Saffman-Taylor fingers with kinetic undercooling
arXiv:1502.02334 · doi:10.1103/PhysRevE.91.023016
Abstract
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.
10 pages, 6 figures, accepted for publication by Physical Review E
References in corpus (6)
- What is the apparent angle of a Kelvin ship wave pattern?
- Saffman-Taylor streamers: mutual finger interaction in an electric breakdown
- Jacobian-free Newton-Krylov methods with GPU acceleration for computing nonlinear ship wave patterns
- Construction and test of a moving boundary model for negative streamer discharges
- Convective stabilization of a Laplacian moving boundary problem with kinetic undercooling
- A moving boundary problem motivated by electric breakdown: I. Spectrum of linear perturbations