Time integration and steady-state continuation for 2D lubrication equations

SIAM Journal on Applied Dynamical Systems

Volumn 9, Issue 2, 2010, Pages 484-518

  Beltrame, Philippe ^a,b   Thiele, Uwe ^c  

^a UNIVERSITY OF AVIGNON   (France)

^b UNIVERSITY OF AUGSBURG   (Germany)

^c LOUGHBOROUGH UNIVERSITY   (United Kingdom)

Author keywords

Continuation; Depinning; Dewetting; Exponential propagation; Krylov reduction; Lubrication equation; Time integration

Indexed keywords

CAYLEY TRANSFORMS; DE-WETTING; DEPINNING; DYNAMIC PROCESS; DYNAMICAL SYSTEMS APPROACH; HOMOCLINIC CONNECTION; HOMOGENEOUS SUBSTRATES; LUBRICATION EQUATION; NUMERICAL PROBLEMS; SCALE SEPARATION; SPACE AND TIME; STEADY STATE SOLUTION; STRUCTURING PROCESS; THIN LIQUID FILM; TIME INTEGRATION; TIME INTEGRATION ALGORITHMS; TIME STEP; TIME-SCALES; TIME-STEPPING;

ALGORITHMS; DYNAMICAL SYSTEMS; LIQUID FILMS; WAVELET TRANSFORMS;

LUBRICATION;

EID: 77953278361     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/080718619     Document Type: Article

References (103)

