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.
@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},
volume = {42},
year = {2008},
pages = {749-775},
doi = {10.1051/m2an:2008028},
mrnumber = {2454622},
zbl = {1147.76038},
language = {en},
url = {http://dml.mathdoc.fr/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, Tome 42 (2008) pp. 749-775. doi : 10.1051/m2an:2008028. http://gdmltest.u-ga.fr/item/M2AN_2008__42_5_749_0/
[1] and , Convergence of a finite-element approximation of surfactant spreading on a thin film in the presence of van der Waals forces. IMA J. Numer. Anal. 24 (2004) 323-363. | MR 2046180 | Zbl 1143.76473
[2] , and , Finite element approximation of surfactant spreading on a thin film. SIAM J. Numer. Anal. 41 (2003) 1427-1464. | MR 2034888 | Zbl 1130.76361
[3] , and , Finite element approximation of soluble surfactant spreading on a thin film. SIAM J. Numer. Anal. 44 (2006) 1218-1247. | MR 2231862 | Zbl pre05167772
[4] , , , , and , Stability of evaporating two-layered liquid film in the presence of surfactant - ii Linear analysis. Chem. Eng. Sci. 53 (1998) 2823-2837.
[5] and , Surfactant spreading on thin viscous films: nonnegative solutions of a coupled degenerate system. SIAM J. Math. Anal. 37 (2006) 2025-2048. | MR 2213404 | Zbl 1102.35056
[6] , On the convergence of entropy consistent schemes for lubrication type equations in multiple space dimensions. Math. Comp. 72 (2003) 1251-1279. | MR 1972735 | Zbl 1084.65093
[7] and , Nonnegativity preserving numerical schemes for the thin film equation. Numer. Math. 87 (2000) 113-152. | MR 1800156 | Zbl 0988.76056
[8] , A singularly perturbed problem related to surfactant spreading on thin films. Nonlinear Anal. 27 (1996) 287-296. | MR 1391438 | Zbl 0862.35091
[9] and , An Introduction to Partial Differential Equations. Springer-Verlag, New York, 1992. | MR 2028503 | Zbl 0917.35001
[10] and , ALBERT-software for scientific computations and applications. Acta Math. Univ. Comenian. (N.S.) 70 (2000) 105-122. | MR 1865363 | Zbl 0993.65134
[11] , Thin liquid films. Adv. Colloid Interface Sci. 1 (1967) 391-464.
[12] and , Positivity preserving numerical schemes for lubrication-type equations. SIAM J. Numer. Anal. 37 (2000) 523-555. | MR 1740768 | Zbl 0961.76060