Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T12:31:30.120Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  17 July 2017

M. V. GNANN ,
H. J. KIM  and
H. KNÜPFER
M. V. GNANN
Affiliation:
Center for Mathematics, Technical University of Munich, Boltzmannstr. 3, 85747 Garching near Munich, Germany email: gnann@ma.tum.de
H. J. KIM
Affiliation:
Institute of Applied Mathematics and IWR, University of Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany emails: kim.hyeonjeong@math.uni-heidelberg.de, knuepfer@uni-heidelberg.de
H. KNÜPFER
Affiliation:
Institute of Applied Mathematics and IWR, University of Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany emails: kim.hyeonjeong@math.uni-heidelberg.de, knuepfer@uni-heidelberg.de
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

PDEs for fluid mechanicstravelling wave solutionsnon-linear fourth-order equationsexistence and uniquenessthin fluid films
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 3 , June 2018 , pp. 369 - 392
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[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), 963–968.Google Scholar
[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), 621–635.Google Scholar
[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, 204–235.Google Scholar
[4] Beretta, E., Hulshof, J. & Peletier, L. A. (1996) On an ode from forced coating flow. J. Differ. Equ. 130 (1), 247–265.CrossRefGoogle Scholar
[5] Bernis, F. & Peletier, L. A. (1996) Two problems from draining flows involving third-order ordinary differential equations. SIAM J. Math. Anal. 27 (2), 515–527.Google Scholar
[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), 217–234.Google Scholar
[7] Bertozzi, A., Shearer, M. & Buckingham, R. (2003) Thin film traveling waves and the Navier slip condition. SIAM J. Appl. Math. 63 (2), 722–744.CrossRefGoogle Scholar
[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), 764–790.CrossRefGoogle Scholar
[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), 253–291.Google Scholar
[10] Bertozzi, A. L. & Shearer, M. (2000) Existence of undercompressive traveling waves in thin-film equations. SIAM J. Math. Anal. 32 (1), 194–213.Google Scholar
[11] Birnie, D. P. III & Manley, M. (1997) Combined flow and evaporation of fluid on a spinning disk. Phys. Fluids (1994-present) 9 (4), 870–875.Google Scholar
[12] Boatto, S., Kadanoff, L. P. & Olla, P. (1993) Traveling-wave solutions to thin-film equations. Phys. Rev. E 48 (6), 4423–4431.CrossRefGoogle ScholarPubMed
[13] Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. (2009) Wetting and spreading. Rev. Mod. Phys. 81 (2), 739–805.Google Scholar
[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), 77–94.Google Scholar
[15] Chiricotto, M. & Giacomelli, L. (2011) Droplets spreading with contact-line friction: Lubrication approximation and traveling wave solutions. Commun. Appl. Ind. Math. 2 (2), 1–16.Google Scholar
[16] Cuesta, C. M. & Velázquez, J. J. L. (2012) Analysis of oscillations in a drainage equation. SIAM J. Math. Anal. 44 (3), 1588–1616.CrossRefGoogle Scholar
[17] De Gennes, P. G. (1985) Wetting: Statics and dynamics. Rev. Mod. Phys. 57 (3), 827.CrossRefGoogle Scholar
[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), 858–862.Google Scholar
[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), 889–922.CrossRefGoogle Scholar
[20] Evans, L. C. (1998) Partial Differential Equations, American Mathematical Society, Providence, RI.Google Scholar
[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, 1–25.Google Scholar
[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), 1491–1504.Google Scholar
[23] Galaktionov, V. A. (2003) On higher-order viscosity approximations of one-dimentional conservation laws.Google Scholar
[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), 15–81.Google Scholar
[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), 735–760.Google Scholar
[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.Google Scholar
[27] Greer, J. B. & Bertozzi, A. L. (2004) Traveling wave solutions of fourth order PDEs for image processing. SIAM J. Math. Anal. 36 (1), 38–68.Google Scholar
[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), 407–425.Google Scholar
[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), 245–254.Google Scholar
[30] Kim, J. S., Kim, S. & Ma, F. (1993) Topographic effect of surface roughness on thin-film flow. J. Appl. Phys. 73 (1), 422–428.Google Scholar
[31] King, J. R. & Taranets, R. M. (2013) Asymmetric travelling waves for the thin-film equation. J. Math. Anal. Appl. 404 (2), 399–419.Google Scholar
[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), 1265–1271.CrossRefGoogle Scholar
[33] Melo, F., Joanny, J. F. & Fauve, S. (1989) Fingering instability of spinning drops. Phys. Rev. Lett. 63 (18), 1958.Google Scholar
[34] Middleman, S. (1987) The effect of induced air-flow on the spin-coating of viscous liquids. J. Appl. Phys. 62 (6), 2530–2532.Google Scholar
[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), 733–742.Google Scholar
[36] Münch, A., Wagner, B. A. & Witelski, T. P. (2005) Lubrication models with small to large slip lengths. J. Eng. Math. 53 (3), 359–383.Google Scholar
[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), 629–635.Google Scholar
[38] Oron, A., Davis, S. H. & Bankoff, S. G. (1997) Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931–980.Google Scholar
[39] Peurrung, L. M. & Graves, D. B. (1991) Film thickness profiles over topography in spin-coating. J. Electrochem. Soc. 138 (7), 2115–2124.CrossRefGoogle Scholar
[40] Schwartz, L. W. & Roy, R. V. (2004) Theoretical and numerical results for spin-coating of viscous liquids. Phys. Fluids 16 (3), 569–584.CrossRefGoogle Scholar
[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), 569–587.Google Scholar
[42] Troy, W. C. (1993) Solutions of third-order differential equations relevant to draining and coating flows. SIAM J. Math. Anal. 24 (1), 155–171.CrossRefGoogle Scholar
[43] Tu, Y.-O. (1983) Depletion and retention of fluid on a rotating disk. J. Tribol. 105 (4), 625–629.Google Scholar
[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), 453–469.Google Scholar
[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), 506–513.Google Scholar
[46] Wu, L. (2006) Spin-coating of thin liquid films on an axisymmetrically heated disk. Phys. Fluids (1994-present) 18 (6), 063602.Google Scholar