An analytical comprehensive solution for the superficial waves appearing in gravity-driven flows of liquid films

A Correction to this article was published on 25 September 2020

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Abstract

This paper is devoted to analytical solutions for the base flow and temporal stability of a liquid film driven by gravity over an inclined plane when the fluid rheology is given by the Carreau–Yasuda model, a general description that applies to different types of fluids. In order to obtain the base state and critical conditions for the onset of instabilities, two sets of asymptotic expansions are proposed, from which it is possible to find four new equations describing the reference flow and the phase speed and growth rate of instabilities. These results lead to an equation for the critical Reynolds number, which dictates the conditions for the onset of the instabilities of a falling film. Different from previous works, this paper presents asymptotic solutions for the growth rate, wavelength and celerity of instabilities obtained without supposing a priori the exact fluid rheology, being, therefore, valid for different kinds of fluids. Our findings represent a significant step toward understanding the stability of gravitational flows of non-Newtonian fluids.

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Change history

  • 25 September 2020

    In the original publication of the article, Eq. (23) was incorrectly published.

References

  1. 1.

    Kapitza, P.L.: Wave flow of thin layers of a viscous liquid. Part I. Free flow. Zh. Eksp. Teor. Fiz 18(1), 3 (1948)

    Google Scholar 

  2. 2.

    Kapitza, P.L., Kapitza, S.P.: Wave flow of thin layers of a viscous fluid. Zh. Eksp. Teor. Fiz 19, 105 (1949)

    MATH  Google Scholar 

  3. 3.

    Benjamin, T.B.: Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2(06), 554 (1957)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Benney, D.: Long waves on liquid films. J. Math. Phys. 45(2), 150 (1966)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Yih, C.S.: Stability of liquid flow down an inclined plane. Phys. Fluids 6(3), 321 (1963)

    Article  Google Scholar 

  6. 6.

    Lin, C.C.: On the stability of two-dimensional parallel flows. Part III. Stability in a viscous fluid. Q. Appl. Math. 3(4), 27 (1946)

    Article  Google Scholar 

  7. 7.

    Orszag, S.A.: Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50(04), 689 (1971)

    Article  Google Scholar 

  8. 8.

    Floryan, J.M., Davis, S.H., Kelly, R.E.: Instabilities of a liquid film flowing down a slightly inclined plane. Phys. Fluids 30(4), 983 (1987)

    Article  Google Scholar 

  9. 9.

    Morrison, F.A.: Understanding Rheology. Oxford University Press, Oxford (2001)

    MATH  Google Scholar 

  10. 10.

    Macosko, C.W.: Rheology: principles, measurements, and applications. Wiley-Vch, Weinheim (1994)

    Google Scholar 

  11. 11.

    Ostwald, V.: Ueber die rechnerische Darstellung des Strukturgebietes der Viskosität. Colloid Polym. Sci. 47(2), 176 (1929)

    Google Scholar 

  12. 12.

    Bingham, E.: An investigation of the laws of plastic flow. US Bur Stand Bull 13, 309 (1916)

    Article  Google Scholar 

  13. 13.

    Cross, M.: Rheology of non-Newtonian fluids: a new flow equation for pseudoplastic systems. J. Coll. Sci. 20(5), 417 (1965)

    Article  Google Scholar 

  14. 14.

    Ng, C.O., Mei, C.C.: Roll waves on a shallow layer of mud modelled as a power-law fluid. J. Fluid Mech. 263, 151 (1994)

    Article  Google Scholar 

  15. 15.

    Ruyer-Quil, S., Chakraborty, C., Dandapat, B.S.: Wavy regime of a power-law film flow. J. Fluid Mech. 692, 220 (2012)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Noble, P., Vila, J.P.: Thin power-law film flow down an inclined plane: consistent shallow-water models and stability under large-scale perturbations. J. Fluid Mech. 735, 29 (2013)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Yasuda, K., Armstrong, R., Cohen, R.: Shear flow properties of concentrated solutions of linear and star branched polystyrenes. Rheol. Acta 20(2), 163 (1981)

    Article  Google Scholar 

  18. 18.

