Skip to main content
SpringerLink
Log in

Effect of an insoluble surfactant on the dynamics of a thin liquid film flowing over a non-uniformly heated substrate

  • Regular Article
  • Published:
The European Physical Journal E Aims and scope Submit manuscript

Abstract.

The stability analysis of a gravity-driven thin liquid film with an insoluble surfactant flowing over a surface with embedded, regularly spaced heaters is investigated. At the leading edge of a heater, the presence of a temperature gradient induces an opposing Marangoni stress at the interface leading to the formation of a capillary ridge. This ridge has been shown to be susceptible to thermocapillary (oscillating in the flow direction) and rivulet (spanwise periodic pattern) instabilities. The presence of an insoluble surfactant is shown to have a stabilizing effect on this system. The governing equations for the evolution of the film thickness and surfactant concentration are obtained within the lubrication approximation. The coupled two-dimensional base solutions for the film thickness and surfactant concentration show that there is no significant change in the height of the capillary ridge at the subsequent heaters downstream. The height of the capillary ridge is reduced by the presence of the surfactant. For very small Peclet number, the presence of multiple heaters has almost no significant effect on the film stability as compared to a single heater and similar trends are observed between the two configurations in the presence of the surfactant as for the case of a clean interface. However, for large Peclet number, the effect was observed on both types of instabilities for certain heater configurations. The Biot number is shown to have a strong effect on the stability results wherein the dominant mode of instability is altered (from rivulet to thermocapillary instability) for a passive or no surfactant case with increase in the Biot number. For an active surfactant thermocapillary instability is found to remain the dominant mode of instability for all the values of the Biot number. It is shown that increasing the number of heaters beyond a couple does not further affect the stability results.

Graphical abstract

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

Access this article

Log in via an institution

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. V.G. Levich, Physicochemical Hydrodynamics (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1962)

  2. S.H. Davis, Annu. Rev. Fluid Mech. 19, 403 (1987)

    Article  ADS  Google Scholar 

  3. M. Dietzel, S.M. Troian, Phys. Rev. Lett. 103, 074501 (2009)

    Article  ADS  Google Scholar 

  4. O.A. Kabov, I.V. Marchuk, V.M. Chupin, Russ. J. Eng. Thermophys. 6, 105 (1996)

    Google Scholar 

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

    Article  Google Scholar 

  6. O.A. Kabov, B. Scheid, I.A. Sharina, J.C. Legros, Int. J. Therm. Sci. 41, 664 (2002)

    Article  Google Scholar 

  7. A.M. Frank, Eur. J. Mech. B/Fluids 22, 445 (2003)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  9. S. Kalliadasis, A. Kiyashko, E.A. Demekhin, J. Fluid Mech. 475, 377 (2003)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  12. Y.O. Kabova, V.V. Kuznetsov, O.A. Kabov, Microgravity Sci. Technol. 19, 53 (2007)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  16. H.H. Katkar, J.M. Davis, J. Fluid Mech. 726, 656 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  17. R. Liu, O.A. Kabov, Int. J. Heat Mass Transfer 65, 23 (2013)

    Article  Google Scholar 

  18. J.B. Grotberg, Annu. Rev. Fluid Mech. 26, 529 (1994)

    Article  ADS  Google Scholar 

  19. A.F.M. Leenaars, J.A.M. Huethorst, J.J. van Oekel, Langmuir 6, 1701 (1990)

    Article  Google Scholar 

  20. A. Sharma, E. Ruckenstein, J. Colloid Interface Sci. 111, 8 (1986)

    Article  ADS  Google Scholar 

  21. D. Edwards, H. Brenner, D.T. Wasan, Interfacial Transport Processes and Rheology (Butterworth-Heinemann, MA, 1991)

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

    Article  ADS  Google Scholar 

  23. R.V. Craster, O.K. Matar, Rev. Mod. Phys. 81, 1131 (2009)

    Article  ADS  Google Scholar 

  24. S. Whitaker, Ind. Eng. Chem. Fundam. 3, 132 (1964)

    Article  Google Scholar 

  25. B.E. Anshus, A. Acrivos, Chem. Eng. Sci. 22, 389 (1967)

    Article  Google Scholar 

  26. S.P. Lin, AIChE J. 16, 375 (1970)

    Article  Google Scholar 

  27. A.D. Wit, D. Gallez, C.I. Christov, Phys. Fluids 6, 3256 (1994)

    Article  ADS  Google Scholar 

  28. K.D. Danov, N. Alleborn, H. Raszillier, F. Durst, Phys. Fluids 10, 131 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  29. K.D. Danov, V.N. Paunov, N. Alleborn, H. Raszillier, F. Durst, Chem. Eng. Sci. 53, 2809 (1998)

    Article  Google Scholar 

  30. K.D. Danov, V.N. Paunov, S.D. Stoyanov, N. Alleborn, H. Raszillier, F. Durst, Chem. Eng. Sci. 53, 2823 (1998)

    Article  Google Scholar 

  31. V.N. Paunov, K.D. Danov, N. Alleborn, H. Raszillier, F. Durst, Chem. Eng. Sci. 53, 2839 (1998)

    Article  Google Scholar 

  32. M.G. Blyth, C. Pozrikidis, J. Fluid Mech. 521, 241 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  33. B.D. Edmonstone, O.K. Matar, J. Colloid Interface Sci. 274, 183 (2004)

    Article  ADS  Google Scholar 

  34. Z. Ding, T.N. Wong, Int. J. Heat Mass Transfer 67, 627 (2013)

    Article  Google Scholar 

  35. W.M. Deen, Analysis of Transport Phenomena (Oxford University Press, New York, 1998)

  36. B.D. Edmonstone, O.K. Matar, R.V. Craster, J. Eng. Math. 50, 141 (2004)

    Article  Google Scholar 

  37. L.G. Leal, Advanced Transport Phenomena (Cambridge University Press, New York, 2007)

Download references

Author information

Authors and Affiliations

  1. Department of Chemical Engineering, Indian Institute of Technology Kanpur, 208016, Kanpur, India

    Ashna Srivastava & Naveen Tiwari

Authors
  1. Ashna Srivastava
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Naveen Tiwari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Naveen Tiwari.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Srivastava, A., Tiwari, N. Effect of an insoluble surfactant on the dynamics of a thin liquid film flowing over a non-uniformly heated substrate. Eur. Phys. J. E 41, 56 (2018). https://doi.org/10.1140/epje/i2018-11664-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epje/i2018-11664-1

Keywords

Search

Navigation