Hostname: page-component-84b7d79bbc-g7rbq Total loading time: 0 Render date: 2024-07-26T12:31:31.076Z Has data issue: false hasContentIssue false
  • English
  • Français

The asymptotics of the moving contact line: cracking an old nut

Published online by Cambridge University Press:  08 January 2015

David N. Sibley ,
Andreas Nold  and
Serafim Kalliadasis
David N. Sibley
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
Andreas Nold
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
Serafim Kalliadasis*
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: s.kalliadasis@imperial.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

For contact line motion where the full Stokes flow equations hold, full matched asymptotic solutions using slip models have been obtained for droplet spreading and more general geometries. These solutions to the singular perturbation problem in the slip length, however, all involve matching through an intermediate region that is taken to be separate from the outer–inner regions. Here, we show that the intermediate region is in fact an overlap region representing extensions of both the outer and the inner region, allowing direct matching to proceed. In particular, we investigate in detail how a previously seen result of the matching of the cubes of the free surface slope is justified in the lubrication setting. We also extend this two-region direct matching to the more general Stokes flow case, offering a new perspective on the asymptotics of the moving contact line problem.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface) Low-Reynolds-number flows: Low-Reynolds-number flows Mathematical Foundations: Mathematical Foundations
Type
Papers
Information
Journal of Fluid Mechanics , Volume 764 , 10 February 2015 , pp. 445 - 462
Check for updates
Copyright
© 2015 Cambridge University Press 

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

Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Dover.Google Scholar
Anderson, D. M., McFadden, G. B. & Wheeler, A. A. 1998 Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165.CrossRefGoogle Scholar
Blake, T. & Haynes, J. 1969 Kinetics of liquid/liquid displacement. J. Colloid Interface Sci. 30 (3), 421–423.Google Scholar
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81, 739–805.Google Scholar
Chan, T. S., Gueudré, T. & Snoeijer, J. H. 2011 Maximum speed of dewetting on a fiber. Phys. Fluids 23, 112103.Google Scholar
Chan, T. S., Snoeijer, J. H. & Eggers, J. 2012 Theory of the forced wetting transition. Phys. Fluids 24, 072104.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
Cox, R. G. 1998 Inertial and viscous effects on dynamic contact angles. J. Fluid Mech. 357, 249–278.Google Scholar
Ding, H. & Spelt, P. D. M. 2007 Inertial effects in droplet spreading: a comparison between diffuse-interface and level-set simulations. J. Fluid Mech. 576, 287–296.Google Scholar
Duffy, B. R. & Wilson, S. K. 1997 A third-order differential equation arising in thin-film flows and relevant to Tanner’s law. Appl. Maths Lett. 10 (3), 63–68.Google Scholar
Eggers, J. 2004 Hydrodynamic theory of forced dewetting. Phys. Rev. Lett. 93, 094502.Google Scholar
Eggers, J. 2005 Existence of receding and advancing contact lines. Phys. Fluids 17 (8), 082106.Google Scholar
de Gennes, P. G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57, 827–863.CrossRefGoogle Scholar
Hocking, L. M. 1983 The spreading of a thin drop by gravity and capillarity. Q. J. Mech. Appl. Maths 36 (1), 55–69.Google Scholar
Hocking, L. M. & Rivers, A. D. 1982 The spreading of a drop by capillary action. J. Fluid Mech. 121, 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 (1), 85–101.CrossRefGoogle Scholar
Peters, I., Snoeijer, J. H., Daerr, A. & Limat, L. 2009 Coexistence of two singularities in dewetting flows: regularizing the corner tip. Phys. Rev. Lett. 103, 114501.Google Scholar
Petrov, P. & Petrov, I. 1992 A combined molecular–hydrodynamic approach to wetting kinetics. Langmuir 8 (7), 1762–1767.Google Scholar
Savva, N. & Kalliadasis, S. 2009 Two-dimensional droplet spreading over topographical substrates. Phys. Fluids 21 (9), 092102.Google Scholar
Savva, N. & Kalliadasis, S. 2012 Influence of gravity on the spreading of two-dimensional droplets over topographical substrates. J. Engng Maths 73 (1), 3–16.Google Scholar
Shikhmurzaev, Y. D. 1997 Moving contact lines in liquid/liquid/solid systems. J. Fluid Mech. 334, 211–249.Google Scholar
Sibley, D. N., Nold, A. & Kalliadasis, S. 2013a Unifying binary fluid diffuse-interface models in the sharp-interface limit. J. Fluid Mech. 736, 5–43.CrossRefGoogle Scholar
Sibley, D. N., Nold, A., Savva, N. & Kalliadasis, S. 2013b The contact line behaviour of solid–liquid–gas diffuse-interface models. Phys. Fluids 25 (9), 092111.Google Scholar
Sibley, D. N., Nold, A., Savva, N. & Kalliadasis, S. 2013c On the moving contact line singularity: Asymptotics of a diffuse-interface model. Eur. Phys. J. E 36, 26.Google Scholar
Sibley, D. N., Nold, A., Savva, N. & Kalliadasis, S. 2014 A comparison of slip, disjoining pressure, and interface formation models for contact line motion through asymptotic analysis of thin two-dimensional droplet spreading. J. Engng Maths; doi:10.1007/s10665-014-9702-9.Google Scholar
Sibley, D. N., Savva, N. & Kalliadasis, S. 2012 Slip or not slip? A methodical examination of the interface formation model using two-dimensional droplet spreading on a horizontal planar substrate as a prototype system. Phys. Fluids 24 (8), 082105.Google Scholar
Snoeijer, J. H. 2006 Free-surface flows with large slopes: beyond lubrication theory. Phys. Fluids 18 (2), 021701.CrossRefGoogle Scholar
Snoeijer, J. H. & Andreotti, B. 2013 Moving contact lines: scales, regimes, and dynamical transitions. Annu. Rev. Fluid Mech. 45, 269–292.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, 244502.Google Scholar
Sui, Y., Ding, H. & Spelt, P. D. M. 2014 Numerical simulations of flows with moving contact lines. Annu. Rev. Fluid Mech. 46, 97–119.Google Scholar
Trefethen, L. N. 2000 Spectral Methods in MATLAB. SIAM.CrossRefGoogle Scholar
Vandre, E., Carvalho, M. S. & Kumar, S. 2013 On the mechanism of wetting failure during fluid displacement along a moving substrate. Phys. Fluids 25 (10), 102103.CrossRefGoogle Scholar
Vellingiri, R., Savva, N. & Kalliadasis, S. 2011 Droplet spreading on chemically heterogeneous substrates. Phys. Rev. E 84, 036305.Google Scholar
Voinov, O. V. 1976 Hydrodynamics of wetting. Fluid Dyn. 11 (5), 714–721.Google Scholar
Weidner, D. E. & Schwartz, L. W. 1994 Contact-line motion of shear-thinning liquids. Phys. Fluids 6, 3535–3538.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