Skip to main content
SpringerLink
Log in

The mechanism of long-wave instability in a shear-thinning film flow on a porous substrate

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

A linear stability analysis of a thin shear-thinning film with a deformable top surface flowing down an inclined porous substrate modelled as a smooth substrate with velocity slip at the wall is examined, and the physical mechanism for the long-wave instability is analysed. Through a phenomenological model, the influence of slip velocity and the shear-thinning rheology on the wave speed of long surface waves on a non-Newtonian shear-thinning film down a substrate with velocity slip is predicted. The viscosity disturbance plays a significant role in the destabilization of the flow system. Indeed, slip at the bottom that accounts for the characteristics of the porous/rough substrate does not affect the physical mechanism of the instability. However, it is shown that slip at the bottom enhances the inertia effects which in turn destabilizes the flow system at smaller Reynolds numbers.

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. Amaouche, M., Mehidi, N., Amatousse, N.: Linear stability of a two-layer film flow down an inclined channel: a second-order weighted residual approach. Phys. Fluids 19(8), 084106 (2007)

    Article  MATH  Google Scholar 

  2. Beavers, G.S., Joseph, D.D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30(1), 197–207 (1967)

    Article  Google Scholar 

  3. Benjamin, T.: Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2(06), 554–573 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bird, R.B., Armstrong, R.C., Hassager, O., Curtiss, C.F.: Dynamics of Polymeric Liquids, vol. 1. Wiley, New York (1977)

    Google Scholar 

  5. Carreau, P., Kee, D., Daroux, M.: An analysis of the viscous behaviour of polymeric solutions. Can. J. Chem. Eng. 57(2), 135–140 (1979)

    Article  Google Scholar 

  6. Chang, H.: Wave evolution on a falling film. Annu. Rev. Fluid Mech. 26(1), 103–136 (1994)

    Article  MathSciNet  Google Scholar 

  7. Chang, M.H.: Thermal convection in superposed fluid and porous layers subjected to a plane poiseuille flow. Phys. Fluids 18(3), 035104 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chang, M.H., Chen, F., Straughan, B.: Instability of poiseuille flow in a fluid overlying a porous layer. J. Fluid Mech. 564, 287–303 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chattopadhyay, G., Ranganathan, U., Millet, S.: Instabilities in viscosity-stratified two-fluid channel flow over an anisotropic-inhomogeneous porous bottom. Phys. Fluids 31(1), 012103 (2019)

    Article  Google Scholar 

  10. Cheremisinoff, N.: Encyclopedia of Fluid Mechanics: Rheology and Non-Newtonian Flows, vol. 7. Gulf Publishing Corp., Houston (1988)

    Google Scholar 

  11. Craster, R., Matar, O.: Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81(3), 1131 (2009)

    Article  Google Scholar 

  12. Danaila, L., Voivenel, L., Varea, E.: Self-similarity criteria in anisotropic flows with viscosity stratification. Phys. Fluids 29(2), 020716 (2017)

    Article  Google Scholar 

  13. Ewing, R.E., Weekes, S.: Numerical methods for contaminant transport in porous media. Adv. Comput. Math. 202, 75–95 (1998)

    MathSciNet  MATH  Google Scholar 

  14. Gao, P., Lu, X.Y.: Mechanism of the long-wave inertialess instability of a two-layer film flow. J. Fluid Mech. 608, 379–391 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ghosh, S., Usha, R.: Stability of viscosity stratified flows down an incline: role of miscibility and wall slip. Phys. Fluids 28(10), 104101 (2016)

    Article  Google Scholar 

  16. Govindarajan, R., Sahu, K.C.: Instabilities in viscosity-stratified flow. Annu. Rev. Fluid Mech. 46, 331–353 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Goyal, H., Kumar, A.A.P., Bandyopadhyay, D., Usha, R., Banerjee, T.: Instabilities of a confined two-layer flow on a porous medium: an Orr–Sommerfeld analysis. Chem. Eng. Sci. 97, 109–125 (2013)

