Skip to main content
SpringerLink
Log in

Contact Lines

  • Chapter
A Celebration of Mathematical Modeling

Abstract

Here we review work on the formulation and evolution of contact line motion problems. Static, steady and unsteady problems will be discussed. Recent numerical investigations of interfacial problems have allowed for a number of interesting and reasonable approximations to the motions of contact lines but these investigations must go hand-in-hand with models which need to be verified.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson, D.M. and Davis, S.H. (1995). The spreading of volatile liquid droplets on heated surfaces. Phys. Fluids A. 7, 248 - 265.

    Article  MATH  Google Scholar 

  2. Bach, P. and Hassager, O. (1984). An algorithm for the use of the Lagrangian specification in Newtonian fluid mechanics and applications to free-surface flow. J. Fluid Mech. 152, 173 - 190.

    Article  Google Scholar 

  3. Berg, J.C. editor (1993). Wettability, Surfactant Science Series 49, New York, NY, Marcel Dekker.

    Google Scholar 

  4. Bertozzi, A.L. (1998). The mathematics of moving contact lines in thin liquid films. Notices AMS June/July, 689 - 687.

    Google Scholar 

  5. Blake, T.D. (1993). Dynamic Contact Angles and Wetting Kinetics, in Wettability, Surfactant Science Series 49, 251-309. ed. J.C. Berg, New York, NY, Marcel Dekker.

    Google Scholar 

  6. Bose, A.. (1993). Wetting by Solutions, in Wettability, Surfactant Science Series 49, 149-181. ed. J.C. Berg, New York, NY, Marcel Dekker.

    Google Scholar 

  7. Clay, M.A., and Miksis, M.J. (2003). Effects of surfactant on droplet spreading. Preprint.

    Google Scholar 

  8. 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.

    Article  MATH  Google Scholar 

  9. Davis., S.H. (1983). Contact-line problem in fluid mechanics. Trans. ASME E: J. Appl. Mech. 50, 977.

    Article  Google Scholar 

  10. de Gennes, P.G. (1985). Wetting: statics and dynamics. Rev. Mod. Physics 57 (3), 827 - 863.

    Article  Google Scholar 

  11. de Ruijter, M.J., Blake, T.D., and de Coninck, J. (1999). Dynamic wetting studied by molecular modeling simulations of droplet spreading. Langmuir 15, 7836.

    Article  Google Scholar 

  12. Dimitrakopoulos, P and Higdon, J.J.L. (1998). On the displacement of three-dimensional droplets from solid surfaces in low-Reynolds number shear flows. J. Fluid Mech. 377, 189 - 222.

    Article  MATH  Google Scholar 

  13. Dussan V., E. B. and Davis, S. H. (1974). On the motion of a fluid-fluid interface along a solid surface. J. Fluid Mech. 65, 7195.

    Google Scholar 

  14. Dussan V., E. B., (1976). The moving contact line: the slip boundary condition. J. Fluid Mech. 77, 665 - 684.

    Article  MATH  Google Scholar 

  15. Dussan V., E. B., (1979). On the spreading of liquids on solid surfaces: Static and dynamic contact lines. Ann. Rev. Fluid Mech.. 11, 371 - 400.

    Article  Google Scholar 

  16. Dussan V., E.B. (1987). On the ability of drops to stick to surfaces of solids. Part 3. The influences of the motion of the surrounding fluid on dislodging drops. J. Fluid Mech. 174, 381 - 387.

    MATH  Google Scholar 

  17. Dussan V., E. B., Rame, E. and Garoff, S. (1991). On identifying the appropriate boundary conditions at a moving contact line: an experimental investigation. J. Fluid Mech. 230, 97 - 116.

    Article  Google Scholar 

  18. Ehrhard, P. (1993). Experiments on isothermal and non-isothermal spreading. J. Fluid Mech. 257, 463 - 483.

    Article  Google Scholar 

  19. Ehrhard, P., and Davis, S.H. (1991). Non-isothermal spreading of liquid drops on horizontal plates. J. Fluid Mech. 229, 365 - 388.

    Article  MATH  Google Scholar 

  20. Fukai, J., Shiiba, Y., Yamamoto, T, Miyatake, O., Poulikalos, D., Megaridis, C.M., and Zhao, Z. (1995). Wetting effects on the spreading of a liquid droplet colliding with a flat surface: Experiment and modeling. Phys. Fluids 7 (2), 236 - 247.

