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Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants

Published online by Cambridge University Press:  30 July 2008

John W. Barrett  and
Linda El Alaoui
John W. Barrett
Affiliation:
Department of Mathematics, Imperial College, London, SW7 2AZ, UK. jwb@ic.ac.uk
Linda El Alaoui
Affiliation:
Department of Mathematics, Imperial College, London, SW7 2AZ, UK. jwb@ic.ac.uk
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Abstract

We consider a system of degenerate parabolic equations modelling a thin film, consisting of two layers of immiscible Newtonian liquids, on a solid horizontal substrate. In addition, the model includes the presence of insoluble surfactants on both the free liquid-liquid and liquid-air interfaces, and the presence of both attractive and repulsive van der Waals forces in terms of the heights of the two layers. We show that this system formally satisfies a Lyapunov structure, and a second energy inequality controlling the Laplacian of the liquid heights. We introduce a fully practical finite element approximation of this nonlinear degenerate parabolic system, that satisfies discrete analogues of these energy inequalities. Finally, we prove convergence of this approximation, and hence existence of a solution to this nonlinear degenerate parabolic system.

Keywords

Thin filmsurfactantbilayerfourth order degenerate parabolic systemfinite elementsconvergence analysis.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 5 , September 2008 , pp. 749 - 775
Copyright
© EDP Sciences, SMAI, 2008

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