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Rigorous results in existence and selection of Saffman–Taylor fingers by kinetic undercooling

Published online by Cambridge University Press:  10 January 2018

XUMING XIE
XUMING XIE*
Affiliation:
Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA email: xuming.xie@morgan.edu
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Abstract

The selection of Saffman–Taylor fingers by surface tension has been extensively investigated. In this paper, we are concerned with the existence and selection of steadily translating symmetric finger solutions in a Hele–Shaw cell by small but non-zero kinetic undercooling (ε2). We rigorously conclude that for relative finger width λ near one half, symmetric finger solutions exist in the asymptotic limit of undercooling ε2 → 0 if the Stokes multiplier for a relatively simple non-linear differential equation is zero. This Stokes multiplier S depends on the parameter $\alpha \equiv \frac{2 \lambda -1}{(1-\lambda)}\epsilon^{-\frac{4}{3}}$ and earlier calculations have shown this to be zero for a discrete set of values of α. While this result is similar to that obtained previously for Saffman–Taylor fingers by surface tension, the analysis for the problem with kinetic undercooling exhibits a number of subtleties as pointed out by Chapman and King (2003, The selection of Saffman–Taylor fingers by kinetic undercooling, Journal of Engineering Mathematics, 46, 1–32). The main subtlety is the behaviour of the Stokes lines at the finger tip, where the analysis is complicated by non-analyticity of coefficients in the governing equation.

Keywords

finger selectionHele–Shawkinetic undercoolingexistenceanalytic solution
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 1 , February 2019 , pp. 63 - 116
Copyright
Copyright © Cambridge University Press 2018 

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