Properties of steady states for thin film equations

European Journal of Applied Mathematics

Volumn 11, Issue 3, 2000, Pages 293-351

  Laugesen, R S ^a   Pugh, M C ^b  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^b UNIVERSITY OF PENNSYLVANIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034380061     PISSN: 09567925     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0956792599003794     Document Type: Article

References (48)

1. DUSSAN V, E. B. (1976) The moving contact line: the slip boundary condition. J. Fluid Mech. 77, 665-684.
2. EHRHARD, P. & DAVIS, S. H. (1991) Non-isothermal spreading of liquid drops on horizontal plates. J. Fluid Mech. 229, 365-388.
3. GREENSPAN, H. P. (1978) On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84, 125-143.
4. HALEY, P. J. & MIKSIS, M. J. (1991) The effect of the contact line on droplet spreading. J. Fluid Mech. 223, 57-81.
5. NEOGI, P. & MILLER, C. A. (1984) Spreading kinetics of a drop on a rough solid surface. J. Colloid Interf. Sci. 92, 338-349.
6. ORON, A., DAVIS, S. H. & BANKOFF, S. G. (1997) Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69(3), 931-980.
7. EHRHARD, P. (1994) The spreading of hanging drops. J. Colloid Interf. Sci. 168, 242-246.
8. DE GENNES, P. G. (1985) Wetting: statics and dynamics. Rev. Mod. Phys. 57(3), 827-863.
9. ISRAELACHVILI, J. N. (1992) Intermolecular and Surface Forces, 2nd ed. Academic Press.
10. WILLIAMS, M. B. & DAVIS, S. H. (1982) Nonlinear theory of film rupture. J. Colloid Interf. Sci. 90(1), 220-228.
11. ORON, A. & ROSENAU, P. (1992) Formation of patterns induced by thermocapillarity and gravity. J. Phys II, 2, 131-146.
12. VANHOOK, S. J., SCHATZ, M. F., SWIFT, J. B., McCoRMICK, W. D. & SWINNEY, H. L. (1997) Long-wavelength surface-tension-driven Bénard convection: experiment and theory. J. Fluid Mech. 345, 45-78.
13. HAMMOND, P. S. (1983) Nonlinear adjustment of a thin annular film of viscous fluid surrounding a thread of another within a circular cylindrical pipe. J. Fluid Mech. 137, 363-384.
14. JENSEN, O. E. (1997) The thin liquid lining of a weakly curved cylindrical tube. J. Fluid Mech. 345, 373-403.
15. GOLDSTEIN, R. E., PESCI, A. I. & SHELLEY, M. J. (1993) Topology transitions and singularities in viscous flows. Phys. Rev. Lett. 17(20), 3043-3046.
16. GOLDSTEIN, R., PESCI, A. & SHELLEY, M. (1998) Instabilities and singularities in Hele-Shaw flow. Phys. Fluids, 10(11), 2701-2723.
17. LEWIS, M. A. (1994) Spatial coupling of plant and herbivore dynamics - the contribution of herbivore dispersal to transient and persistent 'waves' of damage. Theor. Popul. Biol. 45(3), 277-312.
18. BERTOZZI, A. L. & PUGH, M. C. (1998) Long-wave instabilities and saturation in thin film equations. Comm. Pure Appl. Math. 51, 625-661.
19. TUCK, E. O. & SCHWARTZ, L. W. (1991) Thin static drops with a free attachment boundary. J. Fluid Mech. 223, 313-324.
20. CAFLISCH, R. & ORELLANA, O. (1986) Long time existence for a slightly perturbed vortex sheet. Comm. Pure Appl. Math. 39(6), 807-838.
21. BEALE, J. T. & SCHAEFFER, D. G. (1988) Nonlinear behavior of model equations which are linearly ill-posed. Commun. Part. D iff. Eq. 13(4), 423-467.
22. BERETTA, E., BERTSCH, M. & DAL PASSO, R. (1995) Nonnegative solutions of a fourth order nonlinear degenerate parabolic equation. Arch. Ration. Mech. Ann. 129,175-200.
23. BERNIS, F. & FRIEDMAN, A. (1990) Higher order nonlinear degenerate parabolic equations. J. Diff. Eq. 83, 179-206.
