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 
@article{M2AN_2008__42_5_749_0,
     author = {Barrett, John W. and El Alaoui, Linda},
     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://archive.numdam.org/articles/10.1051/m2an:2008028/}
}
TY  - JOUR
AU  - Barrett, John W.
AU  - El Alaoui, Linda
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://archive.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 El Alaoui, Linda
%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://archive.numdam.org/articles/10.1051/m2an:2008028/
%R 10.1051/m2an:2008028
%G en
%F M2AN_2008__42_5_749_0
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://archive.numdam.org/articles/10.1051/m2an:2008028/

[1] J.W. Barrett and R. Nürnberg, 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 | Zbl

[2] J.W. Barrett, H. Garcke and R. Nürnberg, Finite element approximation of surfactant spreading on a thin film. SIAM J. Numer. Anal. 41 (2003) 1427-1464. | MR | Zbl

[3] J.W. Barrett, R. Nürnberg and M.R.E. Warner, Finite element approximation of soluble surfactant spreading on a thin film. SIAM J. Numer. Anal. 44 (2006) 1218-1247. | MR

[4] K.D. Danov, V.N. Paunov, S.D. Stoyanov, N. Alleborn, H. Raszillier and F. Durst, Stability of evaporating two-layered liquid film in the presence of surfactant - ii Linear analysis. Chem. Eng. Sci. 53 (1998) 2823-2837.

[5] H. Garcke and S. Wieland, Surfactant spreading on thin viscous films: nonnegative solutions of a coupled degenerate system. SIAM J. Math. Anal. 37 (2006) 2025-2048. | MR | Zbl

[6] G. Grün, On the convergence of entropy consistent schemes for lubrication type equations in multiple space dimensions. Math. Comp. 72 (2003) 1251-1279. | MR | Zbl

[7] G. Grün and M. Rumpf, Nonnegativity preserving numerical schemes for the thin film equation. Numer. Math. 87 (2000) 113-152. | MR | Zbl

[8] M. Renardy, A singularly perturbed problem related to surfactant spreading on thin films. Nonlinear Anal. 27 (1996) 287-296. | MR | Zbl

[9] M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations. Springer-Verlag, New York, 1992. | MR | Zbl

[10] A. Schmidt and K.G. Siebert, ALBERT-software for scientific computations and applications. Acta Math. Univ. Comenian. (N.S.) 70 (2000) 105-122. | MR | Zbl

[11] A. Sheludko, Thin liquid films. Adv. Colloid Interface Sci. 1 (1967) 391-464.

[12] L. Zhornitskaya and A.L. Bertozzi, Positivity preserving numerical schemes for lubrication-type equations. SIAM J. Numer. Anal. 37 (2000) 523-555. | MR | Zbl

Cité par Sources :