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Multiple finger propagation modes in Hele-Shaw channels of variable depth

Published online by Cambridge University Press:  28 March 2014

Alice B. Thompson ,
Anne Juel  and
Andrew L. Hazel
Alice B. Thompson*
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Anne Juel
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Andrew L. Hazel
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
*
†Present address: Department of Mathematics, Imperial College London, London SW7 2AZ, UK. Email address for correspondence: alice.thompson1@imperial.ac.uk
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Abstract

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We consider the propagation of an air finger into a wide fluid-filled channel with a spatially varying depth profile. Our aim is to understand the origin of the multiple coexisting families of both steady and oscillatory propagating fingers previously observed in experiments in axially uniform channels each containing a centred step-like occlusion. We find that a depth-averaged model can reproduce all the finger propagation modes observed experimentally. In addition, the model reveals new modes for symmetric finger propagation. The inclusion of a spatially variable channel depth in the depth-averaged equations leads to: (i) a variable mobility coefficient within the fluid domain due to variations in viscous resistance of the channel; and (ii) a variable transverse curvature term in the dynamic boundary condition that modifies the pressure jump over the air–liquid interface. We use our model to examine the roles of these two distinct effects and find that both contribute to the steady bifurcation structure, while the transverse curvature term is responsible for the distinctive oscillatory propagation modes.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Drops and Bubbles: Bubble dynamics Low-Reynolds-number flows: Hele-Shaw flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 746 , 10 May 2014 , pp. 123 - 164
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© 2014 Cambridge University Press

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