24. BERTOZZI, A. L. & PUGH, M. (1994) The lubrication approximation for thin viscous films: the moving contact line with a 'porous media' cut off of Van der Waals interactions. Nonlinearity 7, 1535-1564.
25. BERTOZZI, A. L. & PUGH, M. (1996) The lubrication approximation for thin viscous films: regularity and long time behavior of weak solutions. Comm. Pure Appl. Math. 49(2), 85-123.
26. BERNIS, F. (1996) Finite speed of propagation and continuity of the interface for slow viscous flows. Adv. Diff. Eq. 1(3), 337-368.
27. BERNIS, F. (1996) Finite speed of propagation for thin viscous flows when 2 ≤ n < 3. CR. Acad. Sci. I - Math. 322, 1169-1174.
28. DAL PASSO, R., GARCKE, H. & GRÜN, G. (1998) On a fourth order degenerate parabolic equation: Global entropy estimates and qualitative behavior of solutions. SIAM J. Math. Anal. 29(2), 321-342.
29. OTTO, F. (1998) Lubrication approximation with prescribed non-zero contact angle: an existence result. Comm. Part. Diff. Eq. 23, 2077-2164.
30. BERETTA, E. (1997) Selfsimilar source solutions of a fourth order degenerate parabolic equation. Nonlinear Anal.-Theor. 29(7), 741-760.
31. BURELBACH, J. P., BANKOFF, S. G. & DAVIS, S. H. (1988) Nonlinear stability of evaporating/condensing liquid films. J. Fluid Mech. 195, 463-494.
32. SCHAAF, R. (1985) A class of Hamiltonian systems with increasing periods. J. Reine Angew. Math. 363, 96-109.
33. ORON, A. & ROSENAU, P. (1994) On a nonlinear thermocapillary effect in thin liquid layers. J. Fluid Mech. 273, 361-374.
34. NOVICK-COHEN, A. & PELETIER, L. A. (1992) The steady states of one-dimensional Sivashinsky equations. Quart. J. Appl. Math. 50(4), 759-777.
35. NOVICK-COHEN, A. & PELETIER, L. A. (1993) Steady states of the one-dimensional CahnHilliard equation. Proc. Roy. Soc. Edin. A, 123, 1071-1098.
36. TAN, M. J., BANKOFF, S. G. & DAVIS, S. H. (1990) Steady thermocapillary flows of thin liquid layers. I. Theory. Phys. Fluids A - Fluid, 2, 313-321.
37. BURELBACH, J. P., BANKOFF, S. G. & DAVIS, S. H. (1990) Steady thermocapillary flows of thin liquid layers. II. Experiment. Phys. Fluids A - Fluid, 2, 322-333.
38. DEISSLER, R. J. & ORON, A. (1992) Stable localized patterns in thin liquid films. Phys. Rev. Lett. 68(19), 2948-2951.
39. NOVICK-COHEN, A. (1993) A singular minimization problem for droplet profiles. Euro. J. Appl. Math. 4, 399-418.
40. ABRAMOWITZ, M. & STEGUN, I. A. (1972) Handbook of Mathematical Functions. Dover.
41. BERTOZZI, A. L., BRENNER, M. P., DUPONT, T. F & KADANOFF, L. P. (1994) Singularities and similarities in interface flow. In: L. Sirovich, editor, Trends and Perspectives in Applied Mathematics: Applied Mathematical Sciences Vol 100, pp. 155-208. Springer-Verlag.
42. SCHAAF, R. (1990) Global Solution Branches of Two Point Boundary Value Problems. Lecture Notes in Mathematics 1458. Springer-Verlag.
43. CHICONE, C. (1987) The monotonicity of the period function for planar Hamiltonian vector fields. J. Diff. Eq. 69, 310-321.
44. ROTHE, F. (1993) Remarks on periods of planar Hamiltonian systems. SIAM J. Math. Anal. 24, 129-154.
45. FURTER, J. E. & EILBECK, J. C. (1995) Analysis of bifurcations in reaction-diffusion systems with no-flux boundary-conditions - the Selkov model. Proc. Roy. Soc. Edin. A, 125, 413-438.
46. BARENBLATT, G. I. (1996) Scaling, Self-similarity, and Intermediate Asymptotics. Cambridge University Press.
47. BERTOZZI, A. & PUGH, M. (1997) Presented at 1997 APS Division of Fluid Dynamics meeting, November.
48. BOATTO, S., KADANOFF, L. & OLLA, P. (1993) Traveling wave solutions to thin film equations. Phys. Rev. E, 48, 4423-4431.