Convective Stabilization of a Laplacian Moving Boundary Problem with Kinetic Undercooling

Abstract

Abstract. 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 t → ∞. 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.

A moving boundary problem motivated by electric breakdown, I: Spectrum of linear perturbations

“…A simple steady solution to the streamer equations given above is a circle translating with constant velocity determined by E ∞ [35,36]. Though such circles differ from proper streamers, which are growing channels of ionized matter [33,[38][39][40], their front half closely resembles the head of the streamer where the growth takes place.…”

“…In [36], the equation L β = 0 was solved as an initial value problem for the special value = 1. It was found that any initial perturbation β(ω, 0) holomorphic in U ⊃ U ω for τ → ∞ is exponentially convergent to some constant.…”

“…where (36) wheref 3 (ω) is analytic at ω = 1. Using this form of f 3 (ω), we exclude the case 1 − λ ∈ Z + , that will be discussed later.…”

Nonlinear Dynamics of Surface Steps

“…A similar scenario combining linear stability with nonlinear instability was previously found in the problem of two-dimensional void migration [18,25] as well as in the dynamics of ionization fronts [26,27]. The effects of crystalline anisotropy in this problem have not been explored so far.…”

On travelling-wave solutions for a moving boundary problem of Hele-Shaw type

“…Furthermore, (1.1) is also discussed as a model for certain electrical discharge processes [2,8,9]. Such discharge processes are observed in various natural and experimental settings, e.g.…”