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Interfacial wave theory for oscillatory finger formation in a Hele–Shaw cell: a comparison with experiments

Published online by Cambridge University Press:  26 September 2008

Jian-Jun Xu
Jian-Jun Xu
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada H3A 2K6
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Abstract

This paper is devoted to an analysis of the formation of oscillatory viscous fingers in a Hele-Shaw cell on the basis of the interfacial wave theory, previously established for the pattern formation dynamics in dendrite growth, as well as in the classic Saffman–Taylor flow. In particular, we study the problem of selection and persistence of oscillatory fingers with a tiny bubble at the finger tip. We obtain uniformly valid asymptotic solutions for this problem, and derive the linear, global wave instability mechanism for this more complicated system. The global, neutrally stable modes are computed in a large region of parameters, which select the form of oscillatory fingers in the later stage of evolution. We have compared the theoretical predictions with the experimental data by Couder et al. (1986) and by Kopf-Sill & Homsy (1987), and found excellent quantitative agreement.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 7 , Issue 2 , April 1996 , pp. 169 - 199
Copyright
Copyright © Cambridge University Press 1996

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