Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-19T20:23:53.324Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite element approximation of a two-layered liquid filmin the presence of insoluble surfactants

Published online by Cambridge University Press:  30 July 2008

John W. Barrett  and
Linda El Alaoui
John W. Barrett
Affiliation:
Department of Mathematics, Imperial College, London, SW7 2AZ, UK. [email protected]
Linda El Alaoui
Affiliation:
Department of Mathematics, Imperial College, London, SW7 2AZ, UK. [email protected]
Get access

Abstract

We consider a system of degenerate parabolic equations modelling a thin film, consisting of two layers of immiscible Newtonian liquids, ona solid horizontal substrate. In addition, the model includes the presence of insoluble surfactants onboth the free liquid-liquid and liquid-air interfaces,and the presence of both attractive and repulsive van der Waals forcesin 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 Laplacianof the liquid heights.We introduce a fully practical finite element approximation of this nonlinear degenerate parabolic system, that satisfies discrete analoguesof these energy inequalities. Finally, we prove convergence of this approximation,and hence existence of a solutionto this nonlinear degenerate parabolic system.

Keywords

Thin filmsurfactantbilayerfourth order degenerate parabolic systemfiniteelementsconvergence analysis.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 5 , September 2008 , pp. 749 - 775
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barrett, J.W. and Nürnberg, R., 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. CrossRef
Barrett, J.W., Garcke, H. and Nürnberg, R., Finite element approximation of surfactant spreading on a thin film. SIAM J. Numer. Anal. 41 (2003) 1427–1464. CrossRef
Barrett, J.W., Nürnberg, R. and Warner, M.R.E., Finite element approximation of soluble surfactant spreading on a thin film. SIAM J. Numer. Anal. 44 (2006) 1218–1247. CrossRef
Danov, K.D., Paunov, V.N., Stoyanov, S.D., Alleborn, N., Raszillier, H. and Durst, F., Stability of evaporating two-layered liquid film in the presence of surfactant - ii Linear analysis. Chem. Eng. Sci. 53 (1998) 2823–2837. CrossRef
Garcke, H. and Wieland, S., Surfactant spreading on thin viscous films: nonnegative solutions of a coupled degenerate system. SIAM J. Math. Anal. 37 (2006) 2025–2048. CrossRef
Grün, G., On the convergence of entropy consistent schemes for lubrication type equations in multiple space dimensions. Math. Comp. 72 (2003) 1251–1279. CrossRef
Grün, G. and Rumpf, M., Nonnegativity preserving numerical schemes for the thin film equation. Numer. Math. 87 (2000) 113–152.
Renardy, M., A singularly perturbed problem related to surfactant spreading on thin films. Nonlinear Anal. 27 (1996) 287–296. CrossRef
M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations. Springer-Verlag, New York, 1992.
Schmidt, A. and Siebert, K.G., ALBERT—software for scientific computations and applications. Acta Math. Univ. Comenian. (N.S.) 70 (2000) 105–122.
Sheludko, A., Thin liquid films. Adv. Colloid Interface Sci. 1 (1967) 391–464. CrossRef
Zhornitskaya, L. and Bertozzi, A.L., Positivity preserving numerical schemes for lubrication-type equations. SIAM J. Numer. Anal. 37 (2000) 523–555. CrossRef