Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-19T12:59:13.783Z Has data issue: false hasContentIssue false
  • English
  • Français

On the nonlinear stability of slowly varying time-dependent viscous flows

Published online by Cambridge University Press:  20 April 2006

P. Hall
P. Hall
Affiliation:
Department of Mathematics, Imperial College, London SW7 2BZ
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we investigate the manner in which finite-amplitude disturbances are set up in viscous fluid flows that are changing slowly in time. It is shown that, when the appropriate Reynolds or Rayleigh number is slowly increased, then, no matter how slowly this change takes place, there is always a short time interval where a quasi-steady approach breaks down. In this time interval a finite-amplitude solution is set up which ultimately approaches that predicted by a quasi-steady theory. In order to demonstrate our ideas we discuss the Taylor-vortex problem in a situation in which the speed of the inner cylinder changes slowly in time. In particular we discuss the case when the speed of the inner cylinder is modulated slowly in time and it is found that at low frequencies the disturbances of most physical relevance are not periodic solutions of the equations of motion.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 126 , January 1983 , pp. 357 - 368
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

Davey, A., Diprima, R. C. & Stuart, J. T. 1968 J. Fluid Mech. 31, 17.
Davis, S. D. & Rosenblat, S. 1977 Stud. Appl. Maths 57, 59.
Diprima, R. C. & Stuart, J. T. 1973 J. Fluid Mech. 54, 393.
Diprima, R. C. & Stuart, J. T. 1975 J. Fluid Mech. 67, 85.
Donnelly, R. J. 1964 Proc. R. Soc. Lond. A281, 130.
Donnelly, R. J. & Park, K. 1982 Unpublished manuscript.
Hall, P. 1975 J. Fluid Mech. 67, 29.
Hall, P. 1982a J. Fluid Mech. 124, 475.
Hall, P. 1982b J. Inst. Maths Applics (to appear).
Hall, P. & Smith, F. T. 1982 Stud. Appl. Maths (to appear).
Hall, P. & Walton, I. C. 1979 J. Fluid Mech. 90, 377.
Riley, P. J. & Lawrence, R. L. 1976 J. Fluid Mech. 75, 625.
Rosenblat, S. & Herbert, D. M. 1970 J. Fluid Mech. 43, 385.
Stuart, J. T. 1971 Ann. Rev. Fluid Mech. 3, 347.