    Article  Google Scholar 

  21. Freund, J.B. (2003), The atomic detail of a wetting/de-wetting flow. Phys. Fluids. 15 (5), L33 - L36.

    Article  Google Scholar 

  22. Gauss, K.I. (1829). Principia generalia theoriae figurae fluidorum in statuaequilibrii. Gott Gelelirte Anz 1829, 1641-48–Werke, S1287-92, Gottingen K Ges Wiss Gott. 1867, Hildeshem Olms, 1973.

    Google Scholar 

  23. Gokhale, S.J., Plawsky, J.L. and Wayner Jr., P.C. (2003). Experimental investigation of contact angle, curvature, and contact line motion in dropwise condensation and evaporation. J. Colloid Inter. Sci. 259, 354 - 366.

    Article  Google Scholar 

  24. Golovin, A. A., Davis, S.H. and Nepomnyashchy, A.A. (1998) A convective Chan-Hilliard model for the formation of facets and corners in crystal growth. Physica D 122, 202 - 230.

    Article  MathSciNet  MATH  Google Scholar 

  25. Goodwin, R. and Homsy, G. M. (1991). Viscous flow down a slope in the vicinity of a contact line. Phys. Fluids A 3, 515 - 528.

    Article  MATH  Google Scholar 

  26. Greenspan, H.P. (1978). On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84, 125 - 143.

    Article  MATH  Google Scholar 

  27. Greenspan, H.P., and McCay, B.M. (1981). On the wetting of a surface by a very viscous fluid. Studies Appl. Math 64, 95 - 112.

    MathSciNet  MATH  Google Scholar 

  28. Gurtin, M. (1993). Thermomechanics of Evolving Phase Boundaries in the Plane. New York, NY, Oxford University Press.

    Google Scholar 

  29. Haley, P.J. and Miksis, M.J., (1991). The effect of the contact line on droplet spreading. J. Fluid Mech. 223, 57 - 81.

    Article  MathSciNet  MATH  Google Scholar 

  30. Hocking, L. M., (1976). A moving fluid on a rough surface. J. Fluid Mech. 76, 801 - 817.

    Article  MATH  Google Scholar 

  31. Hocking, L.M. (1977). A moving fluid interface. part 2. The removal of the force singularity by a slip flow. J. Fluid Mech. 79, 209 - 229.

    Article  MATH  Google Scholar 

  32. Hocking, L.M. (1981). Sliding and spreading of thin two-dimensional drops. Q.J. Mech. Appl. Math. 34 (1), 37 - 55.

    Article  MathSciNet  MATH  Google Scholar 

  33. Hocking, L.M. and Rivers, A.D. (1982). The spreading of a drop by capillary action. J. Fluid Mech., 121, 425 - 442.

    Article  MathSciNet  MATH  Google Scholar 

  34. Hocking, L.M. (1983). The spreading of a thin drop by gravity and capillarity. Q.J. Mech. Appl. Math. 36 (1), 55 - 69.

    Article  MathSciNet  MATH  Google Scholar 

  35. Hocking, L.M. (1992). Rival contact-angle models and the spreading of drops. J. Fluid Mech. 239, 671 - 681.

    Article  MathSciNet  MATH  Google Scholar 

  36. Hocking, L.M. (1993). The influence of intermolecular forces on thin fluid layers. Phys. Fluids A 5 (4), 793 - 799.

    Article  Google Scholar 

  37. Hocking, L.M. and Davis, S.H. (2002). Inertial effects in time-dependent motion of thin films and drops. J. Fluid Mech. 467, 1 - 17.

    Article  MathSciNet  MATH  Google Scholar 

  38. Huh, C. and Scriven, L.E., (1971). Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid and Interface Science 35, 85 - 101.

    Article  Google Scholar 

  39. Jacqmin, D. (2000) Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402, 57 - 88.

    Article  MathSciNet  MATH  Google Scholar 

  40. Joanny, J.F. (1986). Dynamics of wetting: Interface profile of a spreading liquid. J. Theor. and Applied Mech. Numero special, 249 - 271.