1. D. BANDYOPADHYAY, R. GULABANI, AND A. SHARMA, Stability and dynamics of bilayers, Ind. Eng. Chem. Res., 44 (2005), pp. 1259-1272.
2. D. BANDYOPADHYAY, A. SHARMA, U. THIELE, AND P. D. S. REDDY, Electric field induced interfacial instabilities and morphologies of thin viscous and elastic bilayers, Langmuir, 25 (2009), pp. 9108-9118.
3. J. BECKER, G. GRÜN, R. SEEMANN, H. MANTZ, K. JACOBS, K. R. MECKE, AND R. BLOSSEY, Complex dewetting scenarios captured by thin-film models, Nature Mater., 2 (2003), pp. 59-63.
4. P. BELTRAME, P. HÄNGGI, AND U. THIELE, Depinning of three-dimensional drops from wettability defects, Europhys. Lett., 86 (2009), paper 24006.
5. M. BEN AMAR, L. CUMMINGS, AND Y. POMEAU, Singular points of a moving contact line, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univ ers Sci. Terre, 329 (2001), pp. 277-282.
6. A. L. BERTOZZI AND M. P. BRENNER, Linear stability and transient growth in driven contact lines, Phys. Fluids, 9 (1997), pp. 530-539.
7. A. L. BERTOZZI, G. GRÜN, AND T. P. WITELSKI, Dewetting films: Bifurcations and concentrations, Nonlinearity, 14 (2001), pp. 1569-1592.
8. A. L. BERTOZZI, A. MÜNCH, X. FANTON, AND A. M. CAZABAT, Contact line stability and "undercompressive shocks" in driven thin film flow, Phys. Rev. Lett., 81 (1998), pp. 5169-5173.
9. A. L. BERTOZZI, A. MÜNCH, AND M. SHEARER, Undercompressive shocks in thin film flows, Phys. D, 134 (1999), pp. 431-464.
10. M. BESTEHORN AND K. NEUFFER, Surface patterns of laterally extended thin liquid films in three dimensions, Phys. Rev. Lett., 87 (2001), paper 046101.
11. M. BESTEHORN, A. POTOTSKY, AND U. THIELE, 3D large scale Marangoni convection in liquid films, Eur. Phys. J. B Condens. Matter Phys., 33 (2003), pp. 457-467.
12. D. BONN, J. EGGERS, J. INDEKEU, J. MEUNIER, AND E. ROLLEY, Wetting and spreading, Rev. Modern Phys., 81 (2009), pp. 739-805.
13. H. S. BROWN, I. G. KEVREKIDIS, A. ORON, AND P. ROSENAU, Bifurcations and pattern-formation in the regularized Kuramoto-Sivashinsky equation, Phys. Lett. A, 163 (1992), pp. 299-308.
14. J. M. BURGESS, A. JUEL, W. D. MCCORMICK, J. B. SWIFT, AND H. L. SWINNEY, Suppression of dripping from a ceiling, Phys. Rev. Lett., 86 (2001), pp. 1203-1206.
15. E. BUZANO AND M. GOLUBITSKY, Bifurcation on the hexagonal lattice and the planar Benard problem, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 308 (1983), pp. 617-667.
16. J. W. CAHN AND J. E. HILLIARD, Free energy of a nonuniform system. 1. Interfacial free energy, J. Chem. Phys., 28 (1958), pp. 258-267.
17. A. M. CAZABAT, F. HESLOT, S. M. TROIAN, AND P. CARLES, Fingering instability of thin spreading films driven by temperature gradients, Nature, 346 (1990), pp. 824-826.
18. L. CHEN, L. ZHUANG, P. DESHPANDE, AND S. CHOU, Novel polymer patterns formed by lithographically induced self-assembly (LISA), Langmuir, 21 (2005), pp. 818-821.
19. J. CONWAY, The orbifold notation for surface groups, in Groups, Combinatorics and Geometry, Proceedings of the L. M. S. Durham Symposium, Durham, UK, M. W. Liebeck and J. S., eds., Cambridge University Press, London, 1992.
20. B. P. COOK, A. L. BERTOZZI, AND A. E. HOSOI, Shock solutions for particle-laden thin films, SIAM J. Appl. Math., 68 (2008), pp. 760-783.
21. J. D. CRAWFORD AND E. KNOBLOCH, Symmetry and symmetry-breaking bifurcations in fluid-dynamics, in Ann. Rev. Fluid Mech. 23, Annual Reviews, Palo Alto, CA, 1991, pp. 341-387.
22. M. C. CROSS AND P. C. HOHENBERG, Pattern formation out of equilibrium, Rev. Modern Phys., 65 (1993), pp. 851-1112.
23. Z. DAGAN AND L. M. PISMEN, Marangoni waves induced by a multistable chemical reaction on thin liquid films, J. Colloid Interface Sci., 99 (1984), pp. 215-225.
24. P.-G. DE GENNES, Wetting: Statics and dynamics, Rev. Modern Phys., 57 (1985), pp. 827-863.
25. R. J. DEISSLER AND A. ORON, Stable localized patterns in thin liquid films, Phys. Rev. Lett., 68 (1992), pp. 2948-2951.
