Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-15T18:44:56.054Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite-amplitude stability of a parallel flow with a free surface

Published online by Cambridge University Press:  29 March 2006

S. P. Lin
S. P. Lin
Affiliation:
Mechanical Engineering Department, Clarkson College of Technology, Potsdam, New York
Get access Rights & Permissions [Opens in a new window]

Abstract

The linearized problem of the instability of a layer of liquid flowing down an inclined plane was formulated by Yih (1954) and was solved by Benjamin (1957). It was found that the instability of such a film flow is initially due to long surface waves of infinitesimally small amplitudes. In the present study, a closed-form expression for the non-linear development of these long surface waves is obtained. It is shown that in the neighbourhood of the neutral curve an exponentially growing infinitesimal disturbance may develop into supercritically stable wave motion of small but finite amplitude if the surface tension of the liquid is sufficiently large. Theoretically obtained amplitudes of such waves are consistent with Kapitza's (1949) observation. The approach used in this analysis is a modification of the method used by Reynolds & Potter (1967), who extended the method of Stuart (1960) and Watson (1960) in their study of the non-linear instability of plane Poiseuille and Poiseuille-Couette flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 36 , Issue 1 , 27 March 1969 , pp. 113 - 126
Copyright
© 1969 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.
Benney, D. J. 1961 J. Fluid Mech. 10, 209.
Binnie, A. M. 1957 J. Fluid Mech. 2, 551.
Eckhaus, W. 1965 Studies in Non-Linear Stability Theory. New York: Springer.
Ince, E. L. 1926 Ordinary Differential Equations. New York: Dover.
Kapitza, P. L. 1949 Zh. Eksperim. i Teor. Fiz. 19, 105. (See Collected papers of P. L. Kapitza, vol. II, p. 662. Oxford: Pergamon, 1964.)
Landau, L. D. 1944 C. R. Acad. Sci. U.R.S.S. 44, 311.
Pekeris, C. L. & Shkoller, B. 1967 J. Fluid Mech. 29, 31.
Reynolds, W. C. & Potter, M. C. 1967 J. Fluid Mech. 27, 465.
Schade, H. 1964 Phys. Fluids, 7, 623.
Stuart, J. T. 1960 J. Fluid Mech. 9, 353.
Watson, J. 1960 J. Fluid Mech. 9, 371.
Yih, C. S. 1954 Proc. 2nd U.S. Congr. Appl. Mech. (Amer. Soc. Mech. Engrs.), p. 623.
Yih, C. S. 1963 Phys. Fluids, 6, 321.