Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Critical Rayleigh number in Rayleigh-Bénard convection

Authors: Yan Guo and Yongqian Han
Journal: Quart. Appl. Math. 68 (2010), 149-160
MSC (2000): Primary 35B40, 35B41, 35B45, 35Q35, 35K45
DOI: https://doi.org/10.1090/S0033-569X-09-01179-4
Published electronically: October 28, 2009
MathSciNet review: 2598887
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Abstract: The Rayleigh-Bénard convection is a classical problem in fluid dynamics. In the presence of rigid boundary condition, we identify the critical Rayleigh number $ R_{a}^{\ast }$ by a reduced variational problem. We prove nonlinear asymptotic stability for motionless steady states for $ R_{a}<R_{a}^{\ast },$ and their nonlinear instability for $ R_{a}>R_{a}^{\ast }.$ The dynamic of such instability is determined by the leading growing mode(s) for the corresponding linearized system within the time interval of instability.

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Additional Information

Yan Guo
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
Email: guoy@dam.brown.edu

Yongqian Han
Affiliation: Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, People’s Republic of China

DOI: https://doi.org/10.1090/S0033-569X-09-01179-4
Keywords: Nonlinear instability, Rayleigh-B\'enard convection, Boussinesq approximation.
Received by editor(s): December 31, 2008
Published electronically: October 28, 2009
Dedicated: Dedicated to Prof. W. A. Strauss on the occasion of his 70th birthday
Article copyright: © Copyright 2009 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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