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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]
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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 

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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
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Watson, G. N., Theory of Bessel Functions, Cambridge University Press (1952).Google Scholar