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The stability of finite amplitude cellular convection and its relation to an extremum principle

Published online by Cambridge University Press:  28 March 2006

F. H. Busse
F. H. Busse
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, Los Angeles
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Abstract

The stability of cellular convection flow in a layer heated from below is discussed for Rayleigh number R close to the critical value Rc. It is shown that in this region the stable stationary solution is determined by a minimum of the integral \[ \int_0^{H_0}R(H)\,dH, \] where R(H) is a functional of arbitrary convective velocity fields which satisfy the boundary conditions. For the stationary solutions R(H) is equal to the Rayleigh number. H0 is a given value of the convective heat transport. In a second part of the paper explicit results are derived for the convection problem with deviations from the Boussinesq approximation owing to the temperature dependence of the material properties.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 30 , Issue 4 , 22 December 1967 , pp. 625 - 649
Copyright
© 1967 Cambridge University Press

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References

Busse, F. 1962 Dissertation, University of Munich. English translation by S. H. Davis, Rand Rept. LT-66-19, Rand Corp., Santa Monica, California.
Busse, F. 1967a J. Fluid Mech. 28, 223.
Busse, F. 1967b J. Math. & Phys. 46, 140.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Krishnamurti, R. 1967 Dissertation, University of California, Los Angeles.
Küppers, G. 1966 Diplomarbeit, Institute of Theoretical Physics, University of Munich.
Malkus, W. V. R. & Veronis, G. 1958 J. Fluid Mech. 4, 225.
Malkus, W. V. R. 1959 Astrophys. J. 130, 259.
Palm, E. 1960 J. Fluid Mech. 8, 183.
Palm, E. & Oiann, H. 1964 J. Fluid Mech. 19, 353.
Pellew, A. & Southwell, R. V. 1940 Proc. Roy. Soc. A 176, 312.
Reid, W. H. & Harris, D. L. 1958 Phys. Fluids, 1, 102.
Sani, R. L. 1965 A.I.Ch.E. J. 11, 971.
Schlüter, A., Lortz, D. & Busse, F. 1965 J. Fluid Mech. 23, 129.
Segel, L. A. 1965 J. Fluid Mech. 21, 359.
Segel, L. A. 1966 In Non-Equilibrium Thermodynamics, Variational Techniques and Stability, Ed. by R. J. Donnelly, R. Herman and I. Prigogine. University of Chicago Press.
Silveston, P. S. 1958 Forsch. Ing. Wes. 24, 29, 59.
Von Tippelskirch, H. 1957 Beitr. Phys. Atmosph. 29, 219.
Veronis, G. 1959 J. Fluid Mech. 5, 401.
Veronis, G. 1965 J. Marine Res. 23, 1.