ON A SINGULAR INITIAL-VALUE PROBLEM FOR THE NAVIER–STOKES EQUATIONS. Issue 2 (7th January 2015)

• Record Type: Journal Article
• Authors: Fraenkel, L. E.; Preston, M. D.
• Is Part Of: Mathematika. Volume 61:Issue 2(2015)
• Page Start: 277
• Page End: 294
• Publication Date: 2015-01-07
• Subjects: 76D05, -- 76D17 (primary); Mathematics -- Periodicals; 510.5
• DOI: 10.1112/S0025579314000333
• Languages: English
• ISSNs: 0025-5793

Abstract

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.