Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 5, p. 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 : https://doi.org/10.1051/m2an:2008028
Classification:  65M60,  65M12,  35K55,  35K65,  35K35,  76A20,  76D08
Keywords: thin film, surfactant, bilayer, fourth order degenerate parabolic system, finite elements, convergence analysis
@article{M2AN_2008__42_5_749_0,
     author = {Barrett, John W. and Alaoui, Linda El},
     title = {Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {5},
     year = {2008},
     pages = {749-775},
     doi = {10.1051/m2an:2008028},
     zbl = {1147.76038},
     mrnumber = {2454622},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2008__42_5_749_0}
}
Barrett, John W.; Alaoui, Linda El. Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 5, pp. 749-775. doi : 10.1051/m2an:2008028. http://www.numdam.org/item/M2AN_2008__42_5_749_0/

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