Applied Mathematics and Mechanics

, Volume 31, Issue 1, pp 1–12 | Cite as

Spatio-temporal instability of two-layer liquid film at small Reynolds numbers

  • Zhi-liang Wang (王志亮)Email author
  • S. P. Lin (林松飘)
  • Zhe-wei Zhou (周哲玮)


The onset of instability with respect to the spatio-temporally growing disturbance in a viscosity-stratified two-layer liquid film flow is analyzed. The known results obtained from the temporal theory of instability show that the flow is unstable in the limit of zero Reynolds numbers. The present theory predicts the neutral stability in the same limit. The discrepancy is explained. Based on the mechanical energy equation, a new mechanism of instability is found. The new mechanism is associated with the convective nature of the disturbance that is not Galilei invariant.

Key words

flow instability film coating energy budget low Reynolds number Galilei invariant 

Chinese Library Classification


2000 Mathematics Subject Classification



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  1. [1]
    Yih, C. S. Instability due to viscosity stratification. J. Fluid Mech. 27(2), 337–352 (1967)zbMATHCrossRefGoogle Scholar
  2. [2]
    Renardy, Y. Instability at the interface between two shearing fluids in a channel. Phys. Fluids 28(12), 3441–3443 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Renardy, Y. The thin-layer effect and interfacial stability in a two-layer Couette flow with similar liquids. Phys. Fluids 30(6), 1627–1637 (1987)zbMATHCrossRefGoogle Scholar
  4. [4]
    Chen, K. P. Interfacial instabilities in stratified shear flows involving multiple viscous and viscoelastic fluids. Appl. Mech. Rev. 48(11), 763–776 (1995)CrossRefGoogle Scholar
  5. [5]
    Tilley, B. S., Davis, S. H., and Bankoff, S. G. Linear stability theory of two-layer fluid flow in an inclined channel. Phys. Fluids 6(12), 3906–3922 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Tilley, B. S., Davis, S. H., and Bankoff, S. G. Nonlinear long-wave stability of superposed fluids in an inclined channel. J. Fluid Mech. 277, 55–83 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Hooper, A. P. Long-wave instability at the interface between two viscous fluids: thin layer effects. Phys. Fluids 28(6), 1613–1618 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Hooper, A. P. and Boyd, W. G. C. Shear-flow instability at the interface between two viscous fluids. J. Fluid Mech. 128, 507–528 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Hooper, A. P. and Grimshaw, R. Nonlinear instability at the interface between two viscous fluids. Phys. Fluids 28(1), 37–45 (1985)zbMATHCrossRefGoogle Scholar
  10. [10]
    Kao, T. W. Role of viscosity stratification in the stability of two-layer flow down an incline. J. Fluid Mech. 33, 561–572 (1968)zbMATHCrossRefGoogle Scholar
  11. [11]
    Loewenherz, D. S. and Lawrence, C. J. The effect of viscosity stratification on the stability of a free surface flow at low Reynolds number. Phys. Fluids A 1(10), 1686–1693 (1989)zbMATHCrossRefGoogle Scholar
  12. [12]
    Chen, K. P. Wave formation in the gravity-driven low-Reynolds number flow of two liquid films down an inclined plane. Phys. Fluids A 5(12), 3038–3048 (1993)zbMATHCrossRefGoogle Scholar
  13. [13]
    Wang, C. K., Seaborg, J. J., and Lin, S. P. Instability of multi-layered liquid films. Phys. Fluids 21(10), 1669–1673 (1978)zbMATHCrossRefGoogle Scholar
  14. [14]
    Jiang, W. Y., Helenbrook, B., and Lin, S. P. Inertialess instability of a two-layer liquid film flow. Phys. Fluids 16(3), 652–663 (2004)CrossRefMathSciNetGoogle Scholar
  15. [15]
    Weinstein, S. J. and Kurz, M. R. Long-wavelength instabilities in three-layer flow down an incline. Phys. Fluids A 3(11), 2680–2687 (1991)zbMATHCrossRefGoogle Scholar
  16. [16]
    Weinstein, S. J. and Chen, K. P. Large growth rate instabilities in three-layer flow down an incline in the limit of zero Reynolds number. Phys. Fluids 11(11), 3270–3282 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Kliakhandler, I. L. and Sivashinsky, G. I. Viscous damping and instabilities in stratified liquid film flowing down a slightly inclined plane. Phys. Fluids 9(1), 23–30 (1997)CrossRefGoogle Scholar
  18. [18]
    Pozrikidis, C. Gravity-driven creeping flow of two adjacent layers through a channel and down a plane wall. J. Fluid Mech. 371, 345–376 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    Kliakhandler, I. L. Long interfacial waves in multilayer thin films and coupled Kuramoto-Sivashinsky equations. J. Fluid Mech. 391, 45–65 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    Jiang, W. Y., Helenbrook, B., and Lin, S. P. Low Reynolds number instabilities in three-layer flow down an inclined wall. J. Fluid Mech. 539, 387–416 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    Brooke Benjamin, T. The development of three-dimensional disturbances in an unstable film of liquid flowing down an inclined plane. J. Fluid Mech. 10(3), 401–419 (1961)CrossRefMathSciNetGoogle Scholar
  22. [22]
    Yih, C. S. Stability of two-dimensional parallel flows for three-dimensional disturbances. Quart. Appl. Math. 12, 434–435 (1955)zbMATHMathSciNetGoogle Scholar
  23. [23]
    Squire, H. B. On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls. Proc. R. Soc. Lond. A 142(847), 621–628 (1933)CrossRefGoogle Scholar

Copyright information

© Shanghai University and Springer Berlin Heidelberg 2010

Authors and Affiliations

  • Zhi-liang Wang (王志亮)
    • 1
    • 2
    Email author
  • S. P. Lin (林松飘)
    • 2
  • Zhe-wei Zhou (周哲玮)
    • 1
  1. 1.Shanghai Institute of Applied Mathematics and MechanicsShanghai UniversityShanghaiP. R. China
  2. 2.Department of Mechanical and Aeronautical EngineeringClarkson UniversityPotsdamUSA

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