No Access Submitted: 19 January 2009 Accepted: 14 August 2009 Published Online: 09 September 2009
Physics of Fluids 21, 092102 (2009);
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  • Nikos Savva
  • Serafim Kalliadasis
Contact line motion over a topographical substrate is considered using the spreading of a two-dimensional droplet as a model system. The spreading dynamics is modeled under the assumption of small contact angles where the long-wave expansion in the Stokes-flow regime can be employed to derive a single equation for the evolution of the droplet thickness. The contact line singularity is removed through the Navier slip condition, while the contact angle at the contact points is assumed to remain always equal to its static value. Through a singular perturbation approach, the flow in the vicinity of the contact line is matched asymptotically with the flow in the bulk of the droplet to yield a set of two coupled integrodifferential equations for the location of the two droplet fronts. Our matching procedure is verified through direct comparisons with numerical solutions to the full problem. Analysis of the equations obtained by asymptotic matching reveals a number of intriguing features that are not present when the substrate is flat. In particular, we demonstrate the existence of multiple equilibrium states which allows for a hysteresislike effect on the apparent contact line. Further, we demonstrate a stick-slip-type behavior of the contact line as it moves along the local variations of the substrate shape and the interesting possibility of a relatively brief recession of one of the contact lines.
  1. 1. E. B. Dussan, V, “On the spreading of liquids on solid surfaces: Static and dynamic contact lines,” Annu. Rev. Fluid Mech. 11, 371 (1979)., Google ScholarCrossref, ISI
  2. 2. P. -G. de Gennes, “Wetting: Statics and dynamics,” Rev. Mod. Phys. 57, 827 (1985)., Google ScholarCrossref, ISI
  3. 3. T. D. Blake, in Wettability, edited by J. C. Berg (Dekker, New York, 1993). Google Scholar
  4. 4. D. Bonn, J. Eggers, J. Indekeu, J. Meunier, and E. Rolley, “Wetting and spreading,” Rev. Mod. Phys. 81, 739 (2009)., Google ScholarCrossref, ISI
  5. 5. L. W. Schwartz and R. R. Eley, “Simulation of droplet motion on low-energy and heterogeneous surfaces,” J. Colloid Interface Sci. 202, 173 (1998)., Google ScholarCrossref, ISI
  6. 6. L. W. Schwartz, “Hysteretic effects in droplet motions on heterogeneous substrates: Direct numerical simulation,” Langmuir 14, 3440 (1998)., Google ScholarCrossref
  7. 7. P. Ehrhard and S. H. Davis, “Non-isothermal spreading of liquid drops on horizontal plates,” J. Fluid Mech. 229, 365 (1991)., Google ScholarCrossref
  8. 8. C. Sodtke, V. S. Ajaev, and P. Stephan, “Dynamics of volatile liquid droplets on heated surfaces: Theory versus experiment,” J. Fluid Mech. 610, 343 (2008)., Google ScholarCrossref, ISI
  9. 9. V. S. Ajaev and G. M. Homsy, “Modeling shapes and dynamics of confined bubbles,” Annu. Rev. Fluid Mech. 38, 277 (2006)., Google ScholarCrossref
  10. 10. H. K. Moffatt, “Viscous and resistive eddies near a sharp corner,” J. Fluid Mech. 18, 1 (1964)., Google ScholarCrossref, ISI
  11. 11. C. Huh and L. E. Scriven, “Hydrodynamic model of steady movement of a solid/liquid/fluid contact line,” J. Colloid Interface Sci. 35, 85 (1971)., Google ScholarCrossref, ISI
  12. 12. S. M. Troian, E. Herbolzheimer, S. A. Safran, and J. F. Joanny, “Fingering instabilities of driven spreading films,” Europhys. Lett. 10, 25 (1989)., Google ScholarCrossref