    Article  Google Scholar 

  18. Hill, A.A., Straughan, B.: Poiseuille flow in a fluid overlying a porous medium. J. Fluid Mech. 603, 137–149 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hwang, C.C., Chen, J.L., Wang, J.S., Lin, J.S.: Linear stability of power law liquid film flows down an inclined plane. J. Phys. D: Appl. Phys. 27(11), 2297 (1994)

    Article  Google Scholar 

  20. Iervolino, M., Pascal, J.P., Vacca, A.: Instabilities of a power-law film over an inclined permeable plane: a two-sided model. J. Non-Newton. Mech. 259, 111–124 (2018)

    Article  MathSciNet  Google Scholar 

  21. Kalliadasis, S., Ruyer-Quil, C., Scheid, B., Velarde, M.: Falling Liquid Films. Springer Series on Applied Mathematical Sciences. Springer, London (2011)

    MATH  Google Scholar 

  22. Kelly, R., Goussis, D., Lin, S., Hsu, F.: The mechanism for surface wave instability in film flow down an inclined plane. Phys. Fluids A: Fluid Dyn. 1(5), 819–828 (1989)

    Article  Google Scholar 

  23. Knox, D.J., Duffy, B.R., McKee, S., Wilson, S.K.: Squeeze-film flow between a curved impermeable bearing and a flat porous bed. Phys. Fluids 29(2), 023101 (2017). https://doi.org/10.1063/1.4974521

    Article  Google Scholar 

  24. Kumar, A.A.P., Goyal, H., Banerjee, T., Bandyopadhyay, D.: Instability modes of a two-layer newtonian plane couette flow past a porous medium. Phys. Rev. E 87(6), 063003 (2013)

    Article  Google Scholar 

  25. Liu, R., Liu, Q.: Instabilities of a liquid film flowing down an inclined porous plane. Phys. Rev. E 80(3), 036316 (2009)

    Article  Google Scholar 

  26. Liu, R., Liu, Q.S., Zhao, S.C.: Instability of plane poiseuille flow in a fluid-porous system. Phys. Fluids 20(10), 104105 (2008)

    Article  MATH  Google Scholar 

  27. Maissel, L.I., Glang, R.: Handbook of Thin Film Technology. McGraw-Hill, New York (1970)

    Google Scholar 

  28. Millet, S., Botton, V., Ben Hadid, H., Henry, D., Rousset, F.: Stability of two-layer shear-thinning film flows. Phys. Rev. E 88(4), 043004 (2013). https://doi.org/10.1103/PhysRevE.88.043004

    Article  Google Scholar 

  29. Millet, S., Botton, V., Rousset, F., Ben Hadid, H.: Wave celerity on a shear-thinning fluid film flowing down an incline. Phys. Fluids 20(3), 031701 (2008)

    Article  MATH  Google Scholar 

  30. Myers, T.: Application of non-Newtonian models to thin film flow. Phys. Rev. E 72(6), 066302 (2005)

    Article  MathSciNet  Google Scholar 

  31. Neogi, P., Miller, C.A.: Spreading kinetics of a drop on a rough solid surface. J. Colloid Interface Sci. 92(2), 338–349 (1983)

    Article  Google Scholar 

  32. Nield, D.A., Bejan, A.: Convection in Porous Media, vol. 3. Springer, New York (2006)

    MATH  Google Scholar 

  33. Oron, A., Davis, S.H., Bankoff, S.G.: Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69(3), 931 (1997)

    Article  Google Scholar 

  34. Pascal, J.: Linear stability of fluid flow down a porous inclined plane. J. Phys. D: Appl. Phys. 32(4), 417 (1999)

    Article  MathSciNet  Google Scholar 

  35. Picchi, D., Barmak, I., Ullmann, A., Brauner, N.: Stability of stratified two-phase channel flows of Newtonian/non-Newtonian shear-thinning fluids. Int. J. Multiph. Flow 99, 111–131 (2018)

