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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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