26. B. V. DERJAGUIN, N. V. CHURAEV, AND V. M. MULLER, Surface Forces, Consultants Bureau, New York, 1987.
27. J. A. DIEZ AND L. KONDIC, Computing three-dimensional thin film flows including contact lines, J. Comput. Phys., 183 (2002), pp. 274-306.
28. E. J. DOEDEL, A. R. CHAMPNEYS, T. F. FA IRGRIEVE, Y. A. KUZNETSOV, B. SANDSTEDE, AND X. J. WANG, AUTO97: Continuation and B ifurcation Software for Ordinary Differential Equations, Concordia University, Montreal, 2000.
29. M. H. ERES, L. W. SCHWARTZ, AND R. V. ROY, Fingering phenomena for driven coating films, Phys. Fluids, 12 (2000), pp. 1278-1295.
30. M. FERMIGIER, L. LIMAT, J. E. WESFREID, P. BOUDINET, AND C. QUILLIET, 2-dimensional patterns in Rayleigh-Taylor instability of a thin-layer, J. Fluid Mech., 236 (1992), pp. 349-383.
31. R. A. FRIESNER, L. S. TUCKERMAN, B. C. DORNBLASER, AND T. V. RUSSO, A method for exponential propagation of large systems of stiff nonlinear differential equations, J. Sci. Comput., 4 (1989), pp. 327-354.
32. S. GASNER, P. BLOMGREN, AND A. PALACIOS, Noise-induced intermittency in cellular pattern-forming systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), pp. 2765-2779. (Pubitemid 47479153)
33. K. B. GLASER AND T. P. WITELSKI, Coarsening dynamics of dewetting films, Phys. Rev. E, 67 (2003), paper 016302.
34. A. A. GOLOVIN, A. A. NEPOMNYASHCHY, S. H. DAVIS, AND M. A. ZAKS, Convective Cahn-Hilliard models: From coarsening to roughening, Phys. Rev. Lett., 86 (2001), pp. 1550-1553.
35. G. GRÜN, On the convergence of entropy consistent schemes for lubrication-type equations in multiple space dimensions, Math. Comput., 72 (2003), pp. 1251-1279.
36. G. GRÜN AND M. RUMPF, Simulation of singularities and instabilities arising in thin film flow, European J. Appl. Math., 12 (2001), pp. 293-320.
37. D. HO, Tchebychev acceleration technique for large scale nonsymmetric matrices, Numer. Math., 56 (1990), pp. 721-734.
38. M. HOCHBRUCK AND C. LUBICH, On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 34 (1997), pp. 1911-1925.
39. M. HOCHBRUCK, C. LUBICH, AND H. SELHOFER, Exponential integrators for large systems of differential equations, SIAM J. Sci. Comput., 19 (1998), pp. 1552-1574.
40. J. N. ISRAELACHVILI, Intermolecular and Surface Forces, Academic Press, London, 1992.
41. 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.
42. K. JOHN, M. BÄR, AND U. THIELE, Self-propelled running droplets on solid substrates driven by chemical reactions, Eur. Phys. J. E, 18 (2005), pp. 183-199.
43. K. JOHN, P. HÄNGGI, AND U. THIELE, Ratchet-driven fluid transport in bounded two-layer films of immiscible liquids, Soft Matter, 4 (2008), pp. 1183-1195.
44. S. KALLIADASIS, Nonlinear instability of a contact line driven by gravity, J. Fluid Mech., 413 (2000), pp. 355-378.
45. S. KALLIADASIS AND U. THIELE, EDS., Thin Films of Soft Matter, CISM 490, Springer, Vienna, New York, 2007.
46. D. E. KATAOKA AND S. M. TROIAN, A theoretical study of instabilities at the advancing front of thermally driven coating films, J. Colloid Interface Sci., 192 (1997), pp. 350-362.
47. I. G. KEVREKIDIS, B. NICOLAENKO, AND J. C. SCOVEL, Back in the saddle again: A computer assisted study of the Kuramoto-Sivashinsky equation, SIAM J. Appl. Math., 50 (1990), pp. 760-790.
48. R. KHANNA, A. SHARMA, AND G. REITER, The ABC of pattern evolution in self-destruction of thin polymer films, EPJ direct, E 2 (2000), pp. 1-9.
49. V. LAPUERTA, F. J. MANCEBO, AND J. M. VEGA, Control of Rayleigh-Taylor instability by vertical vibration in large aspect ratio containers, Phys. Rev. E, 64 (2001), paper 016318.
50. N. LE GRAND, A. DAERR, AND L. LIMAT, Shap e and motion of drops sliding down an inclined plane, J. Fluid Mech., 541 (2005), pp. 293-315.
51. J. LIESEN AND P. TICHÝ, Convergence analysis of Krylov subspace methods, GAMM Mitt. Ges. Angew. Math. Mech., 27 (2004), pp. 153-173.
