Saffman-Taylor streamers: mutual finger interaction in spark formation

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 regularization 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 penalize high velocities and prevent blow-up of the unregularized 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 analog 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 analog 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.

Interaction of streamers and stationary corrugated ionization waves in semiconductors

A numerical simulation of evolution of an identical interacting streamers array in semiconductors has been performed using the diffusion-drift approximation and taking into account the impact and tunnel ionization. It has been assumed that the external electric field E0 is static and uniform, the background electrons and holes are absent, the initial avalanches start simultaneously from the nodes of the plane hexagonal lattice, which is perpendicular to the external field, but the avalanches and streamers are axially symmetric within a cylinder of radius R. It has been shown that under certain conditions, the interaction between the streamers leads finally either to the formation of two types of stationary ionization waves with corrugated front or to a stationary plane ionization wave. A diagram of different steady states of this type of waves in the plane of parameter E0,R has been presented, and a qualitative explanation of the plane partition into four different regions has been given. Characteristics of corrugated waves have been studied in detail and discussed in the region of R and E0 large values, in which the maximum field strength at the front is large enough for the tunnel ionization implementation. It has been shown that corrugated waves ionize semiconductors more efficiently than flat ones, especially in relatively weak external fields.

Electron density fluctuations accelerate the branching of positive streamer discharges in air

Branching is an essential element of streamer discharge dynamics. We review the current state of theoretical understanding and recall that branching requires a finite perturbation. We argue that, in current laboratory experiments in ambient or artificial air, these perturbations can only be inherited from the initial state, or they can be due to intrinsic electron-density fluctuations owing to the discreteness of electrons. We incorporate these electron-density fluctuations into fully three-dimensional simulations of a positive streamer in air at standard temperature and pressure. We derive a quantitative estimate for the ratio of branching length to streamer diameter that agrees within a factor of 2 with experimental measurements. As branching without this noise would occur considerably later, if at all, we conclude that the intrinsic stochastic particle noise triggers branching of positive streamers in air at atmospheric pressure.

Moving-boundary approximation for curved streamer ionization fronts: numerical tests

Recently a moving boundary approximation for the minimal model for negative streamer ionization fronts was extended with effects due to front curvature; this was done through a systematic solvability analysis. A central prediction of this analysis is the existence of a nonvanishing electric field in the streamer interior, whose value is proportional to the front curvature. In this paper we compare this result and other predictions of the solvability analysis with numerical simulations of the minimal model.

Moving boundary approximation for curved streamer ionization fronts: solvability analysis

The minimal density model for negative streamer ionization fronts is investigated. An earlier moving boundary approximation for this model consisted of a "kinetic undercooling" type boundary condition in a Laplacian growth problem of Hele-Shaw type. Here we derive a curvature correction to the moving boundary approximation that resembles surface tension. The calculation is based on solvability analysis with unconventional features, namely, there are three relevant zero modes of the adjoint operator, one of them diverging; furthermore, the inner-outer matching ahead of the front must be performed on a line rather than on an extended region; and the whole calculation can be performed analytically. The analysis reveals a relation between the fields ahead and behind a slowly evolving curved front, the curvature and the generated conductivity. This relation forces us to give up the ideal conductivity approximation, and we suggest to replace it by a charge neutrality approximation. This implies that the electric potential in the streamer interior is no longer constant but solves a Laplace equation; this leads to a Muskat-type problem.

Multiple steady bubbles in a Hele-Shaw cell

This paper presents new solutions, in analytical form, for the shapes of an assembly of steady co-travelling bubbles in a Hele-Shaw cell. The associated velocity field is also derived. The assembly can consist of any finite number of bubbles. The solutions are expressed in terms of Schottky–Klein prime functions and are derived by exploiting results on multiply connected conformal slit mappings.

Interaction of streamer discharges in air and other oxygen-nitrogen mixtures

The interaction of streamers in nitrogen-oxygen mixtures such as air is studied. First, an efficient method for fully three-dimensional streamer simulations in multiprocessor machines is introduced. With its help, we find two competing mechanisms how two adjacent streamers can interact: through electrostatic repulsion and through attraction due to nonlocal photoionization. The nonintuitive effects of pressure and of the nitrogen-oxygen ratio are discussed. As photoionization is experimentally difficult to access, we finally suggest to measure it indirectly through streamer interactions.