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Travelling wave solutions for a thin-film equation related to the spin-coating process

  • M. V. GNANN (a1), H. J. KIM (a2) and H. KNÜPFER (a2)
Abstract

We study a problem related to the spin-coating process in which a fluid coats a rotating surface. Our interest lies in the contact-line region for which we propose a simplified travelling wave approximation. We construct solutions to this problem by a shooting method that matches solution branches in the contact-line region and in the interior of the droplet. Furthermore, we prove uniqueness and qualitative properties of the solution connected to the fourth-order nature of the equation, such as a global maximum in the film height close to the contact line, elevated from the average height of the film.

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MVG received funding from the International Max Planck Research School (IMPRS) of the Max Planck Institute for Mathematics in the Sciences (MIS) in Leipzig, the University of Michigan at Ann Arbor, and the National Science Foundation under Grant No. NSF DMS-1054115. MVG is also supported by the Deutsche Forschungsgemeinschaft (grant GN 109/1-1). HJK acknowledges support by NRF Grant(NRF-2015R1A6A3A03020924) provided by the National Research Foundation of Korea.

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[1] Acrivos A., Shah M. J. & Petersen E. E. (1960) On the flow of a non-Newtonian liquid on a rotating disk. J. Appl. Phys. 13 (6), 963968.
[2] Agarwal R. P. & O'regan D. (2001) Singular problems on the infinite interval modelling phenomena in draining flows. IMA J. Appl. Math. 66 (6), 621635.
[3] Belgacem F. B., Gnann M. V. & Kuehn C. (2016) A dynamical systems approach for the contact-line singularity in thin-film flows. Nonlinear Anal.: Theory, Methods Appl. 144, 204235.
[4] Beretta E., Hulshof J. & Peletier L. A. (1996) On an ode from forced coating flow. J. Differ. Equ. 130 (1), 247265.
[5] Bernis F. & Peletier L. A. (1996) Two problems from draining flows involving third-order ordinary differential equations. SIAM J. Math. Anal. 27 (2), 515527.
[6] Bernis F., Peletier L. A. & Williams S. M. (1992) Source type solutions of a fourth order nonlinear degenerate parabolic equation. Nonlinear Anal. 18 (3), 217234.
[7] Bertozzi A., Shearer M. & Buckingham R. (2003) Thin film traveling waves and the Navier slip condition. SIAM J. Appl. Math. 63 (2), 722744.
[8] Bertozzi A. L. & Greer J. B. (2004) Low-curvature image simplifiers: Global regularity of smooth solutions and laplacian limiting schemes. Commun. Pure Appl. Math. 57 (6), 764790.
[9] Bertozzi A. L., Münch A., Shearer M. & Zumbrun K. (2001) Stability of compressive and undercompressive thin film travelling waves. Eur. J. Appl. Math. 12 (03), 253291.
[10] Bertozzi A. L. & Shearer M. (2000) Existence of undercompressive traveling waves in thin-film equations. SIAM J. Math. Anal. 32 (1), 194213.
[11] Birnie D. P. III & Manley M. (1997) Combined flow and evaporation of fluid on a spinning disk. Phys. Fluids (1994-present) 9 (4), 870875.
[12] Boatto S., Kadanoff L. P. & Olla P. (1993) Traveling-wave solutions to thin-film equations. Phys. Rev. E 48 (6), 44234431.
[13] Bonn D., Eggers J., Indekeu J., Meunier J. & Rolley E. (2009) Wetting and spreading. Rev. Mod. Phys. 81 (2), 739805.
[14] Caginalp G. (1990) The dynamics of a conserved phase field system: Stefan-like, hele-shaw, and cahn-hilliard models as asymptotic limits. IMA J. Appl. Math. 44 (1), 7794.
[15] Chiricotto M. & Giacomelli L. (2011) Droplets spreading with contact-line friction: Lubrication approximation and traveling wave solutions. Commun. Appl. Ind. Math. 2 (2), 116.
[16] Cuesta C. M. & Velázquez J. J. L. (2012) Analysis of oscillations in a drainage equation. SIAM J. Math. Anal. 44 (3), 15881616.
[17] De Gennes P. G. (1985) Wetting: Statics and dynamics. Rev. Mod. Phys. 57 (3), 827.
[18] Emslie A. G., Bonner F. T. & Peck L. G. (1958) Flow of a viscous liquid on a rotating disk. J. Appl. Phys. 29 (5), 858862.
[19] Engelberg S. (1996) The stability of the viscous shock profiles of the Burgers equation with a fourth order viscosity. Commun. Partial Differ. Equ. 21 (5–6), 889922.
