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On the distinguished limits of the Navier slip model of the moving contact line problem

Published online by Cambridge University Press:  28 April 2015

Weiqing Ren ,
Philippe H. Trinh Open the ORCID record for Philippe H. Trinh [Opens in a new window]  and
Weinan E
Weiqing Ren
Affiliation:
Department of Mathematics, National University of Singapore, Singapore 119076, Singapore Institute of High Performance Computing, A*STAR, Singapore 138632, Singapore
Philippe H. Trinh*
Affiliation:
Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
Weinan E
Affiliation:
Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544-1000, USA School of Mathematics, Peking University, Beijing 100871, PR China
*
†Email address for correspondence: trinh@maths.ox.ac.uk
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Abstract

When a droplet spreads on a solid substrate, it is unclear what the correct boundary conditions are to impose at the moving contact line. The classical no-slip condition is generally acknowledged to lead to a non-integrable singularity at the moving contact line, which a slip condition, associated with a small slip parameter, ${\it\lambda}$, serves to alleviate. In this paper, we discuss what occurs as the slip parameter, ${\it\lambda}$, tends to zero. In particular, we explain how the zero-slip limit should be discussed in consideration of two distinguished limits: one where time is held constant, $t=O(1)$, and one where time tends to infinity at the rate $t=O(|\!\log {\it\lambda}|)$. The crucial result is that in the case where time is held constant, the ${\it\lambda}\rightarrow 0$ limit converges to the slip-free equation, and contact line slippage occurs as a regular perturbative effect. However, if ${\it\lambda}\rightarrow 0$ and $t\rightarrow \infty$, then contact line slippage is a leading-order singular effect.

JFM classification

Interfacial Flows (free surface): Contact lines Low-Reynolds-number flows: Lubrication theory Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 772 , 10 June 2015 , pp. 107 - 126
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Copyright
© 2015 Cambridge University Press 

