Numerical solution to the SaffmanTaylor finger problem with kinetic undercooling regularisation
Dallaston, Michael C. & McCue, Scott W. (2011) Numerical solution to the SaffmanTaylor finger problem with kinetic undercooling regularisation. ANZIAM Journal, 52, C124C138.

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Abstract
The SaffmanTaylor finger problem is to predict the shape and,in particular, width of a finger of fluid travelling in a HeleShaw 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 SaffmanTaylor finger problem with surface tension to instead include the effect of kinetic undercooling, a regularisation effect important in Stefan meltingfreezing problems, for which HeleShaw 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 

Item Type:  Journal Article 
Refereed:  Yes 
Keywords:  SaffmanTaylor instability, HeleShaw flow, viscous fingering, kinetic undercooling, selection problem, boundary integral method, conformal mapping 
ISSN:  14468735 
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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