# Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants

John W. Barrett; Linda El Alaoui

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

- Volume: 42, Issue: 5, page 749-775
- ISSN: 0764-583X

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topBarrett, John W., and El Alaoui, Linda. "Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants." ESAIM: Mathematical Modelling and Numerical Analysis 42.5 (2008): 749-775. <http://eudml.org/doc/250368>.

@article{Barrett2008,

abstract = {
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.
},

author = {Barrett, John W., El Alaoui, Linda},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Thin film; surfactant; bilayer; fourth order degenerate parabolic system; finite
elements; convergence analysis.; fourth-order degenerate parabolic system; convergence; energy inequality; existence},

language = {eng},

month = {7},

number = {5},

pages = {749-775},

publisher = {EDP Sciences},

title = {Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants},

url = {http://eudml.org/doc/250368},

volume = {42},

year = {2008},

}

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: Mathematical Modelling and Numerical Analysis

DA - 2008/7//

PB - EDP Sciences

VL - 42

IS - 5

SP - 749

EP - 775

AB -
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.

LA - eng

KW - Thin film; surfactant; bilayer; fourth order degenerate parabolic system; finite
elements; convergence analysis.; fourth-order degenerate parabolic system; convergence; energy inequality; existence

UR - http://eudml.org/doc/250368

ER -

## References

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- 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. Zbl1130.76361
- 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. Zbl1301.76044
- 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.
- 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. Zbl1102.35056
- G. Grün, On the convergence of entropy consistent schemes for lubrication type equations in multiple space dimensions. Math. Comp.72 (2003) 1251–1279. Zbl1084.65093
- G. Grün and M. Rumpf, Nonnegativity preserving numerical schemes for the thin film equation. Numer. Math.87 (2000) 113–152. Zbl0988.76056
- M. Renardy, A singularly perturbed problem related to surfactant spreading on thin films. Nonlinear Anal.27 (1996) 287–296. Zbl0862.35091
- M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations. Springer-Verlag, New York, 1992. Zbl0917.35001
- A. Schmidt and K.G. Siebert, ALBERT—software for scientific computations and applications. Acta Math. Univ. Comenian. (N.S.)70 (2000) 105–122. Zbl0993.65134
- A. Sheludko, Thin liquid films. Adv. Colloid Interface Sci.1 (1967) 391–464.
- L. Zhornitskaya and A.L. Bertozzi, Positivity preserving numerical schemes for lubrication-type equations. SIAM J. Numer. Anal.37 (2000) 523–555. Zbl0961.76060

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