Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-12-01T02:43:37.546Z Has data issue: false hasContentIssue false
  • English
  • Français

Gravity-driven thin-film flow on a flexible substrate

Published online by Cambridge University Press:  30 August 2013

P. D. Howell ,
J. Robinson  and
H. A. Stone
P. D. Howell*
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St Giles’, Oxford OX1 3LB, UK
J. Robinson
Affiliation:
Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
H. A. Stone
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the flow of a thin liquid film along a flexible substrate. The flow is modelled using lubrication theory, assuming that gravity is the dominant driving force. The substrate is modelled as an elastic beam that deforms in two dimensions. Steady solutions are found using numerical and perturbation methods, and several different asymptotic regimes are identified. We obtain a complete characterization of how the length and stiffness of the beam and the imposed liquid flux determine the profile of the liquid film and the resulting beam deformation.

JFM classification

Materials Processing Flows: Coating Low-Reynolds-number flows: Lubrication theory Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 732 , 10 October 2013 , pp. 190 - 213
Copyright
©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Boussinesq, J. 1904 Recherches théoriques sur l’écoulement des nappes d’eau infiltrées dans le sol et sur le débit des sources. J. Math. Pure Appl. 10, 5–78.Google Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81 (3), 1131–1198.Google Scholar
Duffy, B. R. & Wilson, S. K. 1999 Thin-film and curtain flows on the outside of a rotating horizontal cylinder. J. Fluid Mech. 394, 29–49.Google Scholar
Gibson, L. J., Ashby, M. F. & Easterling, K. E. 1988 Structure and mechanics of the iris leaf. J. Mater. Sci. 23 (9), 3041–3048.Google Scholar
Halpern, D. & Grotberg, J. B. 1992 Fluid-elastic instabilities of liquid-lined flexible tubes. J. Fluid Mech. 244, 615–632.Google Scholar
Heil, M. & White, J. P. 2002 Airway closure: surface-tension-driven non-axisymmetric instabilities of liquid-lined elastic rings. J. Fluid Mech. 462, 79–109.Google Scholar
Howell, P. D. 2003 Surface-tension-driven flow on a moving curved surface. J. Eng. Math. 45 (3–4), 283–308.Google Scholar
Huppert, H. E. 1982a Flow and instability of a viscous current down a slope. Nature 300 (5891), 427–429.Google Scholar
Huppert, H. E. 1982b The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58.Google Scholar
Jensen, O. E. 1997 The thin liquid lining of a weakly curved cylindrical tube. J. Fluid Mech. 331, 373–403.Google Scholar
Matar, O. K., Craster, R. V. & Kumar, S. 2007 Falling films on flexible inclines. Phys. Rev. E 76 (5), 056301.Google Scholar
Matar, O. K. & Kumar, S. 2007 Dynamics and stability of flow down a flexible incline. J. Eng. Math. 57 (2), 145–158.Google Scholar
Myers, T. G., Charpin, J. P. F. & Chapman, S. J. 2002 The flow and solidification of a thin fluid film on an arbitrary three-dimensional surface. Phys. Fluids 14 (8), 2788–2803.Google Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931–980.Google Scholar
Roy, R. V., Roberts, A. J. & Simpson, M. E. 2002 A lubrication model of coating flows over a curved substrate in space. J. Fluid Mech. 454, 235–261.Google Scholar
Rupp, D. E. & Selker, J. S. 2005 Drainage of a horizontal Boussinesq aquifer with a power law hydraulic conductivity profile. Water Resour. Res. 41 (11), W11422.Google Scholar
Takagi, D. & Huppert, H. E. 2010 Flow and instability of thin films on a cylinder and sphere. J. Fluid Mech. 647, 221–238.Google Scholar
Zheng, Z., Soh, B., Huppert, H. E. & Stone, H. A. 2013 Fluid drainage from the edge of a porous reservoir. J. Fluid Mech. 718, 558–568.Google Scholar