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 $t\to\infty$. 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

Construction and test of a moving boundary model for negative streamer discharges

Starting from the minimal model for the electrically interacting particle densities in negative streamer discharges, we derive a moving boundary approximation for the ionization fronts. The boundary condition on the moving front is found to be of 'kinetic undercooling' type. The boundary approximation, whose first results have been published in [Meulenbroek et al., PRL 95, 195004 (2005)], is then tested against 2-dimensional simulations of the density model. The results suggest that our moving boundary approximation adequately represents the essential dynamics of negative streamer fronts.Comment: 10 pages, 7 figures; submitted to Phys. Rev.

Weakly nonlinear subcritical instability of visco-elastic Poiseuille flow

It is well known that the Poiseuille flow of a visco-elastic polymer fluid between plates or through a tube is linearly stable in the zero Reynolds number limit, although the stability is weak for large Weissenberg numbers. In this paper we argue that recent experimental and theoretical work on the instability of visco-elastic fluids in Taylor-Couette cells and numerical work on channel flows suggest a scenario in which Poiseuille flow of visco-elastic polymer fluids exhibits a nonlinear "subcritical" instability due to normal stress effects, with a threshold which decreases for increasing Weissenberg number. This proposal is confirmed by an explicit weakly nonlinear stability analysis for Poiseuille flow of an UCM fluid. Our analysis yields explicit predictions for the critical amplitude of velocity perturbations beyond which the flow is nonlinearly unstable, and for the wavelength of the mode whose critical amplitude is smallest. The nonlinear instability sets in quite abruptly at Weissenberg numbers around 4 in the planar case and about 5.2 in the cylindrical case, so that for Weissenberg numbers somewhat larger than these values perturbations of the order of a few percent in the wall shear stress suffice to make the flow unstable. We have suggested elsewhere that this nonlinear instability could be an important intrinsic route to melt fracture and that preliminary experiments are both qualitatively and quantitatively in good agreement with these predictions.Comment: 20 pages, 16 figures. Accepted for publication in J. of Non-Newtonian Fluid Mechanic

Rigorous stability results for 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 t ! 1. 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 solution

Regularization of moving boundaries in a Laplacian field by a mixed Dirichlet-Neumann boundary condition : exact results

The dynamics of ionization fronts that generate a conducting body, are in simplest approximation equivalent to viscous fingering without regularization. Going beyond this approximation, we suggest that ionization fronts can be modeled by a mixed Dirichlet-Neumann boundary condition. We derive exact uniformly propagating solutions of this problem in 2D and construct a single partial differential equation governing small perturbations of these solutions. For some parameter value, this equation can be solved analytically which shows that the uniformly propagating solution is linearly convectively stabl

Streamer branching rationalized by conformal mapping techniques

Spontaneous branching of discharge channels is frequently observed, but not well understood. We recently proposed a new branching mechanism based on simulations of a simple continuous discharge model in high fields. We here present analytical results for such streamers in the Lozansky-Firsov limit where they can be modelled as moving equipotential ionization fronts. This model can be analyzed by conformal mapping techniques which allow the reduction of the dynamical problem to finite sets of nonlinear ordinary differential equations. The solutions illustrate that branching is generic for the intricate head dynamics of streamers in the Lozansky-Firsov limit

Regularization of moving boundaries in a Laplacian field by a mixed Dirichlet-Neumann boundary condition: exact results

The dynamics of ionization fronts that generate a conducting body, are in simplest approximation equivalent to viscous fingering without regularization. Going beyond this approximation, we suggest that ionization fronts can be modeled by a mixed Dirichlet-Neumann boundary condition. We derive exact uniformly propagating solutions of this problem in 2D and construct a single partial differential equation governing small perturbations of these solutions. For some parameter value, this equation can be solved analytically which shows that the uniformly propagating solution is linearly convectively stable.Comment: 4 pages, 1 figur

Rigorous stability results for 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 t ! 1. 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 solution

Numerical simulations and conformal analysis of growing and branching negative discharge streamers

The dynamics of an anode-directed streamer can be described by advection-diffusion equations for the charged particles, including a local field-dependent impact ionization term, and coupled to the Poisson equation for the electric field. We present the results of new simulations that use a local uniform grid refinement strategy. Even on very fine grids, provided the electric field is high enough, the streamer appears to branch spontaneously. These results are supported by new analytical solutions based on a moving boundary approximatio