title: Saffman-Taylor fingers with kinetic undercooling.
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 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.
authors.affiliation: Mathematical Sciences, Queensland University of Technology, Brisbane QLD 4001, Australia.
authors.forename: Bennett P J
authors.initials: BP
authors.lastname: Gardiner
authors.affiliation: Mathematical Sciences, Queensland University of Technology, Brisbane QLD 4001, Australia.
authors.forename: Scott W
authors.initials: SW
authors.lastname: McCue
authors.affiliation: Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Oxford OX2 6GG, United Kingdom.
authors.forename: Michael C
authors.initials: MC
authors.lastname: Dallaston
authors.affiliation: Mathematical Sciences, Queensland University of Technology, Brisbane QLD 4001, Australia.
authors.forename: Timothy J
authors.initials: TJ
authors.lastname: Moroney