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References

Anderson, D. M., McFadden, G. B. & Wheeler, A. A. 1998 Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30 (1), 139–165.CrossRefGoogle Scholar
Benilov, E. S., Chapman, S. J., McLeod, J. B., Ockendon, J. R. & Zubkov, V. S. 2010 On liquid films on an inclined plate. J. Fluid Mech. 663, 53–69.Google Scholar
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 (6), 1534–1564.CrossRefGoogle Scholar
Billingham, J. 2008 Gravity-driven thin-film flow using a new contact line model. IMA J. Appl. Maths 73 (1), 4–36.CrossRefGoogle Scholar
Blake, T. D. 1993 Dynamic contact angles and wetting kinetics. In Wettability, Surfactant Science Series, vol. 49, p. 251. Marcel Dekker.Google Scholar
Blake, T. D. 2006 The physics of moving wetting lines. J. Colloid Interface Sci. 299, 1–13.CrossRefGoogle ScholarPubMed
Blake, T. D., Bracke, M. & Shikhmurzaev, Y. D. 1999 Experimental evidence of nonlocal hydrodynamic influence on the dynamic contact angle. Phys. Fluids 11, 1995–2007.Google Scholar
Blake, T. D., Clarke, A., De Coninck, J. & de Ruijter, M. J. 1997 Contact angle relaxation during droplet spreading: comparison between molecular kinetic theory and molecular dynamics. Langmuir 13 (7), 2164–2166.Google Scholar
Blake, T. D. & De Coninck, J. 2002 The influence of solid–liquid interactions on dynamic wetting. Adv. Colloid Interface Sci. 96 (1), 21–36.Google Scholar
Blake, T. D. & Haynes, J. M. 1969 Kinetics of liquid/liquid displacement. J. Colloid Interface Sci. 30 (3), 421–423.Google Scholar
Cox, R. G. 1986 The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech. 168, 169–194.Google Scholar
De Coninck, J. & Blake, T. D. 2008 Wetting and molecular dynamics simulations of simple liquids. Annu. Rev. Mater. Res. 38, 1–22.Google Scholar
Dussan V, E. B. & Davis, S. H. 1974 On the motion of a fluid–fluid interface along a solid surface. J. Fluid Mech. 65 (1), 71–95.Google Scholar
Eggers, J. 2004a Hydrodynamic theory of forced dewetting. Phys. Rev. Lett. 93 (9), 094502.Google Scholar
Eggers, J. 2004b Toward a description of contact line motion at higher capillary numbers. Phys. Fluids 16 (9), 3491–3494.CrossRefGoogle Scholar
Eggers, J. 2005 Existence of receding and advancing contact lines. Phys. Fluids 17 (8), 082106.Google Scholar
Eggers, J. & Stone, H. A. 2004 Characteristic lengths at moving contact lines for a perfectly wetting fluid: the influence of speed on the dynamic contact angle. J. Fluid Mech. 505, 309–321.Google Scholar
Flitton, J. C. & King, J. R. 2004 Surface-tension-driven dewetting of Newtonian and power-law fluids. J. Engng Maths 50 (2–3), 241–266.Google Scholar
Greenspan, H. P. 1978 On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84 (01), 125–143.Google Scholar
Haley, P. J. & Miksis, M. J. 1991 The effect of the contact line on droplet spreading. J. Fluid Mech. 223, 57–81.CrossRefGoogle Scholar
Hocking, L. M. 1981 Sliding and spreading of thin two-dimensional drops. Q. J. Mech. Appl. Maths 34 (1), 37–55.Google Scholar
Hocking, L. M. 1983 The motion of a drop on a rigid surface. In Proceedings of the 2nd Intern. Colloq. on Drops and Bubbles, Monterey, pp. 315–321. JPL Publications.Google Scholar
Hocking, L. M. 1992 Rival contact-angle models and the spreading of drops. J. Fluid Mech. 239 (1), 671–681.Google Scholar
Hocking, L. M. & Rivers, A. D. 1982 The spreading of a drop by capillary action. J. Fluid Mech. 121 (1), 425–442.Google Scholar
Huh, C. & Scriven, L. E. 1971 Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35, 85–101.Google Scholar
Jacqmin, D. 2000 Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402 (1), 57–88.CrossRefGoogle Scholar
King, J. R. & Bowen, M. 2001 Moving boundary problems and non-uniqueness for the thin film equation. Eur. J. Appl. Maths 12 (03), 321–356.Google Scholar
Kistler, S. F. 1993 Hydrodynamics of wetting. In Wettability, Surfactant Science Series, vol. 49, pp. 311–430. Marcel Dekker.Google Scholar
Koplik, J., Banavar, J. R. & Willemsen, J. F. 1988 Molecular dynamics of Poiseuille flow and moving contact lines. Phys. Rev. Lett. 60 (13), 1282–1285.Google Scholar
Koplik, J., Banavar, J. R. & Willemsen, J. F. 1989 Molecular dynamics of fluid flow at solid surfaces. Phys. Fluids A 1, 781–794.Google Scholar
Lacey, A. A. 1982 The motion with slip of a thin viscous droplet over a solid surface. Stud. Appl. Maths 67 (3), 217–230.Google Scholar
Lauga, E., Brenner, M. P. & Stone, H. A. 2007 Microfluidics: the no-slip boundary condition. In Handbook of Experimental Fluid Dynamics (ed. Tropea, C., Yarin, A. & Foss, J. F.), chap. 19 pp. 1219–1240. Springer.Google Scholar
Moriarty, J. A. & Schwartz, L. W. 1992 Effective slip in numerical calculations of moving-contact-line problems. J. Engng Maths 26 (1), 81–86.Google Scholar
Moriarty, J. A., Schwartz, L. W. & Tuck, E. O. 1991 Unsteady spreading of thin liquid films with small surface tension. Phys. Fluids 3 (5), 733–742.Google Scholar
Pismen, L. M. 2002 Mesoscopic hydrodynamics of contact line motion. Colloids Surf. A 206 (1), 11–30.Google Scholar
Pomeau, Y. 2002 Recent progress in the moving contact line problem: a review. C. R. Méc. 330 (3), 207–222.Google Scholar
Qian, T., Wang, X.-P. & Sheng, P. 2003 Molecular scale contact line hydrodynamics of immiscible flows. Phys. Rev. E 68 (1), 016306.Google Scholar
Ren, W. & E, W. 2007 Boundary conditions for the moving contact line problem. Phys. Fluids 19 (2), 022101.Google Scholar
Ren, W., Hu, D. & E, W. 2010 Continuum models for the contact line problem. Phys. Fluids 22 (10), 102103.Google Scholar
Shikhmurzaev, Y. D. 1997 Moving contact lines in liquid/liquid/solid systems. J. Fluid Mech. 334 (1), 211–249.Google Scholar
Shikhmurzaev, Y. D. 2007 Capillary Flows with Forming Interfaces. Chapman and Hall/CRC.Google Scholar
Sibley, D. N., Nold, A. & Kalliadasis, S. 2015 The asymptotics of the moving contact line: cracking an old nut. J. Fluid Mech. 764, 445–462.Google Scholar
Snoeijer, J. H., Delon, G., Fermigier, M. & Andreotti, B. 2006 Avoided critical behavior in dynamically forced wetting. Phys. Rev. Lett. 96 (17), 174504.Google Scholar
Snoeijer, J. H., Ziegler, J., Andreotti, B., Fermigier, M. & Eggers, J. 2008 Thick films of viscous fluid coating a plate withdrawn from a liquid reservoir. Phys. Rev. Lett. 100 (24), 244502.Google Scholar
Thompson, P. A. & Robbins, M. O. 1989 Simulations of contact-line motion: slip and the dynamic contact angle. Phys. Rev. Lett. 63 (7), 766–769.Google Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics: By Milton Van Dyke, Annotated edn. Parabolic Press.Google Scholar
Velarde, M. G. 2011 Discussion and debate: wetting and spreading science – quo vadis? Eur. Phys. J. Special Top. 197 (1), 1–148.Google Scholar
Voinov, O. V. 1976 Hydrodynamics of wetting. Fluid Dyn. 11 (5), 714–721.Google Scholar
Wilson, M. C. T., Summers, J. L., Shikhmurzaev, Y. D., Clarke, A. & Blake, T. D. 2006 Nonlocal hydrodynamic influence on the dynamic contact angle: slip models versus experiment. Phys. Rev. E 73, 041606.Google Scholar
Wilson, S. D. R. 1982 The drag-out problem in film coating theory. J. Engng Maths 16 (3), 209–221.Google Scholar
Yue, P., Zhou, C. & Feng, J. J. 2010 Sharp interface limit of the Cahn–Hilliard model for moving contact lines. J. Fluid Mech. 645, 279–294.Google Scholar