Eltayeb, IA, Hughes, DW orcid.org/0000-0002-8004-8631 and Proctor, MRE (2020) The convective instability of a Maxwell-Cattaneo fluid in the presence of a vertical magnetic field. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 476 (2241). ISSN 1364-5021
Abstract
We study the instability of a Bénard layer subject to a vertical uniform magnetic field, in which the fluid obeys the Maxwell–Cattaneo (MC) heat flux–temperature relation. We extend the work of Bissell (Proc. R. Soc. A 472, 20160649 (doi:10.1098/rspa.2016.0649)) to non-zero values of the magnetic Prandtl number pm. With non-zero pm, the order of the dispersion relation is increased, leading to considerably richer behaviour. An asymptotic analysis at large values of the Chandrasekhar number Q confirms that the MC effect becomes important when C Q1/2 is O(1), where C is the MC number. In this regime, we derive a scaled system that is independent of Q. When CQ1/2 is large, the results are consistent with those derived from the governing equations in the limit of Prandtl number p → ∞ with pm finite; here we identify a new mode of instability, which is due neither to inertial nor induction effects. In the large pm regime, we show how a transition can occur between oscillatory modes of different horizontal scale. For Q ≫ 1 and small values of p, we show that the critical Rayleigh number is non-monotonic in p provided that C > 1/6. While the analysis of this paper is performed for stress-free boundaries, it can be shown that other types of mechanical boundary conditions give the same leading-order results.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2020 The Author(s). This is an author produced version of an article accepted for publication in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Rayleigh–Bénard convection; magnetoconvection; hyperbolic heat flow |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 24 Aug 2020 13:23 |
Last Modified: | 05 Nov 2020 17:16 |
Status: | Published |
Publisher: | The Royal Society |
Identification Number: | 10.1098/rspa.2020.0494 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:164741 |