Hostname: page-component-7c8c6479df-7qhmt Total loading time: 0 Render date: 2024-03-29T00:05:15.248Z Has data issue: false hasContentIssue false

The instability of sheared liquid layers

Published online by Cambridge University Press:  20 April 2006

Marc K. Smith
Affiliation:
Department of Engineering Sciences and Applied Mathematics, The Technological Institute, Northwestern University, Evanston, Illinois 60201, U.S.A.
Stephen H. Davis
Affiliation:
Department of Engineering Sciences and Applied Mathematics, The Technological Institute, Northwestern University, Evanston, Illinois 60201, U.S.A.

Abstract

A prescribed shear stress applied to the free surface of a thin liquid layer sets up a steady shear flow. When the shear flow has a linear velocity, profile, Miles, using asymptotic analysis, finds critical values Rc of the Reynolds number above which unstable travelling waves exist. However, Miles omits a term in the normal-stress boundary condition. We correct this omission and solve the appropriate Orr-Sommer-feld system numerically to obtain the critical conditions. For the case of a zero-surface-tension interface, we find that Rc = 34.2, as compared with Miles’ value of Rc = 203. As surface tension increases, Rc asymptotes to the inviscid limit developed by Miles. The critical Reynolds number, critical wavenumber and critical phase speed are presented as functions of a non-dimensional surface tension. We investigate the mechanism of the instability through an examination of the disturbance-energy equation. When the shear flow has a parabolic velocity profile, we find a long-wave instability at small values of the Reynolds number. Numerical methods are used to extend these results to larger values of the wavenumber. Examination is made of the relation between this long-wave instability and profile curvature.

Type
Research Article
Copyright
© 1982 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

Benjamin, T. B. 1957 J. Fluid Mech. 2, 554.
Benjamin, T. B. 1959 J. Fluid Mech. 6, 161.
Cohen, L. S. & Hanratty, T. J. 1965 A.I.Ch.E. J. 11, 138.
Craik, A. D. D. 1966 J. Fluid Mech. 26, 369.
De Bruin, G. J. 1974 J. Engng Math. 8, 259.
Feldman, S. 1957 J. Fluid Mech. 2, 343.
Howard, L. N. 1961 J. Fluid Mech. 10, 509.
Miles, J. W. 1960 J. Fluid Mech. 8, 593.
Miles, J. W. 1962 J. Fluid Mech. 13, 433.
Potter, M. C. 1966 J. Fluid Mech. 24, 609.
Saric, W. S. & Marshall, B. W. 1971 A.I.A.A. J. 9, 1546.
Scott, M. R. & Watts, H. A. 1975 Sandia Labs, Albuquerque, Rep. SAND75-0198.
Scott, M. R. & Watts, H. A. 1977 SIAM J. Numer. Anal. 14, 40.
Sen, A. K. & Davis, S. H. 1982 J. Fluid Mech. 121, 163.
Smith, M. K. & Davis, S. H. 1981 European Mechanics Colloquium 138, University of Karlsruhe, March 1981.
Yih, C. S. 1963 Phys. Fluids 6, 321.