ON A SINGULAR INITIAL-VALUE PROBLEM FOR THE NAVIER–STOKES EQUATIONS. Issue 2 (7th January 2015) - Fraenkel, L. E.; Preston, M. D.
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 ω of a diffusing vortex circle in a viscous fluid has, for small values of a non-dimensional time, a second approximation ω_A+ω_1 that, although formulated for a fixed, finite Reynolds number λ and exact for λ=0 (then ω=ω_A ), tends to a smooth limiting function as λ∞ . In §§1 and 2 the necessary background and apparatus are described; §3 outlines the new result and its proof.