Journal of Engineering Mathematics

, Volume 42, Issue 3–4, pp 359–372 | Cite as

On the gravity-driven draining of a rivulet of fluid with temperature-dependent viscosity down a uniformly heated or cooled substrate

Article

Abstract

The lubrication approximation is used to investigate the unsteady gravity-driven draining of a thin rivulet of Newtonian fluid with temperature-dependent viscosity down a substrate that is either uniformly hotter or uniformly colder than the surrounding atmosphere. First the general nonlinear evolution equation is derived for a thin film of fluid with an arbitrary dependence of viscosity on temperature. Then it is shown that at leading order in the limit of small Biot number the rivulet is isothermal, as expected, but that at leading order in the limit of large Biot number (in which the rivulet is not isothermal) the governing equation can, rather unexpectedly, always be reduced to that in the isothermal case with a suitable rescaling. These results are then used to give a complete description of steady flow of a slender rivulet in the limit of large Biot number in two situations in which the corresponding isothermal problem has previously been solved analytically, namely non-uniform flow down an inclined plane, and locally unidirectional flow down a slowly varying substrate. In particular, it is found that if a suitably defined integral measure of the fluidity of the film is a decreasing function of the temperature of the atmosphere (as it is for all three specific viscosity models considered) then decreasing the temperature of the atmosphere always has the effect of making the rivulet wider and deeper.

rivulet thin-film flow temperature-dependent viscosity 

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References

  1. 1.
    G. D. Towell and L. B. Rothfeld, Hydrodynamics of rivulet flow. AIChE. J. 12 (1966) 972–980.Google Scholar
  2. 2.
    B. R. Duffy and H. K.Moffatt, Flow of a viscous trickle on a slowly varying incline.Chem. Eng. J. 60 (1995) 141–146.Google Scholar
  3. 3.
    R. F. Allen and C. M. Biggin, Longitudinal flow of a lenticular liquid filament down an inclined plane. Phys. Fluids 17 (1974) 287–291.Google Scholar
  4. 4.
    S. K. Wilson and B. R. Duffy, On the gravity-driven draining of a rivulet of viscous fluid down a slowly varying substrate with variation transverse to the direction of flow. Phys. Fluids 10 (1998) 13–22.Google Scholar
  5. 5.
    S. K. Wilson, B. R. Duffy and A. B. Ross, On the gravity-driven draining of a rivulet of viscoplastic material down a slowly varying substrate. Phys. Fluids 14 (2002) 555–571.Google Scholar
  6. 6.
    P. C. Smith, A similarity solution for slow viscous flow down an inclined plane. J. Fluid Mech. 58 (1973) 275–288.Google Scholar
  7. 7.
    L. W. Schwartz and E. E. Michaelides, Gravity flow of a viscous liquid down a slope with injection. Phys. Fluids 31 (1988) 2739–2741.Google Scholar
  8. 8.
    B. R. Duffy and H. K. Moffatt, A similarity solution for viscous source flow on a vertical plane. Euro. J. Appl. Math. 8 (1997) 37–47.Google Scholar
  9. 9.
    S. K. Wilson, B. R. Duffy and S. H. Davis, On a slender dry patch in a liquid film draining under gravity down an inclined plane. Euro. J. Appl. Math. 12 (2001) 233–252.Google Scholar
  10. 10.
    S. K. Wilson, B. R. Duffy and R. Hunt, A slender rivulet of a power-law fluid driven by either gravity or a constant shear stress at the free surface. To appear in Q. J. Mech. Appl. Math.Google Scholar
  11. 11.
    D. Holland, B. R. Duffy and S. K. Wilson, Thermocapillary effects on a thin viscous rivulet draining steadily down a uniformly heated or cooled slowly varying substrate. J. Fluid Mech. 441 (2001) 195–221.Google Scholar
  12. 12.
    D. A. Goussis and R. E. Kelly, Effects of viscosity variation on the stability of film flow down heated or cooled inclined surfaces: Long-wavelength analysis. Phys. Fluids 28 (1985) 3207–3214.Google Scholar
  13. 13.
    C.-C. Hwang and C.-I. Weng, Non-linear stability analysis of film flow down a heated or cooled inclined plane with viscosity variation. Int. J. Heat Mass Transfer 31 (1988) 1775–1784.Google Scholar
  14. 14.
    B. Reisfeld and S. G. Bankoff, Nonlinear stability of a heated thin liquid film with variable viscosity. Phys. Fluids A 2 (1990) 2066–2067.Google Scholar
  15. 15.
    M.-C. Wu and C.-C. Hwang, Nonlinear theory of film rupture with viscosity variation. Int. Comm. Heat Mass Transfer 18 (1991) 705–713.Google Scholar
  16. 16.
    A. Oron, S. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (1997) 931–980.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Strathclyde, Livingstone TowerGlasgow

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