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

  • L. E. Fraenkel (a1) and M. D. Preston (a2)

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.

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1.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.
2.Saffman, P. G., The velocity of viscous vortex rings. Stud. Appl. Math. 49 1970, 371380.
3.Watson, G. N., Theory of Bessel Functions, Cambridge University Press (1952).
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