Saffman-Taylor fingers with kinetic undercooling

TL;DR

This paper investigates the shape and selection mechanisms of Saffman-Taylor fingers in Hele-Shaw flows with kinetic undercooling, revealing discrete finger shapes and their relation to surface tension effects through numerical analysis.

Contribution

It provides the first detailed numerical computation of Saffman-Taylor fingers with kinetic undercooling, demonstrating a selection mechanism similar to surface tension and addressing the complexities introduced by non-analytic finger shapes.

Findings

Discrete set of finger shapes depending on kinetic undercooling

Finger shape approaches half the channel width as undercooling vanishes

Numerical evidence linking kinetic undercooling and surface tension effects

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.