    Google Scholar 

  41. Johnson, M.F.G., Schluter, R.A., and Bankoff, S.G. (1997). Fluorescent imaging system for free surface flow. Rev. Sci. Instrum., 68, 4097 - 4102.

    Article  Google Scholar 

  42. Johnson, M.F.G., Schluter, R.A., Miksis, M.J., and Bankoff, S.G. (1999). Experimental study of rivulet formation on an inclined plate by fluorescent imaging. J. Fluid Mech., 394, 339 - 354.

    Article  MATH  Google Scholar 

  43. Keller, J.B., and Miksis, M.J. (1983). Surface tension driven flows. SIAM J. Appl. Math. 43 (2), 268 - 277.

    MathSciNet  MATH  Google Scholar 

  44. Kistler, S.F. (1993). Hydrodynamics of Wetting, in Wettability, Surfactant Science Series 49, 311-429. ed. J.C. Berg, New York, NY, Marcel Dekker.

    Google Scholar 

  45. Koch, M., Evans, A., and Brunnschweiler, M. (2000). Microfluidic Technology and Applications. Baldock, Hertfordshire, England: Research Studies Press.

    Google Scholar 

  46. Koplik, J., Banavar, J.R., and Willemsen, J.F. (1989). Moelcular dynamics of fluid flow at solid surfaces. Phys. Fluids A 1, 781 - 794.

    Article  Google Scholar 

  47. Lawrie, J.B. (1990). Surface-tension-driven flow in a wedge. Q.J. Mech. Appl. Math. 43, 251 - 273.

    Article  MathSciNet  MATH  Google Scholar 

  48. Li, X and Pozrikidis, C, (1996). Shear flow over a liquid drop adhering to a solid surface. J. Fluid Mech. 307, 167 - 190.

    Article  MATH  Google Scholar 

  49. Lopez, P.G., Bankoff, S.G., and Miksis, M.J. (1996). Non-isothermal spreading of a thin liquid film on an inclined plane. J. Fluid Mech. 324, 261 - 286.

    Article  MATH  Google Scholar 

  50. Lopez, P.G, Miksis, M.J., and Bankoff, S.G. (1997). Inertial effects on contact line instability in the coating of a dry inclined plate. Phys. Fluids 9 (8), 2177 - 2183

    Article  Google Scholar 

  51. Lopez, P.G, Miksis, M.J., and Bankoff, S.G. (2001). Stability and evolution of a dry spot Phys. Fluids 13 (6), 1601 - 1614

    Article  Google Scholar 

  52. Lowndes, J. (1980) The numerical simulation of the steady movement of a fluid meniscus in a capillary tube. J. Fluid Mech. 101, 631 - 646.

    Article  MATH  Google Scholar 

  53. Merchant, G.J., and Keller, J.B. (1991). Flexural rigidity of a liquid surface. J. Stat. Phys. 63, 1039 - 1051.

    Article  MathSciNet  Google Scholar 

  54. Merchant, G.J., and Keller, J.B. (1992). Contact angles. Phys. Fluids 4 (3), 477 - 485.

    Article  MathSciNet  Google Scholar 

  55. Miksis, M.J. and Davis, S.H. (1994). Slip over rough and coated surfaces. J. Fluid Mech. 273, 125 - 139.

    Article  MATH  Google Scholar 

  56. Milne-Thomson, L.M. (1968). Theoretical Hydrodynamics. 5th edition. London, England. MacMillan Press Ltd.

    MATH  Google Scholar 

  57. Moriarty, J.A., and Schwartz, L.W. (1992). Effective slip in numerical calculation of moving-contact-line problems. J. Eng. Math. 26, 81 - 86.

    Article  Google Scholar 

  58. Oron, A., Davis, S.H., and Bankoff, S.G. (1997). Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931 - 980.

    Article  Google Scholar 

  59. Rayleigh, Lord (1890). On the theory of surface forces. Philos Mag. 30, 285-298, 456-475; Scientific Papers, 3, 398-425, Cambridge University Press, 1902, London.

    Google Scholar 

  60. Renardy, M., Renardy Y. and Li, J. (2001). Numerical simulation of moving contact line problems using a volume of fluid method. J. Comput. Phys. 171, 243 - 263.

