Saffman-Taylor fingers with kinetic undercooling

Bennett P.J. Gardiner, Scott W. McCue, Michael C. Dallaston, Timothy J. Moroney

Research output: Contribution to journalArticle

6 Citations (Scopus)

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.

Original languageEnglish
Article number023016
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume91
Issue number2
DOIs
Publication statusPublished - 23 Feb 2015
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Condensed Matter Physics

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