Saffman-Taylor fingers with kinetic undercooling

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

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


© 2015 American Physical Society. 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 (US)
JournalPhysical Review E
Issue number2
StatePublished - Feb 23 2015
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: SWM acknowledges the support of the Australian Research Council via the Discovery Project DP140100933. MD acknowledges support in part by Award No KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). The authors acknowledge helpful discussions with John King and Jon Chapman.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.


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