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 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: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {749--775}, publisher = {EDP-Sciences}, volume = {42}, number = {5}, year = {2008}, doi = {10.1051/m2an:2008028}, mrnumber = {2454622}, zbl = {1147.76038}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an:2008028/} }
TY - JOUR AU - Barrett, John W. AU - Alaoui, Linda El TI - Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2008 SP - 749 EP - 775 VL - 42 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2008028/ DO - 10.1051/m2an:2008028 LA - en ID - M2AN_2008__42_5_749_0 ER -
%0 Journal Article %A Barrett, John W. %A Alaoui, Linda El %T Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants %J ESAIM: Modélisation mathématique et analyse numérique %D 2008 %P 749-775 %V 42 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2008028/ %R 10.1051/m2an:2008028 %G en %F 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: Modélisation mathématique et analyse numérique, Volume 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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