Skip Navigation

The Quarterly Journal of Mechanics and Applied Mathematics 2000 53(1):27-42; doi:10.1093/qjmam/53.1.27
© 2000 by Oxford University Press
This Article
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Similar articles in ISI Web of Science
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrow Search for citing articles in:
ISI Web of Science (1)
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Ng, B.
Right arrow Articles by Reid, W.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

Asymptotic analysis of a fourth-order turning-point problem in hydrodynamic stability

BS Ng and WH Reid

( Department of Mathematical Sciences, Indiana University-Purdue University, Indianapolis, IN 46202-3216, USA )

An asymptotic analysis is made of the so-called Perkeris modes of the Orr-Sommerfeld problem for plane Poiseuille flow. These are damped modes of the 'centre' or 'fast' type for which cr {uparrow} 0 as {alpha}R -> {infty}. The numerical results obtained by Orszag (J. Fluid Mech. 50 (1971) 689) for {alpha} = 1 and R = 10 000 showed that the eigenvalues for the even and odd modes of this type are very nearly equal, and one of the goals of this paper is to provide an analytical explanation for this rather striking result. Under certain simplifying assumptions we are led to a fourth-order equation which can be viewed as a generalization of Weber's equation for the parabolic cylinder functions. The eigenvalue problem is then posed on an infinite interval and we find that the eigenvalue relations for the even and odd modes are indeed the same even though the underlying analysis is significantly different in the two cases. Explicit results are also given for both the even and odd eigenfunctions. The even eigenfunctions are similar to the Whittaker functions Dn(x) but the odd eigenfunctions involve Dawson's integral and certain polynomials.


Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?




Disclaimer:
Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.