  13. 13. J. A. Moriarty, L. W. Schwartz, and E. O. Tuck, “Unsteady spreading of thin liquid films with small surface tension,” Phys. Fluids A 3, 733 (1991)., Google ScholarScitation
  14. 14. S. Kalliadasis, “Nonlinear instability of a contact line driven by gravity,” J. Fluid Mech. 413, 355 (2000)., Google ScholarCrossref
  15. 15. H. Lamb, Hydrodynamics, 6th ed. (Cambridge University Press, New York, 1975). Google Scholar
  16. 16. H. P. Greenspan, “On the motion of a small viscous droplet that wets a surface,” J. Fluid Mech. 84, 125 (1978)., Google ScholarCrossref
  17. 17. L. M. Hocking, “Sliding and spreading of two-dimensional drops,” Q. J. Mech. Appl. Math. 34, 37 (1981)., Google ScholarCrossref
  18. 18. L. W. Schwartz, R. V. Roy, R. R. Eley, and S. Petrash, “Dewetting patterns in a drying liquid film,” J. Colloid Interface Sci. 234, 363 (2001)., Google ScholarCrossref, ISI
  19. 19. Y. D. Shikhmurzaev, “Moving contact lines in liquid/liquid/solid systems,” J. Fluid Mech. 334, 211 (1997)., Google ScholarCrossref, ISI
  20. 20. E. Lauga, M. P. Brenner, and H. A. Stone, in Springer Handbook of Experimental Fluid Mechanics, edited by C. Tropea, J. F. Foss, and A. Yarin (Springer, New York, 2008), Chap. 19. Google Scholar
  21. 21. O. V. Voinov, “Hydrodynamics of wetting,” Fluid Dyn. 11, 714 (1976)., Google ScholarCrossref
  22. 22. L. M. Hocking, “The spreading of a thin drop by gravity and capillarity,” Q. J. Mech. Appl. Math. 36, 55 (1983)., Google ScholarCrossref
  23. 23. L. M. Hocking, “Rival contact-angle models and the spreading of drops,” J. Fluid Mech. 239, 671 (1992)., Google ScholarCrossref
  24. 24. L. M. Hocking, “The spreading of drops with intermolecular forces,” Phys. Fluids 6, 3224 (1994)., Google ScholarScitation
  25. 25. A. M. Cazabat and M. A. Cohen-Stuart, “Dynamics of wetting: Effects of surface roughness,” J. Phys. Chem. 90, 5845 (1986)., Google ScholarCrossref
  26. 26. A. M. Schwartz and S. B. Tejada, “Studies of dynamic contact angles on solids,” J. Colloid Interface Sci. 38, 359 (1972)., Google ScholarCrossref
  27. 27. S. Semal, T. D. Blake, V. Geskin, M. J. de Ruijter, G. Castelein, and J. de Coninck, “Influence of surface roughness on wetting dynamics,” Langmuir 15, 8765 (1999)., Google ScholarCrossref
  28. 28. R. N. Wenzel, “Resistance of solid surfaces to wetting by water,” Ind. Eng. Chem. 28, 988 (1936)., Google ScholarCrossref
  29. 29. S. J. Hitchcock, N. T. Carroll, and M. G. Nicholas, “Some effects of substrate roughness on wettability,” J. Mater. Sci. 16, 714 (1981)., Google ScholarCrossref
  30. 30. S. Shibuichi, T. Onda, N. Satoh, and K. Tsujii, “Super water-repellent surfaces resulting from fractal structure,” J. Phys. Chem. 100, 19512 (1996)., Google ScholarCrossref
  31. 31. J. Bico, C. Tordeux, and D. Quéré, “Rough wetting,” Europhys. Lett. 55, 214 (2001)., Google ScholarCrossref
  32. 32. R. E. Johnson and R. H. Dettre, “Contact angle hysteresis I. Study of an idealized rough surface,” Adv. Chem. Ser. 43, 112 (1964). Google ScholarCrossref
  33. 33. C. Huh and S. G. Mason, “Effects of surface roughness on wetting (theoretical),” J. Colloid Interface Sci. 60, 11 (1977)., Google ScholarCrossref
  34. 34. R. G. Cox, “The spreading of a liquid on a rough solid surface,” J. Fluid Mech. 131, 1 (1983)., Google ScholarCrossref
