Finite element approximation of surfactant spreading on a thin film

SIAM Journal on Numerical Analysis

Volumn 41, Issue 4, 2003, Pages 1427-1464

  Barrett, John W ^a   Garcke, Harald ^b,c   Nürnberg, Robert ^a  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

^b UNIVERSITY OF BONN   (Germany)

^c UNIVERSITY OF REGENSBURG   (Germany)

Author keywords

Convergence analysis; Finite elements; Fourth order degenerate parabolic system; Surfactant; Thin film flow

Indexed keywords

COATINGS; CONVERGENCE OF NUMERICAL METHODS; COSINE TRANSFORMS; DRYING; FINITE ELEMENT METHOD; FUNCTIONS; ITERATIVE METHODS; LUBRICATION; MONOLAYERS; NONLINEAR EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; SURFACE ACTIVE AGENTS; SURFACE TENSION; VECTORS; VELOCITY; VISCOUS FLOW;

CONVERGENCE ANALYSIS; FINITE ELEMENTS; FOURTH ORDER DEGENERATE PARABOLIC SYSTEM; THIN FILM FLOW;

THIN FILMS;

EID: 4043069669     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S003614290139799X     Document Type: Article

References (32)

1. R. A. ADAMS AND J. FOURNIER, Cone conditions and properties of Sobolev spaces, J. Math. Anal. Appl., 61 (1977), pp. 713-734.
2. J. W. BARRETT AND J. F. BLOWEY, Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy, Numer. Math., 77 (1997), pp. 1-34.
3. J. W. BARRETT AND J. F. BLOWEY, Finite element approximation of a degenerate Allen-Cahn/Cahn-Hilliard system, SIAM J. Numer. Anal., 39 (2001), pp. 1598-1624.
4. J. W. BARRETT, J. F. BLOWEY, AND H. GARCKE, Finite element approximation of a fourth order nonlinear degenerate parabolic equation, Numer. Math., 80 (1998), pp. 525-556.
5. J. W. BARRETT, J. F. BLOWEY, AND H. GARCKE, Finite element approximation of the Cahn-Hilliard equation with degenerate mobility, SIAM J. Numer. Anal., 37 (1999), pp. 286-318.
6. J. W. BARRETT, J. F. BLOWEY, AND H. GARCKE, On fully practical finite element approximations of degenerate Cahn-Hilliard systems, M2AN Math. Model. Numer. Anal., 35 (2001), pp. 713-748.
7. J. W. BARRETT AND R. NÜRNBERG, Finite element approximation of a nonlinear degenerate parabolic system describing bacterial pattern formation, Interfaces Free Bound., 4 (2002), pp. 277-307.
8. F. BERNIS, Viscous flows, fourth order nonlinear degenerate parabolic equations and singular elliptic problems, in Free Boundary Problems: Theory and Applications, Pitman Res. Notes Math. Ser. 323, J. I. Diaz, M. A. Herrero, A. Linan, and J. L. Vazquez, eds., Longman, Harlow, UK, 1995, pp. 40-56.
9. F. BERNIS AND A. FRIEDMAN, Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83 (1990), pp. 179-206.
10. A. L. BERTOZZI, M. P. BRENNER, T. F. DUPONT, AND L. P. KADANOFF, Singularities and similarities in interface flows, in Trends and Perspectives in Applied Mathematics, Appl. Math. Sci. 100, Springer-Verlag, New York, 1994, pp. 155-208.
11. M. BERTSCH, R. DAL PASSO, H. GARCKE, AND G. GRÜN, The thin viscous flow equation in higher space dimensions, Adv. Differential Equations, 3 (1998), pp. 417-440.
12. J. F. BLOWEY AND C. M. ELLIOTT, The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. II. Numerical analysis, European J. Appl. Math., 3 (1992), pp. 147-179.
13. M. S. BORGAS AND J. B. GROTBERG, Monolayer flow on a thin film, J. Fluid Mech., 193 (1988), pp. 151-170.
14. J. W. CAHN, C. M. ELLIOTT, AND A. NOVICK-COHEN, The Cahn-Hilliard equation with a concentration dependent mobility: Motion by minus the Laplacian of the mean curvature, European J. Appl. Math., 7 (1996), pp. 287-301.
15. J. F. CIAVALDINI, Analyse numérique d'un problème de Stefan à deux phases par une méthode d'elements finis, SIAM J. Numer. Anal., 12 (1975), pp. 464-487.
16. C. M. ELLIOTT AND H. GARCKE, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), pp. 404-423.
17. C. M. ELLIOTT AND H. GARCKE, Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix, Phys. D, 109 (1997), pp. 242-256.
18. D. P. GAVER III AND J. B. GROTBERG, The dynamics of a localized surfactant on a thin film, J. Fluid Mech., 213 (1990), pp. 127-148.
19. H. P. GREENSPAN, On the motion of a small viscous droplet that wets a surface, J. Fluid Mech., 84 (1978), pp. 125-143.
20. G. GRÜN, Degenerate parabolic differential equations of fourth order and a plasticity model with nonlocal hardening, Z. Anal. Anwendungen, 14 (1995), pp. 541-574.
21. G. GRÜN, On the convergence of entropy consistent schemes for lubrication type equations in multiple space dimensions, Math. Comp., 72 (2003), pp. 1251-1279.
22. G. GRÜN AND M. RUMPF, Nonnegativity preserving numerical schemes for the thin film equation, Numer. Math., 87 (2000), pp. 113-152.
23. G. GRÜN AND M. RUMPF, Simulation of singularities and instabilities in thin film flow, European J. Appl. Math., 12 (2001), pp. 293-320.
24. O. E. JENSEN AND J. B. GROTBERG, Insoluble surfactant spreading on a thin viscous film: Shock evolution and film rupture, J. Fluid Mech., 240 (1992), pp. 259-288.
25. P. L. LIONS AND B. MERCIER, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), pp. 964-979.
26. O. K. MATAR AND S. M. TROIAN, The development of transient fingering patterns during the spreading of surfactant coated films, Phys. Fluids, 11 (1999), pp. 3232-3246.
27. O. K. MATAR AND S. M. TROIAN, Spreading of a surfactant monolayer on a thin liquid film: Onset and evolution of digitated structures, Chaos, 9 (1999), pp. 141-153.
28. A. ORON, S. H. DAVIS, AND S. G. BANKOFF, Long-scale evolution of thin liquid films, Rev. Modern Phys., 69 (1997), pp. 931-980.
29. M. RENARDY AND R. C. ROGERS, An Introduction to Partial Differential Equations, Springer-Verlag, New York, 1992.
30. A. SCHMIDT AND K. G. SIEBERT, Albert-software for scientific computations and applications, Acta Math. Univ. Comenian (N.S.), 70 (2001), pp. 105-122.
31. S. WIELAND, Modellierung und mathematische Analyse von Oberflächendiffusion auf dünnen Filmen, Ph.D. thesis, University of Bonn, Bonn, Germany, 2003 (to appear).
32. L. ZHORNITSKAYA AND A. L. BERTOZZI, Positivity-preserving numerical schemes for lubrication-type equations, SIAM J. Numer. Anal., 37 (2000), pp. 523-555.