Hostname: page-component-77c89778f8-n9wrp Total loading time: 0 Render date: 2024-07-20T08:26:04.517Z Has data issue: false hasContentIssue false
  • English
  • Français

Mechanism of the long-wave inertialess instability of a two-layer film flow

Published online by Cambridge University Press:  11 July 2008

PENG GAO  and
XI-YUN LU
PENG GAO
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, China
XI-YUN LU*
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, China
*
†Author to whom correspondence should be addressed: xlu@ustc.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper provides an intuitive interpretation of the long-wave inertialess instability of a two-layer film flow. The underlying mechanism is elucidated by inspecting the longitudinal perturbation velocity associated with the surface and interfacial deflections. The velocity is expressed by the composition of three parts, related to the shear stress at the free surface, the continuity condition at the interface, and the pressure disturbance induced by gravity. The effect of each velocity component on the evolutions of the surface and the interface is examined in detail. Specifically, the growth of the free surface is caused by the continuity-induced first-order velocity disturbance associated with an additional phase shift between the surface and interfacial waves, while the growth of the interface is due to the pressure-driven flow. The proposed mechanism gives an alternatively reliable prediction of the wave velocity and growth rate.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 608 , 10 August 2008 , pp. 379 - 391
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Benjamin, T. B. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574.CrossRefGoogle Scholar
Charru, F. & Hinch, E. J. 2000 ‘Phase diagram’ of interfacial instabilities in a two-layer Couette flow and mechanism of the long-wave instability. J. Fluid Mech. 414, 195–223.CrossRefGoogle Scholar
Chen, K. P. 1993 Wave formation in the gravity-driven low-Reynolds number flow of two fluid films down an inclined plane. Phys. Fluids A 5, 3038–3048.CrossRefGoogle Scholar
Gao, P. & Lu, X.-Y. 2006 Effect of surfactants on the long-wave stability of oscillatory film flow. J. Fluid Mech. 562, 345–354.CrossRefGoogle Scholar
Gao, P. & Lu, X.-Y. 2007 Effect of surfactants on the inertialess instability of a two-layer film flow. J. Fluid Mech. 591, 495–507.CrossRefGoogle Scholar
Gao, P. & Lu, X.-Y. 2008 Instability of an oscillatory fluid layer with insoluble surfactants. J. Fluid Mech. 595, 461–490.CrossRefGoogle Scholar
Hu, J., Millet, S., Botton, V., Hadid, H. B. & Henry, D. 2006 Inertialess temporal and spatio-temporal stability analysis of the two-layer film flow with density stratification. Phys. Fluids 18, 104101.CrossRefGoogle Scholar
Huang, C. T. & Khomami, B. 2001 The instability mechanism of single and multilayer Newtonian and viscoelastic flows down an inclined plane. Rheol. Acta 40, 467–484.CrossRefGoogle Scholar
Jiang, W. Y., Helenbrook, B. & Lin, S. P. 2004 Inertialess instability of a two-layer liquid film flow. Phys. Fluids 16, 652–663.CrossRefGoogle Scholar
Jiang, W. Y., Helenbrook, B. T., Lin, S. P. & Weinstein, S. J. 2005 Low-Reynolds-number instabilities in three-layer flow down an inclined wall. J. Fluid Mech. 539, 387–416.CrossRefGoogle Scholar
Kao, T. W. 1965 a Stability of two-layer viscous stratified flow down an inclined plane. Phys. Fluids 8, 812–820.CrossRefGoogle Scholar
Kao, T. W. 1965 b Role of the interface in the stability of stratified flow down an inclined plane. Phys. Fluids 8, 2190–2194.CrossRefGoogle Scholar
Kao, T. W. 1968 Role of viscosity stratification in the stability of two-layer flow down an incline. J. Fluid Mech. 33, 561–572.CrossRefGoogle Scholar
Kelly, R. E., Goussis, D. A., Lin, S. P. & Hsu, F. K. 1989 The mechanism for surface wave instability in film flow down an inclined plane. Phys. Fluids A 1, 819–828.CrossRefGoogle Scholar
Kliakhandler, I. L. 1999 Long interfacial waves in multilayer thin films and coupled Kuramoto–Sivashinsky equations. J. Fluid Mech. 391, 45–65.CrossRefGoogle Scholar
Kliakhandler, I. L. & Sivashinsky, G. I. 1997 Viscous damping and instabilities in stratified liquid film flowing down a slightly inclined plane. Phys. Fluids 9, 23–30.CrossRefGoogle Scholar
Loewenherz, D. S. & Lawrence, C. J. 1989 The effect of viscosity stratification on the stability of a free surface flow at low Reynolds number. Phys. Fluids A 1, 1686–1693.CrossRefGoogle Scholar
Renardy, Y. 1985 Instability at the interface between two shearing fluids in a channel. Phys. Fluids 28, 3441–3443.CrossRefGoogle Scholar
Smith, M. K. 1989 The axisymmetric long-wave instability of a concentric two-phase pipe flow. Phys. Fluids A 1, 494–506.CrossRefGoogle Scholar
Smith, M. K. 1990 The mechanism for the long-wave instability in thin liquid films. J. Fluid Mech. 217, 469–485.CrossRefGoogle Scholar
Wang, C. K., Seaborg, J. J. & Lin, S. P. 1978 Instability of multi-layered liquid films. Phys. Fluids 21, 1669–1673.CrossRefGoogle Scholar
Wei, H.-H. 2005 On the flow-induced Marangoni instability due to the presence of surfactant. J. Fluid Mech. 544, 173–200.CrossRefGoogle Scholar
Weinstein, S. J. & Chen, K. P. 1999 Large growth rate instabilities in three-layer flow down an incline in the limit of zero Reynolds number. Phys. Fluids 11, 3270–3282.CrossRefGoogle Scholar
Weinstein, S. J. & Kurz, M. R. 1991 Long-wavelength instabilities in three-layer flow down an incline. Phys. Fluids A 3, 2680–2687.CrossRefGoogle Scholar
Weinstein, S. J. & Ruschak, K. J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 29–53.CrossRefGoogle Scholar
Yih, C. S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334.CrossRefGoogle Scholar
Yih, C. S. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27, 337–352.CrossRefGoogle Scholar
Yih, C. S. 1968 Instability of unsteady flows or configurations. Part 1. Instability of a horizontal liquid layer on an oscillating plane. J. Fluid Mech. 31, 737–751.CrossRefGoogle Scholar