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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access published online on March 3, 2008
The Quarterly Journal of Mechanics and Applied Mathematics, doi:10.1093/qjmam/hbn001
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© The author 2008. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

SURFACE TENSION EFFECTS ON INTERACTION BETWEEN TWO FLUIDS NEAR A WALL

J.-M. Vanden-Broeck{dagger}

( (School of Mathematics, University of East Anglia, Norwich NR4 7TJ) )

F. T. Smith

( (Mathematics Department, University College London, Gower Street, London WC1E 6BT) )

{dagger} < broeck{at}math.ucl.ac.uk>

Received 12 February 2007. Revise 5 October 2007. Accepted 7 October 2007.


  Abstract

Interaction between two fluids near a fixed solid surface is modelled, with surface tension acting as an important influence on the assumed planar motion. The two fluids are immiscible, incompressible and have small density and viscosity ratios; the heavier more viscous body of fluid is approaching the solid surface and the other fluid is lying as a thin layer in between. In the so-called supercritical range where, for both fluids, inviscid forces dominate over viscous ones, a pair of pressure–shape relations is found which leads to a nonlinear integro-differential equation for the unknown interface shape. Analysis, computation and comparisons are applied to the equation. Travelling-state solutions are found of periodic and non-periodic form, including interesting cases which exhibit parabolic growth of the layer thickness in the far field.


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