Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants
ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 5, pp. 749-775.

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.

DOI : 10.1051/m2an:2008028
Classification : 65M60, 65M12, 35K55, 35K65, 35K35, 76A20, 76D08
Mots-clés : thin film, surfactant, bilayer, fourth order degenerate parabolic system, finite elements, convergence analysis 
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     title = {Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {749--775},
     publisher = {EDP-Sciences},
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Barrett, John W.; El Alaoui, Linda. Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 5, pp. 749-775. doi : 10.1051/m2an:2008028. http://www.numdam.org/articles/10.1051/m2an:2008028/

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