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Modelling the suppression of viscous fingering in elastic-walled Hele-Shaw cells

Published online by Cambridge University Press:  14 August 2013

Draga Pihler-Puzović ,
Raphaël Périllat ,
Matthew Russell ,
Anne Juel  and
Matthias Heil
Draga Pihler-Puzović*
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Raphaël Périllat
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Matthew Russell
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Anne Juel
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Matthias Heil
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
*
†Email address for correspondence: [email protected]
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Abstract

Recent experiments by Pihler-Puzovic et al. (Phys. Rev. Lett., vol. 108, 2012, article 074502) have shown that the onset of viscous fingering in circular Hele-Shaw cells in which an air bubble displaces a viscous fluid is delayed considerably when the top boundary of the cell is replaced by an elastic membrane. Non-axisymmetric instabilities are only observed at much larger flow rates, and the large-amplitude fingers that develop are fundamentally different from the highly branched fingers in rigid-walled cells. We explain the mechanism for the suppression of the instability using a combination of linear stability analysis and direct numerical simulations, based on a theoretical model that couples a depth-averaged lubrication equation for the fluid flow to the Föppl–von Kármán equations, which describe the deformation of the elastic membrane. We show that fluid–structure interaction affects the instability primarily via two changes to the axisymmetric base flow: the axisymmetric inflation of the membrane prior to the onset of any instabilities slows down the expansion of the air bubble and forces the air–liquid interface to propagate into a converging fluid-filled gap. Both of these changes reduce the destabilizing viscous effects that drive the fingering instability in a rigid-walled cell. In contrast, capillary effects only play a very minor role in the suppression of the instability.

JFM classification

Interfacial Flows (free surface): Fingering instability Low-Reynolds-number flows: Hele-Shaw flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 731 , 25 September 2013 , pp. 162 - 183
Copyright
©2013 Cambridge University Press 

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