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Boundary effects and the onset of Taylor vortices

Rucklidge, A. M. and Champneys, A. R. (2004) Boundary effects and the onset of Taylor vortices. Physica D, 191. pp. 282-296. ISSN 0167-2789

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Abstract

It is well established that the onset of spatially periodic vortex states in the Taylor–Couette flow between rotating cylinders occurs at the value of Reynolds number predicted by local bifurcation theory. However, the symmetry breaking induced by the top and bottom plates means that the true situation should be a disconnected pitchfork. Indeed, experiments have shown that the fold on the disconnected branch can occur at more than double the Reynolds number of onset. This leads to an apparent contradiction: why should Taylor vortices set in so sharply at the Reynolds number predicted by the symmetric theory, given such large symmetry-breaking effects caused by the boundary conditions? This paper offers a generic explanation. The details are worked out using a Swift–Hohenberg pattern formation model that shares the same qualitative features as the Taylor–Couette flow. Onset occurs via a wall mode whose exponential tail penetrates further into the bulk of the domain as the driving parameter increases. In a large domain of length L, we show that the wall mode creates significant amplitude in the centre at parameter values that are O(L^−2) away from the value of onset in the problem with ideal boundary conditions. We explain this as being due to a Hamiltonian Hopf bifurcation in space, which occurs at the same parameter value as the pitchfork bifurcation of the temporal dynamics. The disconnected anomalous branch remains O(1) away from the onset parameter since it does not arise as a bifurcation from the wall mode.

(c) 2003 Elsevier B.V. All rights reserved.

Item Type: Article
Copyright, Publisher and Additional Information: Copy on deposit is a post-print
Keywords: Pattern formation; Boundary effects; Taylor–Couette flow; Anomalous modes
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds)
Depositing User: A. M. Rucklidge
Date Deposited: 10 Sep 2004
Last Modified: 08 Feb 2013 17:01
Status: Published
Refereed: Yes
Identification Number: 10.1016/j.physd.2003.12.003
URI: http://eprints.whiterose.ac.uk/id/eprint/94

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