Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-26T21:57:28.435Z Has data issue: false hasContentIssue false

ON A SINGULAR INITIAL-VALUE PROBLEM FOR THE NAVIER–STOKES EQUATIONS

Published online by Cambridge University Press:  07 January 2015

L. E. Fraenkel
Affiliation:
Department of Mathematics, University of Bath, Bath BA2 7AY, U.K. email [email protected]
M. D. Preston
Affiliation:
National Institute for Biological Standards and Control (NIBSC), Blanche Lane, South Mimms, Potters Bar, Hertfordshire EN6 3QG, U.K. email [email protected]
Get access

Abstract

This paper presents a recent result for the problem introduced eleven years ago by Fraenkel and McLeod [A diffusing vortex circle in a viscous fluid. In IUTAM Symposium on Asymptotics, Singularities and Homogenisation in Problems of Mechanics, Kluwer (2003), 489–500], but described only briefly there. We shall prove the following, as far as space allows. The vorticity ${\it\omega}$ of a diffusing vortex circle in a viscous fluid has, for small values of a non-dimensional time, a second approximation ${\it\omega}_{A}+{\it\omega}_{1}$ that, although formulated for a fixed, finite Reynolds number ${\it\lambda}$ and exact for ${\it\lambda}=0$ (then ${\it\omega}={\it\omega}_{A}$), tends to a smooth limiting function as ${\it\lambda}\uparrow \infty$. In §§1 and 2 the necessary background and apparatus are described; §3 outlines the new result and its proof.

Type
Research Article
Copyright
Copyright © University College London 2015 

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

Fraenkel, L. E. and McLeod, J. B., A diffusing vortex circle in a viscous fluid. In IUTAM Symposium on Asymptotics, Singularities and Homogenisation in Problems of Mechanics (ed. Movchan, A. B.), Kluwer (2003), 489500.Google Scholar
Saffman, P. G., The velocity of viscous vortex rings. Stud. Appl. Math. 49 1970, 371380.CrossRefGoogle Scholar
Watson, G. N., Theory of Bessel Functions, Cambridge University Press (1952).Google Scholar