    Nouar, A., Bottaro, C., Brancher, J.P.: Delaying transition to turbulence in channel flow: revisiting the stability of shear-thinning fluids. J. Fluid Mech. 592, 177 (2007)

    Article  Google Scholar 

  19. 19.

    Weinstein, S.J.: Wave propagation in the flow of shear-thinning fluids down an incline. AIChE J. 36(12), 1873 (1990)

    Article  Google Scholar 

  20. 20.

    Rousset, F., Millet, S., Botton, V., Hadid, H.B.: Temporal stability of Carreau fluid flow down an incline. J. Fluid Eng. Trans. ASME 129(7), 913 (2007)

    Article  Google Scholar 

  21. 21.

    Squire, H.B.: On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls. Proc. R. Soc. Lond. A 142(847), 621 (1933)

    Article  Google Scholar 

  22. 22.

    Yih, C.S.: Stability of two-dimensional parallel flows for three-dimensional disturbances. Q. Appl. Math. 12(4), 434 (1955)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Hesla, F.R., Pranckh, T.I., Preziosi, L.: Squire’s theorem for two stratified fluids. Phys. Fluids 29(9), 2808 (1986)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Gupta, A.S., Rai, L.: Note on the stability of a visco-elastic liquid film flowing down an inclined plane. J. Fluid Mech. 33(1), 87 (1968)

    Article  Google Scholar 

  25. 25.

    Van Dyke, M.: Perturbation Methods in Fluid Mechanics. The Parabolic Press, Stanford (1975)

    MATH  Google Scholar 

  26. 26.

    Chimetta, B.P., Franklin, E.M.: Files of Wolfram Mathematica and MATLAB scripts for solving the Orr–Sommerfeld equation for a Carreau–Yasuda fluid over an inclined plane are available on Mendeley Data. https://doi.org/10.17632/yjs9889ytf.1 (2019)

  27. 27.

    See Supplementary Material for a additional graphics concerning the base state and critical conditions

  28. 28.

    Peralta, J., Meza, B., Zorrilla, S.: Analytical solutions for the free-draining flow of a Carreau–Yasuda fluid on a vertical plate. Chem. Eng. Sci. 168, 391 (2017)

    Article  Google Scholar 

  29. 29.

    Khechiba, K., Mamou, M., Hachemi, M., Delenda, N., Rebhi, R.: Effect of Carreau–Yasuda rheological parameters on subcritical Lapwood convection in horizontal porous cavity saturated by shear-thinning fluid. Phys. Fluids 29(6), 063101 (2017)

    Article  Google Scholar 

  30. 30.

    Japper-Jaafar, A., Escudier, M., Poole, R.: Turbulent pipe flow of a drag-reducing rigid “rod-like” polymer solution. J. Non Newton. Fluid 161(1–3), 86 (2009)

    Article  Google Scholar 

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Acknowledgements

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. Bruno Pelisson Chimetta is grateful to CAPES, and Erick de Moraes Franklin would like to express his gratitude to FAPESP (Grant No. 2018/14981-7), CNPq (Grant No. 400284/2016-2) and FAEPEX/UNICAMP (Grant No. 2112/19) for the financial support they provided.

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Correspondence to Erick Franklin.

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This study was funded by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, Fundação de Amparo à Pesquisa do Estado de São Paulo - FAPESP (Grant No. 2018/14981-7), Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq (Grant No. 400284/2016-2) and Fundo de Apoio ao Ensino, Pesquisa e Extensão da Unicamp - FAEPEX/UNICAMP (Grant No. 2112/19).

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The original version of this article was revised: Eq. (23) was incorrectly published and it has been corrected in this version.

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Chimetta, B.P., Franklin, E. An analytical comprehensive solution for the superficial waves appearing in gravity-driven flows of liquid films. Z. Angew. Math. Phys. 71, 122 (2020). https://doi.org/10.1007/s00033-020-01349-x

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Keywords

  • Gravity-driven flow
  • Generalized Newtonian fluid
  • Carreau–Yasuda model
  • Temporal stability
  • Asymptotic method

Mathematics Subject Classification

  • 76E17