Volume 42, Number 5, September-October 2008
|Page(s)||749 - 775|
|Published online||30 July 2008|
Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants
Department of Mathematics, Imperial College, London, SW7 2AZ, UK. email@example.com
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.
© EDP Sciences, SMAI, 2008
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