52. Z. LIN, T. KERLE, S. M. BAKER, D. A. HOAGLAND, E. SCHÄFFER, U. STEINER, AND T. P. RUSSELL, Electric field induced instabilities at liquid/liquid interfaces, J. Chem. Phys., 114 (2001), pp. 2377-2381.
53. T. A. MANTEUFFEL, The Tchebychev iteration for nonsymmetric linear systems, Numer. Math., 28 (1977), pp. 307-327.
54. O. K. MATAR AND R. V. CRASTER, Models for Marangoni drying, Phys. Fluids, 13 (2001), pp. 1869-1883.
55. K. MEERBERGEN AND D. ROOSE, Matrix transformations for computing rightmost eigenvalues of large sparse non-symmetric eigenvalue problems, IMA J. Numer. Anal., 16 (1996), pp. 297-346.
56. D. MERKT, A. POTOTSKY, M. BESTEHORN, AND U. THIELE, Long-wave theory of bounded two-layer films with a free liquid-liquid interface: Short-and long-time evolution, Phys. Fluids, 17 (2005), paper 064104.
57. V. S. MITLIN, Dewetting of solid surface: Analogy with spinodal decomposition, J. Colloid Interface Sci., 156 (1993), pp. 491-497.
58. A. MÜNCH, Pinch-off transition in Marangoni-driven thin films, Phys. Rev. Lett., 91 (2003), paper 016105.
59. A. MÜNCH, Dewetting rates of thin liquid films, J. Phys. Condens. Matter, 17 (2005), pp. S309-S318.
60. A. ORON, Nonlinear dynamics of three-dimensional long-wave Marangoni instability in thin liquid films, Phys. Fluids, 12 (2000), pp. 1633-1645.
61. A. ORON, Three-dimensional nonlinear dynamics of thin liquid films, Phys. Rev. Lett., 85 (2000), pp. 2108-2111.
62. A. ORON, S. H. DAVIS, AND S. G. BANKOFF, Long-scale evolution of thin liquid films, Rev. Modern Phys., 69 (1997), pp. 931-980.
63. A. ORON AND P. ROSENAU, Formation of patterns induced by thermocapillarity and gravity, J. Phys. II France, 2 (1992), pp. 131-146.
64. A. ORON AND P. ROSENAU, On a nonlinear thermocapillary effect in thin liquid layers, J. Fluid Mech., 273 (1994), pp. 361-374.
65. A. PEREIRA, P. M. J. TREVELYAN, U. THIELE, AND S. KALLIADASIS, Dynamics of a horizontal thin liquid film in the presence of reactive surfactants, Phys. Fluids, 19 (2007), paper 112102.
66. T. PODGORSKI, J.-M. FLESSELLES, AND L. LIMAT, Corners, cusps, and pearls in running drops, Phys. Rev. Lett., 87 (2001), paper 036102.
67. A. POTOTSKY, M. BESTEHORN, D. MERKT, AND U. THIELE, Alternative pathways of dewetting for a thin liquid two-layer film, Phys. Rev. E, 70 (2004), paper 025201 (R).
68. A. POTOTSKY, M. BESTEHORN, D. MERKT, AND U. THIELE, Morphology changes in the evolution of liquid two-layer films, J. Chem. Phys., 122 (2005), paper 224711.
69. A. POTOTSKY, M. BESTEHORN, D. MERKT, AND U. THIELE, 3d surface patterns in liquid two-layer films, Europhys. Lett., 74 (2006), pp. 665-671.
70. G. REITER, Dewetting of thin polymer films, Phys. Rev. Lett., 68 (1992), pp. 75-78.
71. G. REITER AND A. SHARMA, Auto-optimization o f dewetting rates by rim instabilities in slipping polymer films, Phys. Rev. Lett., 87 (2001), paper 166103.
72. E. RUCKENSTEIN AND R. K. JAIN, Spontaneous rupture of thin liquid films, J. Chem. Soc. Faraday Trans. II, 70 (1974), pp. 132-147.
73. Y. SAAD, Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 29 (1992), pp. 209-228.
74. M. SADKANE, A block Arnoldi-Chebyshev method for computing the leading eigenpairs of large sparse unsymmetric matrices, Numer. Math., 64 (1993), pp. 181-193.
75. R. SEEMANN, S. HERMINGHAUS, C. NETO, S. SCHLAGOWSKI, D. PODZIMEK, R. KONRAD, H. MANTZ, AND K. JACOBS, Dynamics and structure fo rmation in thin polymer melt films, J. Phys. Condens. Matter, 17 (2005), pp. S267-S290.
76. R. SEYDEL, Practical Bifurcation and Stability A nalysis, Springer-Verlag, New York, 1994.
77. A. SHARMA, Relationship of thin film stability and morphology to macroscopic parameters of wetting in the apolar and polar systems, Langmuir, 9 (1993), pp. 861-869.
78. A. SHARMA AND R. KHANNA, Pattern formation in unstable thin liquid films, Phys. Rev. Lett., 81 (1998), pp. 3463-3466.
