Skip to main content
Log in

Stability analysis of a gravity-driven thermoviscous liquid film flowing over a heated flat substrate

The European Physical Journal E Aims and scope Submit manuscript

Abstract.

Two-dimensional steady-state solutions and their stability analysis are presented for a gravity-driven thin film of a thermoviscous liquid. The governing equations and boundary conditions are simplified using the lubrication approximation. The analytically obtained film thickness evolution equation consists of various dimensionless parameters such as the Marangoni number, Biot number and thermoviscosity number. The viscosity of the liquid is assumed as an exponential function of temperature. The viscosity decreases within the liquid film as the temperature increases. Due to localized heating interfacial temperature gradients generate surface tension gradient which results into thermocapillary or Marangoni stress. The Marangoni stress opposes the fluid flow at the leading edge of heater leading to an increase in the film thickness locally. This locally thick structure becomes unstable beyond critical values of the parameters that leads to formation of rivulets in the transverse direction. Using the linear stability analysis it is found that the Marangoni stress and the thermoviscous effect have a destabilizing effect on the thin-film flow. At much higher values of the thermoviscosity number another mode of instability appears which is known as thermocapillary instability which leads to oscillating film profiles. For streamwise perturbations, the destabilizing effect of the thermoviscosity number for localized and uniform heating remains consistent.

Graphical abstract

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D.L. DeVoe, Sens. Actuators A: Phys. 88, 263 (2001)

    Article  Google Scholar 

  2. S.G. Bankoff, J. Heat Transfer 116, 10 (1994)

    Article  Google Scholar 

  3. K. Wantanaa, P. Aniwatc, S. Bunluec, T. Alongkotd, K. Anusite, K. Pisiste, Mater. Today: Proc. 4, 6626 (2017)

    Google Scholar 

  4. A. Oron, S.H. Davis, S.G. Bankoff, Rev. Mod. Phys. 69, 931 (1997)

    Article  ADS  Google Scholar 

  5. R. Craster, O. Matar, Rev. Mod. Phys. 81, 1131 (2009)

    Article  ADS  Google Scholar 

  6. S.W. Joo, S.H. Davis, S.G. Bankoff, J. Fluid Mech. 230, 117 (1991)

    Article  ADS  Google Scholar 

  7. S. Kalliadasis, E.A. Demekhin, C. Ruyer-Quil, M.G. Velarde, J. Fluid Mech. 492, 303 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  8. P.M.J. Trevelyan, S. Kalliadasis, J. Eng. Math. 50, 177 (2004)

    Article  Google Scholar 

  9. B. Scheid, A. Oron, P. Colinet, U. Thiele, J.C. Legros, Phys. Fluids 14, 4130 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  10. J.M. Skotheim, U. Thiele, B. Scheid, J. Fluid Mech. 475, 1 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  11. N. Tiwari, Z. Mester, J.M. Davis, Phys. Rev. E 76, 056306 (2007)

    Article  ADS  Google Scholar 

  12. N. Tiwari, J.M. Davis, Phys. Fluids 21, 022105 (2009)

    Article  ADS  Google Scholar 

  13. N. Tiwari, A. Awasthi, J.M. Davis, Phys. Fluids 26, 042105 (2014)

    Article  ADS  Google Scholar 

  14. N. Tiwari, J.M. Davis, Phys. Fluids 21, 102101 (2009)

    Article  ADS  Google Scholar 

  15. S. Kalliadasis, C. Bielarz, G.M. Homsy, Phys. Fluids 12, 1889 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  16. S. Kalliadasis, G.M. Homsy, J. Fluid Mech. 448, 387 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  17. C. Bielarz, S. Kalliadasis, Phys. Fluids 15, 2512 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  18. J.M. Davis, S.M. Troian, Phys. Fluids 17, 0721031 (2005)

    Google Scholar 

  19. S. Saprykin, P.M.J. Trevelyan, R.J. Koopmans, S. Kalliadasis, Phys. Rev. E 75, 026306 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  20. A.M. Cazabat, F. Heslot, S.M. Troian, P. Carles, Nature 346, 824 (1990)

    Article  ADS  Google Scholar 

  21. A.M. Frank, O.A. Kabov, Phys. Fluids 18, 032107 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  22. K.A. Smith, J. Fluid Mech. 24, 401 (1966)

    Article  ADS  Google Scholar 

  23. S. Sreenivasan, S.P. Lin, Int. J. Heat Mass Transfer 21, 1517 (1978)

    Article  Google Scholar 

  24. D.A. Goussis, R.E. Kelly, Int. J. Heat Mass Transfer 33, 2237 (1990)

    Article  Google Scholar 

  25. D.A. Goussis, R.E. Kelly, J. Fluid Mech. 223, 25 (1991)

    Article  ADS  Google Scholar 

  26. O.A. Kabov, Heat transfer from a heater with a small linear dimension to a free falling liquid film, in Proceedings of the 1st Russian National Conference on Heat Transfer, Vol. 6 (Energoatomizdat, Moscow, 1994) pp. 90--95

  27. O.A. Kabov, Heat Transfer Res. 27, 221 (1996)

    Google Scholar 

  28. O.A. Kabov, J.C. Legros, I. Marchuk, B. Scheid, Fluid Dyn. 36, 521 (2001)

    Article  Google Scholar 

  29. N. Tiwari, J.M. Davis, J. Colloid Interface Sci. 355, 243 (2011)

    Article  ADS  Google Scholar 

  30. D. Goussis, R.E. Kelly, Phys. Fluids 28, 3207 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  31. S.M. Yih, R.C. Seagrave, Am. Inst. Chem. Eng. J. 24, 803 (1978)

    Article  Google Scholar 

  32. C.-C. Hwang, C.-I. Weng, Int. J. Heat Mass Transfer 31, 1775 (1988)

    Article  Google Scholar 

  33. S.K. Wilson, B.R. Duffy, J. Eng. Math. 42, 359 (2002)

    Article  Google Scholar 

  34. G.A. Leslie, S.K. Wilson, B.R. Duffy, Phys. Fluids 23, 062101 (2011)

    Article  ADS  Google Scholar 

  35. G.A. Leslie, S.K. Wilson, B.R. Duffy, Q. J. Mech. Appl. Math. 65, 483 (2012)

    Article  Google Scholar 

  36. T.C. Kumawat, N. Tiwari, Phys. Fluids 30, 032103 (2018)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Naveen Tiwari.

Additional information

Publisher’s Note

The EPJ Publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Srivastava, A., Kumawat, T.C. & Tiwari, N. Stability analysis of a gravity-driven thermoviscous liquid film flowing over a heated flat substrate. Eur. Phys. J. E 42, 54 (2019). https://doi.org/10.1140/epje/i2019-11820-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epje/i2019-11820-1

Keywords

Navigation