Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-27T13:33:38.569Z Has data issue: false hasContentIssue false
  • English
  • Français

On the non-linear Lamb-Taylor instability

Published online by Cambridge University Press:  29 March 2006

Ali Hasan Nayfeh
Ali Hasan Nayfeh
Affiliation:
Aerotherm Corporation, Mountain View, California
Get access Rights & Permissions [Opens in a new window]

Abstract

A non-linear analysis of the inviscid stability of the common surface of two superposed fluids is presented. One of the fluids is a liquid layer with finite thickness having one surface adjacent to a solid boundary whereas the second surface is in contact with a semi-infinite gas of negligible density. The system is accelerated by a force normal to the interface and directed from the liquid to the gas. A second-order expansion is obtained using the method of multiple time scales. It is found that standing as well as travelling disturbances with wave-numbers greater than

$K^{\prime}_c = k_c[1+\frac{3}{8}a^2k^2_c + \frac{51}{512}a^4k^4_c]^{\frac{1}{2}}$

where a is the disturbance amplitude and kc is the linear cut-off wave-number, oscillate and are stable. However, the frequency in the case of standing waves and the wave velocity in the case of travelling waves are amplitude dependent. Below this cut-off wave-number disturbances grow in amplitude. The cut-off wave-number is independent of the layer thickness although decreasing the layer thickness decreases the growth rate. Although standing waves can be obtained by the superposition of travelling waves in the linear case, this is not true in the non-linear case because the amplitude dependences of the wave speed and frequency are different. A mechanism is proposed to explain the overstability behaviour observed by Emmons, Chang & Watson (1960).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 38 , Issue 3 , 18 September 1969 , pp. 619 - 631
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

Allred, J. C., Blount, G. H. & Miller, J. H. 1953 Experimental studies of Taylor instability. Report of Los Alamos Scientific Laboratory of the University of California, no. LA-1600.Google Scholar
Bellman, R. & Pennington, R. H. 1954 Effects of surface tension and viscosity on Taylor instability Quart. J. Appl. Math. 12, 151–162.Google Scholar
Emmons, H. W., Chang, C. T. & Watson, B. C. 1960 Taylor instability of finite surface waves J. Fluid Mech. 7, 177–193.Google Scholar
Ingraham, R. L. 1954 Taylor instability of the interface between superposed fluids. Proc. Phys. Soc. Lond. B 67, 748–752.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th ed. New York: Dover.
Lewis, D. J. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. II. Proc. Roy. Soc. Lond A 202, 81–96.Google Scholar
Lighthill, M. J. 1949 Phil. Mag. 40, 1179–1201.
Nayfeh, A. H. 1965 A perturbation method for treating nonlinear oscillation problems J. Math. & Phys. 44, 368–374.Google Scholar
Nayfeh, A. H. 1968 Forced oscillations of the van der Pol oscillator with delayed amplitude limiting IEEE Transactions on Circuit Theory, 15, 192–200.Google Scholar
Peirce, B. O. & Foster, R. M. 1956 A Short Table of Integrals. New York: Ginn.
Rajappa, N. R. 1967 A non-linear theory of Taylor instability of superposed fluids. Ph.D. thesis, Stanford University.
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. Roy. Soc. Lond A 201, 192–196.Google Scholar
Whittaker, E. T. & Watson, G. N. 1962 A Course of Modern Analysis. Cambridge University Press.