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On a singular initial-value problem for the Navier-Stokes equations. (English) Zbl 1320.35256

Summary: This paper presents a recent result for the problem introduced eleven years ago by L. E. Fraenkel and J. B. McLeod [“A diffusing vortex circle in a viscous fluid”, in: IUTAM Symposium on Asymptotics, Singularities and Homogenisation in Problems of Mechanics, 489–500 (2003; Zbl 1131.74003)], but described only briefly there. We shall prove the following, as far as space allows. The vorticity \(\omega\) of a diffusing vortex circle in a viscous fluid has, for small values of a non-dimensional time, a second approximation \(\omega_{A}+\omega_{1}\) that, although formulated for a fixed, finite Reynolds number \(\lambda\) and exact for \(\lambda = 0\) (then \(\omega = \omega_{A}\)), tends to a smooth limiting function as \(\lambda \uparrow \infty\). In \({\S}{\S}1\) and 2 the necessary background and apparatus are described; \({\S}3\) outlines the new result and its proof.

MSC:

35Q30 Navier-Stokes equations
35B25 Singular perturbations in context of PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
76D17 Viscous vortex flows

Citations:

Zbl 1131.74003
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References:

[1] Watson, Theory of Bessel Functions (1952)
[2] Fraenkel, IUTAM Symposium on Asymptotics, Singularities and Homogenisation in Problems of Mechanics pp 489– (2003)
[3] Saffman, Stud. Appl. Math. 49 pp 371– (1970) · Zbl 0224.76032 · doi:10.1002/sapm1970494371
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