[20] Evans L. C. (1998) Partial Differential Equations, American Mathematical Society, Providence, RI.
[21] Evans P. L., King J. R. & Münch A. (2006) Intermediate-asymptotic structure of a dewetting rim with strong slip. Appl. Math. Res. Express 2006, Article ID 25262, 125.
[22] Fraysse N. & Homsy G. M. (1994) An experimental study of rivulet instabilities in centrifugal spin coating of viscous newtonian and non-newtonian fluids. Phys. Fluids 6 (4), 14911504.
[23] Galaktionov V. A. (2003) On higher-order viscosity approximations of one-dimentional conservation laws.
[24] Giacomelli L., Gnann M., Knüpfer H. & Otto F. (2014) Well-posedness for the Navier-slip thin-film equation in the case of complete wetting. J. Differ. Equ. 257 (1), 1581.
[25] Giacomelli L., Gnann M. & Otto F. (2013) Regularity of source-type solutions to the thin-film equation with zero contact angle and mobility exponent between 3/2 and 3. Eur. J. Appl. Math. 24 (5), 735760.
[26] Giacomelli L., Gnann M. V. & Otto F. (2016) Rigorous asymptotics of traveling-wave solutions to the thin-film equation and Tanners law. Nonlinearity 29 (9), 2497.
[27] Greer J. B. & Bertozzi A. L. (2004) Traveling wave solutions of fourth order PDEs for image processing. SIAM J. Math. Anal. 36 (1), 3868.
[28] Guidotti P. & Longo K. (2011) Well-posedness for a class of fourth order diffusions for image processing. NoDEA Nonlinear Differ. Equ. Appl. 18 (4), 407425.
[29] Holloway K. E., Tabuteau H. & De Bruyn J. R. (2010) Spreading and fingering in a yield-stress fluid during spin-coating. Rheol. Acta 49 (3), 245254.
[30] Kim J. S., Kim S. & Ma F. (1993) Topographic effect of surface roughness on thin-film flow. J. Appl. Phys. 73 (1), 422428.
[31] King J. R. & Taranets R. M. (2013) Asymmetric travelling waves for the thin-film equation. J. Math. Anal. Appl. 404 (2), 399419.
[32] Ma F. & Hwang J. H. (1990) The effect of air shear on the flow of a thin liquid film over a rough rotating disk. J. Appl. Phys. 68 (3), 12651271.
[33] Melo F., Joanny J. F. & Fauve S. (1989) Fingering instability of spinning drops. Phys. Rev. Lett. 63 (18), 1958.
[34] Middleman S. (1987) The effect of induced air-flow on the spin-coating of viscous liquids. J. Appl. Phys. 62 (6), 25302532.
[35] Moriarty J. A., Schwartz L. W. & Tuck E. O. (1991) Unsteady spreading of thin liquid films with small surface tension. Phys. Fluids A: Fluid Dyn. (1989–1993) 3 (5), 733742.
[36] Münch A., Wagner B. A. & Witelski T. P. (2005) Lubrication models with small to large slip lengths. J. Eng. Math. 53 (3), 359383.
[37] Myers T. G. & Charpin J. P. F. (2001) The effect of the coriolis force on axisymmetric rotating thin-film flows. Int. J. Non-linear Mech. 36 (4), 629635.
[38] Oron A., Davis S. H. & Bankoff S. G. (1997) Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931980.
[39] Peurrung L. M. & Graves D. B. (1991) Film thickness profiles over topography in spin-coating. J. Electrochem. Soc. 138 (7), 21152124.
[40] Schwartz L. W. & Roy R. V. (2004) Theoretical and numerical results for spin-coating of viscous liquids. Phys. Fluids 16 (3), 569584.
[41] Tritscher P. & Broadbridge P. (1995) Grain boundary grooving by surface diffusion: An analytic nonlinear model for a symmetric groove. Proceedings: Math. Phys. Sci. 450 (1940), 569587.
[42] Troy W. C. (1993) Solutions of third-order differential equations relevant to draining and coating flows. SIAM J. Math. Anal. 24 (1), 155171.
[43] Tu Y.-O. (1983) Depletion and retention of fluid on a rotating disk. J. Tribol. 105 (4), 625629.
[44] Tuck E. O. & Schwartz L. W. (1990) A numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows. SIAM Rev. 32 (3), 453469.
[45] Wang J. & Zhang Z. (1998) A boundary value problem from draining and coating flows involving a third-order ordinary differential equation. Zeitschrift für Angewandte Mathematik und Physik ZAMP 49 (3), 506513.
[46] Wu L. (2006) Spin-coating of thin liquid films on an axisymmetrically heated disk. Phys. Fluids (1994-present) 18 (6), 063602.
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European Journal of Applied Mathematics
  • ISSN: 0956-7925
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