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Viscous-fingering mechanisms under a peeling elastic sheet

Published online by Cambridge University Press:  14 February 2019

Gunnar G. Peng Open the ORCID record for Gunnar G. Peng [Opens in a new window]  and
John R. Lister Open the ORCID record for John R. Lister [Opens in a new window]
Gunnar G. Peng*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
John R. Lister
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: [email protected]
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Abstract

We study the mechanisms affecting the viscous-fingering instability in an elastic-walled Hele-Shaw cell by considering the stability of steady states of unidirectional peeling-by-pulling and peeling-by-bending. We demonstrate that the elasticity of the wall influences the steady base state but has a negligible direct effect on the behaviour of linear perturbations, which thus behave like in the ‘printer’s instability’ with rigid walls. Moreover, the geometry of the cell can be very well approximated as a triangular wedge in the stability analysis. We identify four distinct mechanisms – surface tension acting on the horizontal and the vertical interfacial curvatures, kinematic compression in the longitudinal base flow, and the films deposited on the cell walls – that each contribute to stabilizing the system. The vertical curvature is the dominant stabilizing mechanism for small capillary numbers, but all four mechanisms have a significant effect in a large region of parameter space.

JFM classification

Interfacial Flows (free surface): Fingering instability Low-Reynolds-number flows: Hele-Shaw flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 864 , 10 April 2019 , pp. 1177 - 1207
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Copyright
© 2019 Cambridge University Press 

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References

Al-Housseiny, T. T., Christov, I. C. & Stone, H. A. 2013 Two-phase fluid displacement and interfacial instabilities under elastic membranes. Phys. Rev. Lett. 111, 034502.Google Scholar
Al-Housseiny, T. T., Tsai, P. A. & Stone, H. A. 2012 Control of interfacial instabilities using flow geometry. Nat. Phys. 8 (10), 747–750.Google Scholar
Balmforth, N. J., Craster, R. V. & Hewitt, I. J. 2014 The speed of an inclined ruck. Proc. R. Soc. Lond. A 471, 20140740.Google Scholar
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10 (2), 166–188.Google Scholar
Chuoke, R. L., Van Meurs, P. & van der Poel, C. 1959 The instability of slow, immiscible, viscous liquid–liquid displacements in permeable media. Trans. Amer. Inst. Mining Metall. Engrs 216, 188–194.Google Scholar
Ducloué, L., Hazel, A. L., Pihler-Puzović, D. & Juel, A. 2017 Viscous fingering and dendritic growth under an elastic membrane. J. Fluid Mech. 826, R2.Google Scholar
Grenfell-Shaw, J. C. & Woods, A. W. 2017 The instability of a moving interface in a narrow tapering channel of finite length. J. Fluid Mech. 831, 252–270.Google Scholar
Halpern, D. & Gaver, D. P. 1994 Boundary element analysis of the time-dependent motion of a semi-infinite bubble in a channel. J. Comput. Phys. 115 (2), 366–375.Google Scholar
Hewitt, I. J., Balmforth, N. J. & De Bruyn, J. R. 2015 Elastic-plated gravity currents. Eur. J. Appl. Maths 26 (1), 1–31.Google Scholar
Hill, S. 1952 Channelling in packed columns. Chem. Engng Sci. 1 (6), 247–253.Google Scholar
Homsy, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271–311.Google Scholar
Hosoi, A. E. & Mahadevan, L. 2004 Peeling, healing, and bursting in a lubricated elastic sheet. Phys. Rev. Lett. 93, 137802.Google Scholar
Jensen, O. E., Horsburgh, M. K., Halpern, D. & Gaver, D. P. 2002 The steady propagation of a bubble in a flexible-walled channel: asymptotic and computational models. Phys. Fluids 14 (2), 431–442.Google Scholar
Lehoucq, R. B. & Sorensen, D. C. 1995 Deflation techniques for an implicitly restarted Arnoldi iteration. SIAM J. Matrix Anal. Applics. 17, 789–821.Google Scholar
Lister, J. R., Peng, G. G. & Neufeld, J. A. 2013 Viscous control of peeling an elastic sheet by bending and pulling. Phys. Rev. Lett. 111, 154501.Google Scholar
McEwan, A. D. & Taylor, G. I. 1966 The peeling of a flexible strip attached by a viscous adhesive. J. Fluid Mech. 26 (1), 1–15.Google Scholar
Pearson, J. R. A. 1960 The instability of uniform viscous flow under rollers and spreaders. J. Fluid Mech. 7 (4), 481–500.Google Scholar
Peng, G. G., Pihler-Puzović, D., Juel, A., Heil, M. & Lister, J. R. 2015 Displacement flows under elastic membranes. Part 2. Analysis of interfacial effects. J. Fluid Mech. 784, 512–547.Google Scholar
Pihler-Puzović, D., Illien, P., Heil, M. & Juel, A. 2012 Suppression of complex fingerlike patterns at the interface between air and a viscous fluid by elastic membranes. Phys. Rev. Lett. 108, 074502.Google Scholar
Pihler-Puzović, D., Juel, A. & Heil, M. 2014 The interaction between viscous fingering and wrinkling in elastic-walled Hele-Shaw cells. Phys. Fluids 26, 1–24.Google Scholar
Pihler-Puzović, D., Juel, A., Peng, G. G., Lister, J. R. & Heil, M. 2015 Displacement flows under elastic membrances. Part 1. Experiments and direct numerical simulations. J. Fluid Mech. 784, 487–511.Google Scholar
Pihler-Puzović, D., Peng, G. G., Lister, J. R., Heil, M. & Juel, A. 2018 Viscous fingering in a radial elastic-walled Hele-Shaw cell. J. Fluid Mech. 849, 163–191.Google Scholar
Pihler-Puzović, D., Périllat, R., Russell, M., Juel, A. & Heil, M. 2013 Modelling the suppression of viscous fingering in elastic-walled Hele-Shaw cells. J. Fluid Mech. 731, 162–183.Google Scholar
Reinelt, D. A. & Saffman, P. G. 1985 The penetration of a finger into a viscous fluid in a channel and tube. SIAM J. Sci. Stat. Comput. 6 (3), 542–561.Google Scholar
Reynolds, O. 1886 IV. On the theory of lubrication and its application to Mr. Beauchamp Tower’s experiments, including an experimental determination of the viscosity of olive oil. Phil. Trans. R. Soc. Lond. 177, 157–234.Google Scholar
Ruschak, K. J. 1985 Coating flows. Annu. Rev. Fluid Mech. 17 (1), 65–89.Google Scholar
Saffman, P. G. & Taylor, G. I. 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245 (1242), 312–329.Google Scholar