Issue ESAIM: M2AN Volume 42, Number 5, September-October 2008 Page(s) 749 - 775 DOI 10.1051/m2an:2008028 Published online 30 July 2008

ESAIM: M2AN 42 (2008) 749-775
DOI: 10.1051/m2an:2008028

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

John W. Barrett and Linda El Alaoui

Department of Mathematics, Imperial College, London, SW7 2AZ, UK. jwb@ic.ac.uk

Received April 24, 2007. Published online July 30, 2008.

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.

Mathematics Subject Classification. 65M60, 65M12, 35K55, 35K65, 35K35, 76A20, 76D08.

Key words: Thin film, surfactant, bilayer, fourth order degenerate parabolic system, finite elements, convergence analysis.


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