Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-29T02:58:08.795Z Has data issue: false hasContentIssue false
  • English
  • Français

The critical layer in stratified shear flow

Published online by Cambridge University Press:  26 February 2010

P. Baldwin  and
P. H. Roberts
P. Baldwin
Affiliation:
The Department of Engineering Mathematics, The University, Newcastle upon Tyne.
P. H. Roberts
Affiliation:
The School of Mathematics, The University, Newcastle upon Tyne.
Get access

Summary

The study of linear stability of a layer of stratified fluid in horizontal shearing motion leads, in the absence of diffusive effects, to a second order differential equation, often called the Taylor-Goldstein equation. This equation possesses a singularity at any critical point, i.e. at any point at which the flow speed, U, is equal to the wave speed, c. If c is complex, a similar singularity arises at any point at which the analytic extension of U into the complex plane is equal to c. Assuming the stratification is thermal in origin, the introduction of a small viscosity and heat conductivity removes this singularity, but leads to a governing equation of sixth order, four solutions being of rapidly varying WKBJ form. The circumstances in which the remaining two solutions can be uniformly represented in the limit of small viscosity and conductivity by the solutions of the Taylor-Goldstein equation are examined in this paper.

Type
Research Article
Information
Mathematika , Volume 17 , Issue 1 , June 1970 , pp. 102 - 119
Copyright
Copyright © University College London 1970

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

Booker, J. R. and Bretherton, F. P., 1967, J. Fluid Mech., 27, 513.CrossRefGoogle Scholar
Gage, K. S. and Reid, W. H., 1968, J. Fluid Mech., 33, 21.CrossRefGoogle Scholar
Hazel, P., 1967, J. Fluid Mech., 14, 257.Google Scholar
Koppel, D., 1964, J. Math. Phys., 5, 963.CrossRefGoogle Scholar
Slater, L. J., 1960, Confluent hypergeometric functions, (Cambridge University Press).Google Scholar
Whittaker, E. T. and Watson, G. N., 1927, A course of modern analysis, 4th Ed. (Cambridge University Press).Google Scholar