Spreading of a Thin Liquid Drop Under the Influence of Gravity, Rotation and Non-Uniform Surface Tension

  • Ebrahim Momoniat


Figure 1 shows an axisymmetric thin liquid drop that is spreading under the influence of gravity, rotation and non-uniform surface tension. This process is modeled using the Lie group method. Functional forms of the non-uniform surface tension ∑(x) are obtained. A fourth-order nonlinear partial differential equation describing the evolution of the free surface, u(t,x), of the liquid drop is derived by imposing the thin-film approximation on the Navier-Stokes equations and solving the resulting system of equations subject to boundary conditions on the surface of the rotating disk and on the free surface of the liquid drop. Exact and approximate Lie point symmetry generators admitted by the free surface equation are determined. These symmetry generators are imposed on the moving boundary, x=R(t), where R(t) is the radius of the foot of the liquid drop at time t and x is the radial coordinate fixed on the rotating disk. Functional forms of R(t) are determined. The rate of spreading dR(t)/dt can then be easily calculated.


Coriolis Force Liquid Drop Symmetry Generator Approximate Symmetry Pressure Gradient Force 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D.E. Bornside, C.W. Macosko, and L.E. Scriven, On the modeling of spin coating, J. Imaging Tech. 13: 122(1987).Google Scholar
  2. 2.
    W.W. Flack, D.S. Soong, AT. Bell and D.W. Hess, A mathematical model for spin coating of polymer resists, J. Appl. Phys. 56: 1199 (1984).CrossRefGoogle Scholar
  3. 3.
    C.J. Lawrence, The mechanics of spin coating of polymer films,Phys. Fluids, 31:2786 (1988).CrossRefGoogle Scholar
  4. 4.
    D.J. Acheson, Elementary Fluid Dynamics, Clarendon Press, Oxford (1990).Google Scholar
  5. 5.
    G. K. Batchelor, An Introduction to Fluid Mechanics, Cambridge University Press, Cambridge (1967).Google Scholar
  6. 6.
    T.G. Myers, Thin films with high surface tension, SIAM Review, 40:441 (1998).CrossRefGoogle Scholar
  7. 7.
    H.E. Huppert, The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface, J. Fluid Mech. 121:43 (1982).CrossRefGoogle Scholar
  8. 8.
    S. Middleman, Modeling axisymmetric flows: dynamics of films, jets and drops, Academic Press, San Diego (1995).Google Scholar
  9. 9.
    A.G. Emslie, F.T. Bonner, and L.G. Peck, Flow of a viscous liquid on a rotating disk,J. Appl. Phys. 29:858 (1958).CrossRefGoogle Scholar
  10. 10.
    E. Momoniat and D.P. Mason, Investigation of the effect of the Coriolis force on a thin fluid film on a rotating disk, Int. J. Non-Linear Mech. 33:1069 (1998).CrossRefGoogle Scholar
  11. 11.
    L.V. Ovsiannikov, Group Properties of Differential Equations, Izdat. Sibirsk. Otdel. ANSSSR, Novosibirsk (1962). English translation by G. Bluman (1967).Google Scholar
  12. 12.
    L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York (1982).Google Scholar
  13. 13.
    P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York (1986).Google Scholar
  14. 14.
    H. Stephani,Differential Equations: Their Solution Using Symmetries, Cambridge University Press, Cambridge (1989).Google Scholar
  15. 15.
    G.W. Bluman, and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, Berlin (1989).Google Scholar
  16. 16.
    N.H. Ibragimov, Lie Group Analysis of Differential Equations, Vol. 1, CRC Press Inc, Boca Raton, Florida (1994).Google Scholar
  17. 17.
    N.H. Ibragimov, Lie Group Analysis of Differential Equations, Vol. 2, CRC Press Inc, Boca Raton, Florida (1995).Google Scholar
