution method
similar to that used to demonstrate the selection of discrete fingers by
surface tension. With kinetic undercooling, a continuum of corner-free
travelling fingers exists for any finger width above a critical value, which
goes to zero as the kinetic undercooling vanishes. We have not been able to
compute the discrete family of analytic solutions, predicted by previous
asymptotic analysis, because the numerical scheme cannot distinguish between
solutions characterised by analytic fingers and those which are corner-free but
non-analytic
A moving boundary problem motivated by electric breakdown: I. Spectrum of linear perturbations
An interfacial approximation of the streamer stage in the evolution of sparks
and lightning can be written as a Laplacian growth model regularized by a
`kinetic undercooling' boundary condition. We study the linear stability of
uniformly translating circles that solve the problem in two dimensions. In a
space of smooth perturbations of the circular shape, the stability operator is
found to have a pure point spectrum. Except for the zero eigenvalue for
infinitesimal translations, all eigenvalues are shown to have negative real
part. Therefore perturbations decay exponentially in time. We calculate the
spectrum through a combination of asymptotic and series evaluation. In the
limit of vanishing regularization parameter, all eigenvalues are found to
approach zero in a singular fashion, and this asymptotic behavior is worked out
in detail. A consideration of the eigenfunctions indicates that a strong
intermediate growth may occur for generic initial perturbations. Both the
linear and the nonlinear initial value problem are considered in a second
paper.Comment: 37 pages, 6 figures, revised for Physica
A moving boundary model motivated by electric breakdown: II. Initial value problem
An interfacial approximation of the streamer stage in the evolution of sparks
and lightning can be formulated as a Laplacian growth model regularized by a
'kinetic undercooling' boundary condition. Using this model we study both the
linearized and the full nonlinear evolution of small perturbations of a
uniformly translating circle. Within the linear approximation analytical and
numerical results show that perturbations are advected to the back of the
circle, where they decay. An initially analytic interface stays analytic for
all finite times, but singularities from outside the physical region approach
the interface for , which results in some anomalous relaxation at
the back of the circle. For the nonlinear evolution numerical results indicate
that the circle is the asymptotic attractor for small perturbations, but larger
perturbations may lead to branching. We also present results for more general
initial shapes, which demonstrate that regularization by kinetic undercooling
cannot guarantee smooth interfaces globally in time.Comment: 44 pages, 18 figures, paper submitted to Physica