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On the dynamics of a thin viscous film spreading between a permeable horizontal plate and an elastic sheet

Published online by Cambridge University Press:  01 March 2018

F. Box Open the ORCID record for F. Box [Opens in a new window] ,
Jerome A. Neufeld  and
Andrew W. Woods Open the ORCID record for Andrew W. Woods [Opens in a new window]
F. Box*
Affiliation:
BP Institute for Multiphase Flow, University of Cambridge, Cambridge CB3 0EZ, UK Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
Jerome A. Neufeld
Affiliation:
BP Institute for Multiphase Flow, University of Cambridge, Cambridge CB3 0EZ, UK Department of Earth Sciences, University of Cambridge, Cambridge CB3 0EZ, UK Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK
Andrew W. Woods
Affiliation:
BP Institute for Multiphase Flow, University of Cambridge, Cambridge CB3 0EZ, UK Department of Earth Sciences, University of Cambridge, Cambridge CB3 0EZ, UK
*
†Email address for correspondence: finn.box@maths.ox.ac.uk
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Abstract

The two-dimensional dynamics of a thin film of viscous fluid spreading between a permeable horizontal plate and an overlying thin elastic sheet is explored. We use a lubrication model to describe the balance between the elastic stress, the hydrostatic pressure gradient and the viscous resistance of the flow, as fluid spreads laterally from a source and simultaneously drains through the plate. A family of asymptotic solutions are described in which the flow is dominated by either the hydrostatic pressure gradient or the elastic stress associated with the deformation of the sheet. In these solutions, although the deformation of the sheet above the porous plate arises from the fluid flow below the sheet, the fluid typically separates from the sheet a short distance upstream of the full extent of the draining zone, with the region of flow being driven purely by the hydrostatic pressure gradient. As a result, an air gap develops below the sheet up to the point where it touches back down onto the plate. With a very light or stiff elastic sheet, this touchdown point may extend far beyond the fluid draining zone, but otherwise it is similar to the extent of the draining zone.

JFM classification

Geophysical and Geological Flows: Gravity currents Low-Reynolds-number flows: Hele-Shaw flows Low-Reynolds-number flows: Low-Reynolds-number flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 841 , 25 April 2018 , pp. 989 - 1011
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Copyright
© 2018 Cambridge University Press 

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References

Acton, J. M., Huppert, H. E. & Worster, M. G. 2001 Two-dimensional viscous gravity currents flowing over a deep porous medium. J. Fluid Mech. 440, 359–380.CrossRefGoogle Scholar
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
Bunger, A. P. & Cruden, A. R. 2011 Modeling the growth of laccoliths and large mafic sills: role of magma body forces. J. Geophys. Res. 116, B02203.Google Scholar
Detournay, E. 2016 Mechanics of hydraulic fractures. Annu. Rev. Fluid Mech. 48, 311–339.Google Scholar
Flitton, J. C. & King, J. R. 2004 Moving-boundary and fixed-domain problems for a sixth-order thin-film equation. Eur. J. Appl. Maths 15, 713–754.Google Scholar
Garagash, D. & Detournay, E. 2000 The tip region of a fluid-driven fracture in an elastic medium. J. Appl. Mech. 67, 183–192.Google Scholar
Garagash, D. I., Detournay, E. & Adachi, J. I. 2011 Multiscale tip asymptotics in hydraulic fracture with leak-off. J. Fluid Mech. 669, 260–297.Google Scholar
Gordeliy, E. & Detournay, E. 2011 A fixed grid algorithm for simulating the propagation of a shallow hydraulic fracture with a fluid lag. Intl J. Numer. Anal. Meth. Geomech. 35, 602–629.Google Scholar
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 14, 121–147.Google Scholar
Hewitt, I. J., Balmforth, N. J. & De Bruyn, J. R. 2015 Elastic-plated gravity currents. Eur. J. Appl. Maths 26, 1–31.Google Scholar
Hosoi, A. & 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, 443–457.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
Michaut, C. 2011 Dynamics of magmatic intrusions in the upper crust: theory and applications to laccoliths on Earth and the Moon. J. Geophys. Res. 116, B05205.CrossRefGoogle 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., Peng, G. G., Lister, J. R. & Heil, M. 2015 Displacement flows under elastic membranes. Part 1. Experiments and direct numerical simulations. J. Fluid Mech. 784, 487–511.Google Scholar
Pihler-Puzović, D., Périllat, R., Russel, 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
Pritchard, D., Woods, A. W. & Hogg, A. J. 2001 On the slow draining of a gravity current moving through a layered permeable medium. J. Fluid Mech. 444, 23–47.Google Scholar
Spannuth, M. J., Neufeld, J. A., Wettlaufer, J. S. & Worster, M. G. 2009 Axisymmetric viscous gravity currents flowing over a porous medium. J. Fluid Mech. 622, 135–144.Google Scholar