    Article  MATH  Google Scholar 

  61. Reznik, S. N. and Yarin, A. L. (2002) Spreading of a viscous drop due to gravity and capillarity on a horizontal or an inclined dry wall. Phys. Fluids 14, 118 - 132.

    Article  MathSciNet  Google Scholar 

  62. Richardson, S., (1973). On the no-slip boundary condition. J. Fluid Mech. 59, 707 - 719.

    Article  MATH  Google Scholar 

  63. Rosenblat, S. and Davis, D. H. (1985). How do liquid drops spread on solids? in Frontiers in Fluid Mechanics, ed. S.H. Davis and J.L. Lumley, New York, NY, Springer, 171 - 183.

    Chapter  Google Scholar 

  64. Schleizer, A.D. and Bonnecaze, R.T. (1999). Displacement of a two-dimensional immiscible droplet adhering to a wall in shear and pressure-driven flows. J. Fluid Mech. 383, 29 - 54.

    Article  MATH  Google Scholar 

  65. Shikhmurzaev, Y.D. (1997). Moving contact lines in liquid/liquid/solid systems. J. Fluid Mech. 334, 211 - 249.

    Article  MathSciNet  MATH  Google Scholar 

  66. Siegel, M., Miksis, M.J., and Voorhees, P.W. (2003). Evolution of material voids for highly anisotropic surface energy. J. Mech. Phys. Solids, submitted.

    Google Scholar 

  67. Somalinga, S. and Bose, A. (2000) Numerical investigation of boundary conditions for moving contact line problems. Phys. Fluids 12, 499 - 510.

    Article  MATH  Google Scholar 

  68. Spaid, M.A. and Homsy, G.M. (1996). Stability of newtonian and viscoelastic dynamic contact lines. Phys. Fluids 8, 460 - 478.

    Article  MathSciNet  MATH  Google Scholar 

  69. Tanner, L.H. (1979). The spreading of silicone oil on horizontal surfaces. J. Phys. D: Appl. Phys. 12, 1473

    Article  Google Scholar 

  70. Ting, C.-L., and Perlin, M. (1995). Boundary conditions in the vicinity of the contact line at a vertically oscillating upright plate: an experimental investigation. J. Fluid Mech. 295, 263 - 300.

    Article  Google Scholar 

  71. Thompson, P.A. and Troian, S.M. (1997). A general boundary condition for liquid flow at solid surfaces. Nature 389, 360 - 362.

    Article  Google Scholar 

  72. Troian, S.M., Herbolzheimer, E., Safran, S.A., and Joanny, J.F. (1989). Fingering instabilities of driven spreading films. Europhysics Letters 10 (1), 25 - 30.

    Article  Google Scholar 

  73. Unverdi, S. O. and Tryggvason, G. (1992). A front tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys. 100, 25 - 37.

    Article  MATH  Google Scholar 

  74. vanden Broeck, J.M. (1991). Cavitating flow of a fluid with surface tension past a circular cylinder. Phys. Fluids A 3, 263 - 266.

    Article  MATH  Google Scholar 

  75. vanden Broeck, J.-M., and Keller, J.B. (1987). Weir flows, J. Fluid Mech. 176, 283 - 293.

    Google Scholar 

  76. Young, T. (1805). An essay on the cohesion of fluids. Philos. Trans. R. Soc. London 95, 65 - 87.

    Article  Google Scholar 

  77. Zhang, J. (2003). The dynamics of a viscous drop with a moving contact line. Ph.D. thesis, Northwestern University.

    Google Scholar 

  78. Zhang, J., Miksis, M.J., and Bankoff, S.G. (2003). Motion of a viscous drop with a moving contact line. Preprint.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois, 60208, USA

    Michael J. Miksis

Authors
  1. Michael J. Miksis
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Aerospace Engineering, Technion — Israel Institute of Technology, Haifa, Israel

    Dan Givoli

  2. Department of Mathematics, University of Basel, Basel, Switzerland

    Marcus J. Grote

  3. Department of Mathematics, Stanford University, Stanford, California, USA

    George C. Papanicolaou

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Miksis, M.J. (2004). Contact Lines. In: Givoli, D., Grote, M.J., Papanicolaou, G.C. (eds) A Celebration of Mathematical Modeling. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0427-4_9

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-0427-4_9

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-6526-1

  • Online ISBN: 978-94-017-0427-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation