Saffman-Taylor fingers with kinetic undercooling

Gardiner, Bennett P.J., McCue, Scott W., Dallaston, Michael C., & Moroney, Timothy J. (2015) Saffman-Taylor fingers with kinetic undercooling. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 91, 023016-1.

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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.

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ID Code: 82018
Item Type: Journal Article
Refereed: Yes
Keywords: Saffman-Taylor finger, Hele-Shaw flow, viscous fingering, kinetic undercooling, streamer discharges
DOI: 10.1103/PhysRevE.91.023016
ISSN: 1539-3755
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200) > Theoretical and Applied Mechanics (010207)
Divisions: Current > Institutes > Institute for Future Environments
Current > Schools > School of Mathematical Sciences
Current > QUT Faculties and Divisions > Science & Engineering Faculty
Funding:
Copyright Owner: Copyright 2015 American Physical Society
Deposited On: 25 Feb 2015 22:43
Last Modified: 28 Feb 2015 11:08

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