    Article  MathSciNet  Google Scholar 

  36. Rousset, F., Millet, S., Botton, V., Ben Hadid, H.: Temporal stability of Carreau fluid flow down an incline. J. Fluids Eng.—Trans. ASME 129(7), 913–920 (2007)

    Article  Google Scholar 

  37. Sadiq, I.M.R., Usha, R.: Thin Newtonian film flow down a porous inclined plane: stability analysis. Phys. Fluids 20(2), 022105 (2008)

    Article  MATH  Google Scholar 

  38. Sadiq, I.M.R., Usha, R.: Long-wave instabilities in a non-Newtonian film on a nonuniformly heated inclined plane. J. Fluids Eng. 131(3), 031202 (2009)

    Article  Google Scholar 

  39. Saikia, B., Ramachandran, A., Sinha, K., Govindarajan, R.: Effects of viscosity and conductivity stratification on the linear stability and transient growth within compressible Couette flow. Phys. Fluids 29(2), 024105 (2017)

    Article  Google Scholar 

  40. Schmid, P.J., Henningson, D.S.: Stability and Transition in Shear Flows. Springer, New York (2001)

    Book  MATH  Google Scholar 

  41. Smith, M.K.: The mechanism for the long-wave instability in thin liquid films. J. Fluid Mech. 217, 469–485 (1990)

    Article  MATH  Google Scholar 

  42. Thiele, U., Goyeau, B., Velarde, M.G.: Stability analysis of thin film flow along a heated porous wall. Phys. Fluids 21(1), 014103 (2009)

    Article  MATH  Google Scholar 

  43. Usha, R., Millet, S., BenHadid, H., Rousset, F.: Shear-thinning film on a porous substrate: stability analysis of a one-sided model. Chem. Eng. Sci. 66(22), 5614–5627 (2011)

    Article  Google Scholar 

  44. Usha, R., Tammisola, O., Govindarajan, R.: Linear stability of miscible two-fluid flow down an incline. Phys. Fluids 25(10), 104102 (2013)

    Article  MATH  Google Scholar 

  45. Vempati, B., Oztekin, A., Neti, S.: Stability of two-layered fluid flows in an inclined channel. Acta Mech. 209(3–4), 187–199 (2010)

    Article  MATH  Google Scholar 

  46. Weinstein, S.: Wave propagation in the flow of shear-thinning fluids down an incline. AIChE J. 36(12), 1873–1889 (1990)

    Article  Google Scholar 

  47. Yih, C.S.: Stability of liquid flow down an inclined plane. Phys. Fluids 6(3), 321–334 (1963)

    Article  MATH  Google Scholar 

  48. Yih, C.S.: Stability of a non-Newtonian liquid film flowing down an inclined plane. Phys. Fluids 8(7), 1257–1262 (1965)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mécanique des Fluides et d’Acoustique UMR CNRS 5509, École Centrale de Lyon, INSA-Lyon, Université Lyon 1, Université de Lyon, 69621, Lyon, France

    S. Millet & V. Botton

  2. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600 036, India

    R. Usha

  3. INSA Euro-Méditerranée, Université Euro-Méditerranéenne de Fès, Route de Meknès, BP51, Fes, Morocco

    V. Botton

  4. UMR5008, CETHIL, INSA-Lyon, CNRS, Université de Lyon, 69621, Villeurbanne, France

    F. Rousset

Authors
  1. S. Millet
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Usha
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. Botton
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. F. Rousset
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Millet.

Additional information

Publisher's Note

Springer Nature remains 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

Millet, S., Usha, R., Botton, V. et al. The mechanism of long-wave instability in a shear-thinning film flow on a porous substrate. Acta Mech 230, 2201–2220 (2019). https://doi.org/10.1007/s00707-019-02376-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-019-02376-0

Search

Navigation