Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-18T04:46:08.957Z Has data issue: false hasContentIssue false
  • English
  • Français

Shear-flow instability at the interface between two viscous fluids

Published online by Cambridge University Press:  20 April 2006

A. P. Hooper  and
W. G. C. Boyd
A. P. Hooper
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW Present address: Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia.
W. G. C. Boyd
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the linear stability of the cocurrent flow of two fluids of different viscosity in an infinite region (the viscous analogue of the classical Kelvin-Helmholtz problem). Attention is confined to the simplest case, Couette flow, and we solve the problem using both numerical and asymptotic techniques. We find that the flow is always unstable (in the absence of surface tension). The instability arises at the interface between the two fluids and occurs for short wavelengths, when viscosity rather than inertia is the dominant physical effect.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 128 , March 1983 , pp. 507 - 528
Copyright
© 1983 Cambridge University Press

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

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Charles, M. E. & Lilleleht, L. U. 1965 An experimental investigation of stability and interfacial waves in co-current flow of two liquids J. Fluid Mech. 22, 217.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Feldman, S. 1957 On the hydrodynamic stability of two viscous fluids in parallel shearing motion J. Fluid Mech. 2, 343.Google Scholar
Hickox, C. E. 1971 Stability of two fluids in a pipe Phys. Fluids 14, 251.Google Scholar
Hooper, A. P. 1981 Two fluid flows: variable channel problems and stability problems. Ph.D. thesis, University of Bristol.
Kao, T. W. & Park, C. 1972 Experimental investigations of the stability of channel flows. Part 2. 2-layered co-current flow in a rectangular region J. Fluid Mech. 52, 401.Google Scholar
Li, C. H. 1969 Instability of 3-layer viscous stratified fluid Phys. Fluids 12, 2473.Google Scholar
Marcus, P. S. & Press, W. H. 1977 On the Green's functions for small disturbances of plane Couette flow J. Fluid Mech. 79, 525.Google Scholar
Olver, F. W. J. 1974 Asymptotics and Special Functions. Academic.
Réaid, W. H. 1979 An exact solution of the Orr-Sommerfeld equation for plane Couette flow Stud. Appl. Math. 61, 83.Google Scholar
Schulten, Z., Anderson, D. G. M. & Gordon, R. G. 1979 An algorithm for the evaluation of the complex Airy function J. Comp. Phys. 31, 60.Google Scholar
Tsahalis, D. T. 1979 The hydrodynamic stability of two viscous incompressible fluids in parallel shearing motion Trans. A.S.M.E. E. J. Appl. Mech. 46, 499.Google Scholar
Yih, C. S. 1963 Stability of liquid flow down an inclined plane Phys. Fluids 6, 321.Google Scholar
Yih, C. S. 1967 Instability due to viscous stratification J. Fluid Mech. 27, 337.Google Scholar