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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. O. V. Besov, V. P. Il'in, S. M. Nikolskii. Integral Representations of Functions and Imbedding Theorems, Vol. I, John Wiley and Sons, New York, 1978. MR 519341 (80f:46030a)
  • 2. F. H. Busse, Transition to turbulence in Rayleigh-Bénard convection, in Hydrodynamic Instabilities and the Transition to Turbulence, 2nd ed., H. L. Swinney and J. P. Gollub, eds., Springer-Verlag, Berlin, 1985, 467-475. MR 796816
  • 3. S. Chandrasekhar, Hyrodynamic and Hydromagnetic Stability, The Clarendon Press, Oxford, UK, 1961. MR 0128226 (23:B1270)
  • 4. S. H. Davis, On the principle of exchange of stabilities, Proc. Roy. Soc. London A 310, 1969, 341-358. MR 0278615 (43:4345)
  • 5. G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Volume I, Springer-Verlag, New York, 1994. MR 1284205 (95i:35216a)
  • 6. G. P. Galdi, B. Straughan, Exchange of stabilities, symmetry, and nonlinear stability, Arch. Rational Mech. Anal. 89, 1985, 211-228. MR 786547 (86j:35012)
  • 7. D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, Heidelberg, 2001. MR 1814364 (2001k:35004)
  • 8. Y. Guo, C. Hallstrom, D. Spirn. Dynamics near unstable, interfacial fluids, Comm. Math. Phys. 270, 2007, 635-689. MR 2276460 (2008b:76080)
  • 9. Y. Guo, W. Strauss, Instability of periodic BGK equilibria, Comm. Pure Appl. Math. 48, 1995, 861-894. MR 1361017 (96j:35252)
  • 10. I. H. Herron, On the principle of exchange of stabilities in Rayleigh-Bénard convection, SIAM J. Appl. Math. 61(4), 2000, 1362-1368. MR 1813684 (2002e:76021)
  • 11. H. Jeffreys, The stability of a layer of fluid heated from below, Phil. Mag. 2, 1926, 833-844.
  • 12. D. D. Joseph, On the stability of the Boussinesq equations, Arch. Rational Mech. Anal. 20, 1965, 59-71. MR 0182243 (31:6466)
  • 13. D. D. Joseph, Nonlinear stability of the Boussinesq equations by the method of energy, Arch. Rational Mech. Anal. 22, 1966, 163-184. MR 0192725 (33:950)
  • 14. A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications 16. Basel, Boston, Berlin: Birkhäuser, 1995. MR 1329547 (96e:47039)
  • 15. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44. Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983. MR 710486 (85g:47061)
  • 16. A. Pellew, R. V. Southwell, On maintained convective motion in a fluid heated from below, Proc. Roy. Soc. London A 176, 1940, 312-343. MR 0003752 (2:266a)
  • 17. B. Straughan, The Energy Method, Stability, and Nonlinear Convection, Springer-Velag, New York, 1992. MR 1140924 (93e:76027)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 35B40, 35B41, 35B45, 35Q35, 35K45

Retrieve articles in all journals with MSC (2000): 35B40, 35B41, 35B45, 35Q35, 35K45

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.

American Mathematical Society