79. A. SHARMA AND G. REITER, Instability of thin polymer films on coated substrates: Rupture, dewetting and drop formation, J. Colloid Interface Sci., 178 (1996), pp. 383-399.
80. J. H. SNOEIJER, E. RIO, N. LE GRAND, AND L. LIMAT, Self-similar flow and contact line geometry at the rear of cornered drops, Phys. Fluids, 17 (2005), paper 072101.
81. M. A. SPAID AND G. M. HOMSY, Stability of N ewtonian and viscoelastic dynamic contact lines, Phys. Fluids, 8 (1996), pp. 460-478.
82. J. SUR, A. L. BERTOZZI, AND R. P. BEHRINGER, Reverse undercompressive shock structures in driven thin film flow, Phys. Rev. Lett., 90 (2003), paper 126105.
83. U. THIELE, Open questions and promising new fields in dewetting, Eur. Phys. J. E, 12 (2003), pp. 409-416.
84. U. THIELE, Structure formation in thin liquid films, in Thin Films of Soft Matter, S. Kalliadasis and U. Thiele, eds., Springer, Vienna, 2007, pp. 25-93.
85. U. THIELE, Thin film evolution equations from (evaporating) dewetting liquid layers to epitaxial growth, J. Phys. Condens. Matter, 22 (2010), paper 084019.
86. U. THIELE, L. BRUSCH, M. BESTEHORN, AND M. BÄR, Modelling thin-film dewetting on structured substrates and templates: Bifurcation analysi s and numerical simulations, Eur. Phys. J. E, 11 (2003), pp. 255-271.
87. U. THIELE, B. GOYEAU, AND M. G. VELARDE, Film flow on a porous substrate, Phys. Fluids, 21 (2009), paper 014103.
88. U. THIELE, K. JOHN, AND M. BÄR, Dynamica l model for chemically driven running droplets, Phys. Rev. Lett., 93 (2004), paper 027802.
89. U. THIELE AND E. KNOBLOCH, Front and back instability of a liquid film on a slightly inclined plate, Phys. Fluids, 15 (2003), pp. 892-907.
90. U. THIELE AND E. KNOBLOCH, Thin liquid films on a slightly inclined heated plate, Phys. D, 190 (2004), pp. 213-248.
91. U. THIELE AND E. KNOBLOCH, Driven drops on heterogeneous substrates: Onset of sliding motion, Phys. Rev. Lett., 97 (2006), paper 204501.
92. U. THIELE AND E. KNOBLOCH, On the depinni ng of a driven drop on a heterogeneous substrate, New J. Phys., 8 (2006), paper 313.
93. U. THIELE, I. VANCEA, A. J. ARCHER, M. J. ROBBINS, L. FRASTIA, A. STANNARD, E. PAULIAC-VAUJOUR, C. P. MARTIN, M. O. BLUNT, AND P. J. MORIARTY, Modelling approaches to the dewetting of evaporating thin films of nanoparticle suspensions, J. Phys. Condens. Matter, 21 (2009), paper 264016.
94. U. THIELE, J. M. VEGA, AND E. KNOBLOCH, Long-wave Marangoni instability with vibration, J. Fluid Mech., 546 (2006), pp. 61-87.
95. U. THIELE, M. G. VELARDE, AND K. NEUFFER, Dewetting: Film rupture by nucleation in the spinodal regime, Phys. Rev. Lett., 87 (2001), paper 016104.
96. U. THIELE, M. G. VELARDE, K. NEUFFER, M. BESTEHORN, AND Y. POMEAU, Sliding drops in the diffuse interface model coupled to hydrodynamics, Phys. Rev. E, 64 (2001), paper 061601.
97. U. THIELE, M. G. VELARDE, K. NEUFFER, AND Y. POMEAU, Film rupture in the diffuse interface model coupled to hydrodynamics, Phys. Rev. E, 64 (2001), paper 031602.
98. M. TOKMAN, Efficient integration of large stiff systems of ODEs with exponential propagation (epi) methods, J. Comput. Phys., 213 (2006), pp. 748-776.
99. L. S. TUCKERMAN AND D. BARKLEY, Bifurcation analysis for timesteppers, in Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, IMA Vol. Math. Appl. 119, E. Doedel and L. Tuckerman, eds., Springer, New York, 2000, pp. 466-543.
100. R. VERMA, A. SHARMA, K. KARGUPTA, AND J. BHAUMIK, Electric field induced instability and pattern formation in thin liquid films, Langmuir, 21 (2005), pp. 3710-3721.
101. P. WHEELER AND D. BARKLEY, Computation of spiral spectra, IAM J. Appl. Dyn. Syst., 5 (2006), pp. 157-177.
102. T. WITELSKI AND M. BOWEN, Adi schemes for h igher-order nonlinear diffusion equations, Appl. Numer. Math., 45 (2003), pp. 331-351.
103. L. ZHORNITSKAYA AND A. L. BERTOZZI, Positivity-preserving numerical schemes for lubrication-type equations, SIAM J. Numer. Anal., 37 (2000), pp. 523-555.