The development of viscous fingers in circular Hele-Shaw cells is a classical and widely studied fluid mechanical problem. The introduction of wall elasticity (via the replacement of one of the bounding plates by an elastic membrane) can weaken or even suppress the fingering instability, but it also makes the system susceptible to additional solid-mechanical instabilities. We show that in elastic-walled Hele-Shaw cells that are bounded by sufficiently thin elastic sheets the (fluid-based) viscous fingering instability can arise concurrently with a (solid-based) wrinkling instability. We study the interaction between these distinct instabilities, using a theoretical model that couples the depth-averaged lubrication equations for the fluid flow to the Föppl-von Kármán equations, which describe the deformation of the thin elastic sheet. We employ a linear stability analysis to determine the growth rate of non-axisymmetric perturbations to the axisymmetrically expanding bubble, and perform direct numerical simulations to study the nonlinear interactions between the instabilities. We show that the system's behaviour may be characterised by a non-dimensional parameter that indicates the strength of the fluid-structure interaction. For small [large] values of this parameter, the system's behaviour is dominated by viscous fingering [wrinkling], with strong interactions between the two instabilities arising in an intermediate regime.
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February 2014
Research Article|
February 10 2014
The interaction between viscous fingering and wrinkling in elastic-walled Hele-Shaw cells
D. Pihler-Puzović;
D. Pihler-Puzović
Manchester Centre for Nonlinear Dynamics and School of Mathematics,
University of Manchester
, Oxford Road, Manchester M13 9PL, United Kingdom
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A. Juel;
A. Juel
Manchester Centre for Nonlinear Dynamics and School of Mathematics,
University of Manchester
, Oxford Road, Manchester M13 9PL, United Kingdom
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M. Heil
M. Heil
Manchester Centre for Nonlinear Dynamics and School of Mathematics,
University of Manchester
, Oxford Road, Manchester M13 9PL, United Kingdom
Search for other works by this author on:
D. Pihler-Puzović
A. Juel
M. Heil
Manchester Centre for Nonlinear Dynamics and School of Mathematics,
University of Manchester
, Oxford Road, Manchester M13 9PL, United Kingdom
Physics of Fluids 26, 022102 (2014)
Article history
Received:
July 03 2013
Accepted:
January 11 2014
Citation
D. Pihler-Puzović, A. Juel, M. Heil; The interaction between viscous fingering and wrinkling in elastic-walled Hele-Shaw cells. Physics of Fluids 1 February 2014; 26 (2): 022102. https://doi.org/10.1063/1.4864188
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