# Numerical solution to the Saffman-Taylor finger problem with kinetic undercooling regularisation

Dallaston, Michael C. & McCue, Scott W.
(2011)
Numerical solution to the Saffman-Taylor finger problem with kinetic undercooling regularisation.
*ANZIAM Journal*, *52*, C124-C138.

## Abstract

The Saffman-Taylor finger problem is to predict the shape and,in particular, width of a finger of fluid travelling in a Hele-Shaw cell filled with a different, more viscous fluid. In experiments the width is dependent on the speed of propagation of the finger, tending to half the total cell width as the speed increases. To predict this result mathematically, nonlinear effects on the fluid interface must be considered; usually surface tension is included for this purpose. This makes the mathematical problem suffciently diffcult that asymptotic or numerical methods must be used. In this paper we adapt numerical methods used to solve the Saffman-Taylor finger problem with surface tension to instead include the effect of kinetic undercooling, a regularisation effect important in Stefan melting-freezing problems, for which Hele-Shaw flow serves as a leading order approximation when the specific heat of a substance is much smaller than its latent heat. We find the existence of a solution branch where the finger width tends to zero as the propagation speed increases, disagreeing with some aspects of the asymptotic analysis of the same problem. We also find a second solution branch, supporting the idea of a countably infinite number of branches as with the surface tension problem.

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ID Code: | 41536 |
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Item Type: | Journal Article |

Keywords: | Saffman-Taylor instability, Hele-Shaw flow, viscous fingering, kinetic undercooling, selection problem, boundary integral method, conformal mapping |

ISSN: | 1446-8735 |

Subjects: | Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200) > Theoretical and Applied Mechanics (010207) Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > NUMERICAL AND COMPUTATIONAL MATHEMATICS (010300) > Numerical Solution of Differential and Integral Equations (010302) |

Divisions: | Past > QUT Faculties & Divisions > Faculty of Science and Technology Past > Schools > Mathematical Sciences |

Copyright Owner: | Copyright 2011 Please consult the authors. |

Deposited On: | 04 May 2011 01:40 |

Last Modified: | 01 May 2012 00:30 |

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