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ON A SINGULAR INITIAL‐VALUE PROBLEM FOR THE NAVIER–STOKES EQUATIONS
Author(s) -
Fraenkel L. E.,
Preston M. D.
Publication year - 2015
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579314000333
Subject(s) - mathematics , gravitational singularity , vortex , vorticity , reynolds number , limiting , space (punctuation) , mathematical analysis , stokes flow , geometry , flow (mathematics) , mechanics , physics , linguistics , philosophy , turbulence , engineering , mechanical engineering
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 approximationω A + ω 1that, 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.