  18. 18.
    J.M. Hill, Similarity solutions for nonlinear diffusion-A new integration procedure, J. Eng. Math. 23:141 (1989).CrossRefGoogle Scholar
  19. 19.
    D.L. Hill and J.M. Hill, Similarity solutions for nonlinear diffusion-Further exact solutions,J. Eng. Math. 24:109(1990).CrossRefGoogle Scholar
  20. 20.
    E. Momoniat and F. M. Mahomed, The existence of contact transformations for evolution-type equations, J. Phys. A: Math. Gen. 32: 8721 (1999).CrossRefGoogle Scholar
  21. 21.
    G. Baumann, Symmetry analysis with Mathematica, in: Modern Group Analysis VI, N.H. Ibragimov and F.M. Mahomed, eds., New Age International Publishers, New Delhi (1996).Google Scholar
  22. 22.
    F.S. Sherman, Viscous Flow, McGraw-Hill, New-York (1990).Google Scholar
  23. 23.
    V.A. Baikov, R.K. Gazizov, and N.H. Ibragimov, Approximate symmetries of equations with a small parameter,Mat. Sbornik. 136: 435 (1988). English trans, in Math. USSR Sbornik, 64: 427.Google Scholar
  24. 24.
    W.I. Fushchich, and W. M. Shtelen, On approximate symmetry and approximate solutions on the nonlinear wave equation with a small parameter, J. Phys. A: Math. Gen. 22: L887 (1989).CrossRefGoogle Scholar
  25. 25.
    M. Euler, N. Euler, and A. Köhler, On the construction of approximate solutions for a multidimensional nonlinear heat equation, J. Phys. A: Math. Gen, 27:2083 (1994).CrossRefGoogle Scholar
  26. 26.
    N. Euler, M.W. Shul’ga, and W.H. Steeb, Approximate symmetries and approximate solutions for a multi-dimensional Landau-Ginzburg equation, J. Phys. A: Math. Gen. 25:L1095 (1992).CrossRefGoogle Scholar
  27. 27.
    N. Euler, and M. Euler, Symmetry properties of the approximations of multi-dimensional generalized Van der Pol equations, Nonlinear Mathematical Physics, 1:41 (1994).CrossRefGoogle Scholar
  28. 28.
    R.K. Gazizov, Symmetry of equations with a small parameter, in: Proceedings on Modern Group Analysis VII, N.H. Ibragimov, R.K. Naqvi and E. Straume, eds., MARS Publishers, Norway (1999).Google Scholar
  29. 29.
    E.V. Dussan, and S.H. Davis, On the motion of a fluid-fluid interface along a solid surface, J. Fluid Mech. 65:71 (1974).CrossRefGoogle Scholar
  30. 30.
    E. V. Dussan, On the spreading of liquids on solid surfaces: static and dynamic contact lines, Ann. Rev. Fluid Mech. 11:371 (1979).CrossRefGoogle Scholar
  31. 31.
    L.M. Hocking, A moving fluid interface on a rough surface, J. Fluid Mech. 76: 801 (1976).CrossRefGoogle Scholar
  32. 32.
    L.M. Hocking, A moving fluid interface. Part 2. The removal of the force singularity by a slip flow,J. Fluid Mech. 79: 209(1977).CrossRefGoogle Scholar
  33. 33.
    L.M. Hocking and A.D. Rivers, The spreading of a drop by capillary action, J. Fluid Mech. 121: 425 (1982).CrossRefGoogle Scholar
  34. 34.
    L.M. Hocking, The spreading of a thin drop by gravity and capillarity, Q. J. Mech. Appl. Math. 36: 55 (1983).CrossRefGoogle Scholar
  35. 35.
    L.H. Tanner, The spreading of silicone oil drops on horizontal surfaces, J. Phys. D Appl. Phys. 12: 1473 (1979).CrossRefGoogle Scholar
  36. 36.
    J.D. Chen, Experiments on a spreading drop and its contact angle on a solid,J. Coll. Interf. Sci. 122: 60 (1988).CrossRefGoogle Scholar
  37. 37.
    S.K. Wilson, R. Hunt and B.R. Duffy, The rate of spreading in spin coating, J. Fluid. Mech. 413: 65 (2000).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Ebrahim Momoniat
    • 1
  1. 1.Department of Computational and Applied MathematicsUniversity of the Witwatersrand, WitsJohannesburgSouth Africa

Personalised recommendations