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Skew-varicose instability in two-dimensional generalized Swift-Hohenberg equations

Weliwita, JA, Rucklidge, AM and Tobias, SM (2011) Skew-varicose instability in two-dimensional generalized Swift-Hohenberg equations. Physical Review E, 84 (3). ISSN 1539-3755

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Abstract

We apply analytical and numerical methods to study the linear stability of stripe patterns in two generalizations of the two-dimensional Swift-Hohenberg equation that include coupling to a mean flow. A projection operator is included in our models to allow exact stripe solutions. In the generalized models, stripes become unstable to the skew-varicose, oscillatory skew-varicose, and cross-roll instabilities, in addition to the usual Eckhaus and zigzag instabilities. We analytically derive stability boundaries for the skew-varicose instability in various cases, including several asymptotic limits. We also use numerical techniques to determine eigenvalues and hence stability boundaries of other instabilities. We extend our analysis to both stress-free and no-slip boundary conditions and we note a crossover from the behavior characteristic of no-slip to that of stress-free boundaries as the coupling to the mean flow increases or as the Prandtl number decreases. Close to the critical value of the bifurcation parameter, the skew-varicose instability has the same curvature as the Eckhaus instability provided the coupling to the mean flow is greater than a critical value. The region of stable stripes is completely eliminated by the cross-roll instability for large coupling to the mean flow.

Item Type: Article
Copyright, Publisher and Additional Information: © 2011,American Physical Society. This is an author produced version of a paper published in Physical Review E. Uploaded in accordance with the publisher's self-archiving policy.
Keywords: Rayleigh-Benard convection, spiral-defect chaos, stress-free boundaries, Prandtl number fluid, pattern-formation, rolls, stability, threshold, model
Institution: The University of Leeds
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds)
Depositing User: Symplectic Publications
Date Deposited: 13 Dec 2012 15:06
Last Modified: 08 Jun 2014 21:34
Published Version: http://dx.doi.org/10.1103/PhysRevE.84.036201
Status: Published
Publisher: American Physical Society
Identification Number: 10.1103/PhysRevE.84.036201
URI: http://eprints.whiterose.ac.uk/id/eprint/74791

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