Skip to main content Accessibility help

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

  • F. Box (a1) (a2), Jerome A. Neufeld (a1) (a3) (a4) and Andrew W. Woods (a1) (a3)


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.


Corresponding author

Email address for correspondence:


Hide All
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, 359380.
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.
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.
Detournay, E. 2016 Mechanics of hydraulic fractures. Annu. Rev. Fluid Mech. 48, 311339.
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, 713754.
Garagash, D. & Detournay, E. 2000 The tip region of a fluid-driven fracture in an elastic medium. J. Appl. Mech. 67, 183192.
Garagash, D. I., Detournay, E. & Adachi, J. I. 2011 Multiscale tip asymptotics in hydraulic fracture with leak-off. J. Fluid Mech. 669, 260297.
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, 602629.
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 14, 121147.
Hewitt, I. J., Balmforth, N. J. & De Bruyn, J. R. 2015 Elastic-plated gravity currents. Eur. J. Appl. Maths 26, 131.
Hosoi, A. & Mahadevan, L. 2004 Peeling, healing, and bursting in a lubricated elastic sheet. Phys. Rev. Lett. 93, 137802.
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, 443457.
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.
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.
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, 512547.
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.
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, 487511.
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, 162183.
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, 2347.
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, 135144.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

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

  • F. Box (a1) (a2), Jerome A. Neufeld (a1) (a3) (a4) and Andrew W. Woods (a1) (a3)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.