Undercompressive shocks in thin film flows

Physica D: Nonlinear Phenomena

Volumn 134, Issue 4, 1999, Pages 431-464

  Bertozzi, A L ^a   Munch A ^a   Shearer, M ^b  

^a DUKE UNIVERSITY   (United States)

^b NORTH CAROLINA STATE UNIVERSITY   (United States)

Author keywords

Capillarity; Fourth order diffusion; Hyperbolic conservation law; Marangoni stress; Numerical simulation; Thin films; Undercompressive shock

Indexed keywords

EID: 0038919593     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(99)00134-7     Document Type: Article

References (47)

1. R. Abeyaratne, J.K. Knowles, Implications of viscosity and strain gradient effects for the kinetics of propagating phase boundaries in solids, SIAM J. Appl. Math. 51 (1991) 1205-1221.
2. E. Beretta, M. Berstch, R.D. Passo, Nonnegative solutions of a fourth order nonlinear degenerate parabolic equation, Arch. Rational Mech. Anal. 129 (1995) 175-200.
3. A.L. Bertozzi, M.P. Brenner, Linear stability and transient growth in driven contact lines, Phys. Fluids 9 (3) (1997) 530-539.
4. A.L. Bertozzi, A. Munch, X. Fanton, A.M. Cazabat, Contact line stability and 'undercompressive shocks' in driven thin film flow, 1998.
5. A.L. Bertozzi, M. Pugh, The lubrication approximation for thin viscous films; the moving contact line with a 'porous media' cut off of Van der Waals interactions, Nonlinearity 7 (1994) 1535-1564.
6. A.L. Bertozzi, M. Pugh, The lubrication approximation for thin viscous films; regularity and long time behavior of weak solutions, Comm. Pur. Appl. Math. 49 (2) (1996) 85-123.
7. A.L. Bertozzi, The mathematics of moving contact lines in thin liquid films, Notices of the American Math. Soc. 45 (6) (1998) 697-698.
8. A.L. Bertozzi, M.P. Brenner, T.F. Dupont, L.P. Kadanoff, Singularities and similarities in interface flow, in: L. Sirovich (Ed.), Trends and Perspectives in Applied Mathematics, vol. 100, Applied Mathematical Sciences, Springer, New York, 1994, pp. 155-208.
9. P. Carles, A.-M. Cazabat, The thickness of surface-tension-gradient-driven spreading films, J. Coll. Interfac. Sci. 157 (1993) 196-201.
10. A.M. Cazabat, F. Heslot, S.M. Troian, P. Carles, Finger instability of this spreading films driven by temperature gradients, Nature 346 (6287) (1990) 824-826.
11. J.R. deBruyn, Growth of fingers at a driven three-phase contact line, Phys. Rev. A 46 (8) (1992) R4500-R4503.
12. R.G. deGennes, Wetting; Statics and dynamics, Rev. Mod. Phys. 57 (1985) 827.
13. J.J. Dodd, Convective stability of shock profile solutions of a modified KdV-Burgers equation, Ph.D. Thesis, Univ. Maryland, 1996.
14. E.B. Dussan, V.S. Davis, On the motion of a fluid-fluid interface along a solid surface, J. Fluid Mech. 65 (1974) 71-95.
15. S. Engelberg, The stability of the viscous shock profiles of the Burgers equation with a fourth order viscosity, Commun, in Partial Differential Equations 21 (5-6) (1996) 889-992.
16. X. Fanton, A.M. Cazabat, D. Quéré, Thickness and shape of films driven by a Marangoni flow, Langmuir 12 (24) (1996) 5875-5880.
17. X. Fanton, Etalement et instabilités de films de mouillage en presence de gradients de tension superficielle, Ph.D. Thesis, Université Paris 6 Pierre et Marie Curie, Paris, 1998.
18. R.A. Gardner, K. Zumbrun, The gap lemma and geometric criteria for stability of viscous shock profiles, Comm. Pure Appl. Math. 51 (1998) 789-847.
19. J. Goodman, A. Majda, The validity of the modified equation for nonlinear shock waves, J. Comp. Phys. 58 (1985) 336-348.
20. H.P. Greenspan, On the motion of a small viscous droplet that wets a surface, J. Fluid Mech. 84 (1978) 125-143.
21. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, Berlin, 1986.
22. P.J. Haley, M.J. Miksis, The effect of the contact line on droplet spreading, J. Fluid Mech. 223 (1991) 57-81.
23. B. Hayes, M. Shearer, Undercompressive shocks for scalar conservation laws with non-convex fluxes. Proc. A Royal Soc. Edinburgh, to appear.
24. B.T. Hayes, P.G. LeFloch, Nonclassical shock waves; scalar conservation laws, Arch. Rat. Mech. Anal. 139 (1997) 1-56.
25. A.C. Hindmarsh, Odepack, a systematized collection of ode solvers, in: R.S. Stepleman et al. (Eds.), Scientific Computing, North-Holland, Amsterdam, 1983, pp. 55-64.
26. C. Huh, L.E. Scriven, Hydrodynamic model of steady movement of a solid /liquid/fluid contact line, J. Colloid Interface Sci. 35 (1971) 85-101.
27. H. Huppert, Flow and instability of a viscous current down a slope, Nature 300 (1982) 427-429.
28. E. Isaacson, D. Marchesin, B. Plohr, Transitional waves for conservation laws, SIAM J. Math. Anal. 21 (1990) 837-866.
29. D. Jacobs, B. McKinney, M. Shearer, Travelling wave solutions of the modified Korteweg-deVries-Burgers equation, J. Diff. Equ. 116 (2) (1995) 448-467.
30. M.F.G. Johnson, R.A. Schluter, M.J. Miksis, S.G. Bankoff, 1998. Northwestern Univ. Applied Math. Technical Report No. 9604.
31. C.K.R.T. Jones, R.A. Gardner, T. Kapitula, Stability of travelling waves for nonconvex scalar viscous conservation laws, Comm. Pure Appl. Math. 46 (1993) 505-526.
32. T. Kapitula, B. Sandstede, Stability of bright solitary-wave solutions to perturbed nonlinear Schrodinger equations, 1997.
33. D.E. Kataoka, S.M. Troian, A theoretical study of instabilities at the advancing front of thermally driven coating films, J. Coll. Int. Sci. 192 (1997) 350-362.
34. D.E. Kataoka, S.M. Troian, Stabilizing the advancing front of thermally driven climbing films, J. Coll. Int. Sci. 203 (1998) 335-344.
35. N. Kopell, L.N. Howard, Bifurcations and trajectories joining critical points, Adv. Math. 18 (1975) 306-358 .
36. P.D. Lax, Hyperbolic systems of conservation laws ii, Comm. Pure Appl. Math. 10 (1957) 537-566.
37. P.D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, vol. 11, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 1973.
38. T.-P. Liu, K. Zumbrun, Nonlinear stability of general undercompressive shock waves, Comm. Math. Phys. 174 (1995) 319-345.
39. V. Ludviksson, E.N. Lightfoot, The dynamics of thin liquid films in the presence of surface-tension gradients, Am. Inst. Chem. Engrs. J. 17 (5) (1971) 1166-1173.
40. 2 - 1, J. Math. Anal. Appl. 114 (1986) 584-592.
41. D. Michelson, Strong viscous shocks for systems of conservation laws with a high order of dissipatio, J. Diff. Eqs. 71 (1988) 246-254.
42. M. Renardy, A singularly perturbed problem related to surfactant spreading on thin films, Applications 27(3) (1996) 287-296.
43. M. Shearer, D.G. Schaeffer, D. Marchesin, P. Paes-Leme, Solution of the Riemann problem for a prototype 2x2 system of non-strictly hyperbolic conservation laws, Arch. Rat. Mech. Anal. 97 (1987) 299-320.
44. N. Silvi, E.B. Dussan V, On the rewetting of an inclined solid surface by a liquid, Physics of Fluids 28 (1985) 5-7.
45. M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Rat. Mech. Anal. 81 (1983) 301-315.
46. S.M. Troian, E. Herbolzheimer, S.A. Safran, J.F. Joanny, Fingering instabilities of driven spreading films, Europhys. Lett. 10 (1) (1989) 25-30.
47. C.C. Wu, New theory of MHD shock waves, in: M. Shearer (Ed.), Viscous Profiles and Numerical Methods for Shock Waves, SIAM, 1991.