  35. 35. J. F. Oliver, C. Huh, and S. G. Mason, “The apparent contact angle of liquids on finely-grooved solid surfaces—A SEM study,” J. Adhes. 8, 223 (1976)., Google ScholarCrossref
  36. 36. P. H. Gaskell, P. K. Jimack, M. Sellier, and H. M. Thompson, “Efficient and accurate time adaptive multigrid simulations of droplet spreading,” Int. J. Numer. Meth. Fluids 45, 1161 (2004)., Google ScholarCrossref
  37. 37. C. M. Gramlich, A. Mazouchi, and G. M. Homsy, “Time-dependent free surface Stokes flow with a moving contact line. II. Flow over wedges and trenches,” Phys. Fluids 16, 1660 (2004)., Google ScholarScitation
  38. 38. S. Kalliadasis, C. Bielarz, and G. M. Homsy, “Steady free-surface thin film flows over topography,” Phys. Fluids 12, 1889 (2000)., Google ScholarScitation, ISI
  39. 39. S. Kalliadasis and G. M. Homsy, “Stability of free-surface thin-film flows over topography,” J. Fluid Mech. 448, 387 (2001)., Google ScholarCrossref
  40. 40. P. D. Howell, “Surface-tension-driven flow on a moving curved substrate,” J. Eng. Math. 45, 283 (2003)., Google ScholarCrossref
  41. 41. P. H. Gaskell, P. K. Jimack, M. Sellier, H. M. Thompson, and M. C. T. Wilson, “Gravity-driven flow of continuous thin liquid films on non-porous substrates with topography,” J. Fluid Mech. 509, 253 (2004)., Google ScholarCrossref, ISI
  42. 42. J. M. Davis and S. M. Troian, “Generalized linear stability of noninertial coating flows over topographical substrates,” Phys. Fluids 17, 072103 (2005)., Google ScholarScitation
  43. 43. J. F. Oliver, C. Huh, and S. G. Mason, “Resistance to spreading of liquids by sharp edges,” J. Colloid Interface Sci. 59, 568 (1977)., Google ScholarCrossref, ISI
  44. 44. L. M. Pismen and J. Eggers, “Solvability condition for the moving contact line,” Phys. Rev. E 78, 056304 (2008)., Google ScholarCrossref
  45. 45. S. D. R. Wilson, “The drag-out problem in film coating theory,” J. Eng. Math. 16, 209 (1982)., Google ScholarCrossref
  46. 46. B. R. Duffy and S. K. Wilson, “A third-order differential equation arising in thin-film flows and relevant to Tanner’s law,” Appl. Math. Lett. 10, 63 (1997)., Google ScholarCrossref, ISI
  47. 47. J. Eggers, “Existence of receding and advancing contact lines,” Phys. Fluids 17, 082106 (2005)., Google ScholarScitation, ISI
  48. 48. G. McHale, M. I. Newton, S. M. Rowan, and M. Banerjee, “The spreading of small viscous stripes of oil,” J. Phys. D: Appl. Phys. 28, 1925 (1995)., Google ScholarCrossref
  49. 49. P. J. Haley and M. J. Miksis, “The effect of the contact line on droplet spreading,” J. Fluid Mech. 223, 57 (1991)., Google ScholarCrossref
  50. 50. D. Quéré, in Thin Films of Soft Matter, edited by S. Kalliadasis and U. Thiele (Springer-CISM, Wien, 2007). Google Scholar
  51. 51. Y. C. Lee, H. M. Thompson, and P. H. Gaskell, “An efficient adaptive multigrid algorithm for predicting thin film flow on surfaces containing localised topographic features,” Comput. Fluids 36, 838 (2007)., Google ScholarCrossref
  52. 52. Y. Y. Koh, Y. C. Lee, P. H. Gaskell, P. K. Jimack, and H. M. Thompson, “Droplet migration: Quantitative comparisons with experiment,” Eur. Phys. J. Spec. Top. 166, 117 (2009)., Google ScholarCrossref
  53. 53. M. Sellier, Y. C. Lee, H. M. Thompson, and P. H. Gaskell, “Thin film flow on surfaces containing arbitrary occlusions,” Comput. Fluids 38, 171 (2009)., Google ScholarCrossref
  54. 54. L. N. Trefethen, Spectral methods in MATLAB (SIAM, Philadelphia, 2000). Google ScholarCrossref
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