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

  • Weiqing Ren (a1) (a2), Philippe H. Trinh (a3) and Weinan E (a4) (a5)


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.


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Anderson, D. M., McFadden, G. B. & Wheeler, A. A. 1998 Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30 (1), 139165.
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, 5369.
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), 15341564.
Billingham, J. 2008 Gravity-driven thin-film flow using a new contact line model. IMA J. Appl. Maths 73 (1), 436.
Blake, T. D. 1993 Dynamic contact angles and wetting kinetics. In Wettability, Surfactant Science Series, vol. 49, p. 251. Marcel Dekker.
Blake, T. D. 2006 The physics of moving wetting lines. J. Colloid Interface Sci. 299, 113.
Blake, T. D., Bracke, M. & Shikhmurzaev, Y. D. 1999 Experimental evidence of nonlocal hydrodynamic influence on the dynamic contact angle. Phys. Fluids 11, 19952007.
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), 21642166.
Blake, T. D. & De Coninck, J. 2002 The influence of solid–liquid interactions on dynamic wetting. Adv. Colloid Interface Sci. 96 (1), 2136.
Blake, T. D. & Haynes, J. M. 1969 Kinetics of liquid/liquid displacement. J. Colloid Interface Sci. 30 (3), 421423.
Cox, R. G. 1986 The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech. 168, 169194.
De Coninck, J. & Blake, T. D. 2008 Wetting and molecular dynamics simulations of simple liquids. Annu. Rev. Mater. Res. 38, 122.
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), 7195.
Eggers, J. 2004a Hydrodynamic theory of forced dewetting. Phys. Rev. Lett. 93 (9), 094502.
Eggers, J. 2004b Toward a description of contact line motion at higher capillary numbers. Phys. Fluids 16 (9), 34913494.
Eggers, J. 2005 Existence of receding and advancing contact lines. Phys. Fluids 17 (8), 082106.
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, 309321.
Flitton, J. C. & King, J. R. 2004 Surface-tension-driven dewetting of Newtonian and power-law fluids. J. Engng Maths 50 (2–3), 241266.
Greenspan, H. P. 1978 On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84 (01), 125143.
Haley, P. J. & Miksis, M. J. 1991 The effect of the contact line on droplet spreading. J. Fluid Mech. 223, 5781.
Hocking, L. M. 1981 Sliding and spreading of thin two-dimensional drops. Q. J. Mech. Appl. Maths 34 (1), 3755.
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. 315321. JPL Publications.
Hocking, L. M. 1992 Rival contact-angle models and the spreading of drops. J. Fluid Mech. 239 (1), 671681.
Hocking, L. M. & Rivers, A. D. 1982 The spreading of a drop by capillary action. J. Fluid Mech. 121 (1), 425442.
Huh, C. & Scriven, L. E. 1971 Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35, 85101.
Jacqmin, D. 2000 Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402 (1), 5788.
King, J. R. & Bowen, M. 2001 Moving boundary problems and non-uniqueness for the thin film equation. Eur. J. Appl. Maths 12 (03), 321356.
Kistler, S. F. 1993 Hydrodynamics of wetting. In Wettability, Surfactant Science Series, vol. 49, pp. 311430. Marcel Dekker.
Koplik, J., Banavar, J. R. & Willemsen, J. F. 1988 Molecular dynamics of Poiseuille flow and moving contact lines. Phys. Rev. Lett. 60 (13), 12821285.
Koplik, J., Banavar, J. R. & Willemsen, J. F. 1989 Molecular dynamics of fluid flow at solid surfaces. Phys. Fluids A 1, 781794.
Lacey, A. A. 1982 The motion with slip of a thin viscous droplet over a solid surface. Stud. Appl. Maths 67 (3), 217230.
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.
Moriarty, J. A. & Schwartz, L. W. 1992 Effective slip in numerical calculations of moving-contact-line problems. J. Engng Maths 26 (1), 8186.
Moriarty, J. A., Schwartz, L. W. & Tuck, E. O. 1991 Unsteady spreading of thin liquid films with small surface tension. Phys. Fluids 3 (5), 733742.
Pismen, L. M. 2002 Mesoscopic hydrodynamics of contact line motion. Colloids Surf. A 206 (1), 1130.
Pomeau, Y. 2002 Recent progress in the moving contact line problem: a review. C. R. Méc. 330 (3), 207222.
Qian, T., Wang, X.-P. & Sheng, P. 2003 Molecular scale contact line hydrodynamics of immiscible flows. Phys. Rev. E 68 (1), 016306.
Ren, W. & E, W. 2007 Boundary conditions for the moving contact line problem. Phys. Fluids 19 (2), 022101.
Ren, W., Hu, D. & E, W. 2010 Continuum models for the contact line problem. Phys. Fluids 22 (10), 102103.
Shikhmurzaev, Y. D. 1997 Moving contact lines in liquid/liquid/solid systems. J. Fluid Mech. 334 (1), 211249.
Shikhmurzaev, Y. D. 2007 Capillary Flows with Forming Interfaces. Chapman and Hall/CRC.
Sibley, D. N., Nold, A. & Kalliadasis, S. 2015 The asymptotics of the moving contact line: cracking an old nut. J. Fluid Mech. 764, 445462.
Snoeijer, J. H., Delon, G., Fermigier, M. & Andreotti, B. 2006 Avoided critical behavior in dynamically forced wetting. Phys. Rev. Lett. 96 (17), 174504.
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.
Thompson, P. A. & Robbins, M. O. 1989 Simulations of contact-line motion: slip and the dynamic contact angle. Phys. Rev. Lett. 63 (7), 766769.
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics: By Milton Van Dyke, Annotated edn. Parabolic Press.
Velarde, M. G. 2011 Discussion and debate: wetting and spreading science – quo vadis? Eur. Phys. J. Special Top. 197 (1), 1148.
Voinov, O. V. 1976 Hydrodynamics of wetting. Fluid Dyn. 11 (5), 714721.
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.
Wilson, S. D. R. 1982 The drag-out problem in film coating theory. J. Engng Maths 16 (3), 209221.
Yue, P., Zhou, C. & Feng, J. J. 2010 Sharp interface limit of the Cahn–Hilliard model for moving contact lines. J. Fluid Mech. 645, 279294.
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On the distinguished limits of the Navier slip model of the moving contact line problem

  • Weiqing Ren (a1) (a2), Philippe H. Trinh (a3) and Weinan E (a4) (a5)


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