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

Abstract

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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.

How to cite

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Barrett, 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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  8. M. Renardy, A singularly perturbed problem related to surfactant spreading on thin films. Nonlinear Anal.27 (1996) 287–296.  Zbl0862.35091
  9. M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations. Springer-Verlag, New York, 1992.  Zbl0917.35001
  10. 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
  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.  Zbl0961.76060

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