On the Stationary Convection of Thermohaline Problems of Veronis and Stern Types

Joginder S. Dhiman, Praveen K. Sharma, Poonam Sharma

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DOI: 10.4236/am.2010.15052                

Abstract

The stability of thermohaline convection problems of Veronis and Stern types for stationary convection is studied for quite general nature of boundaries. It is shown by means of an appropriately chosen linear transformation, that in case of stationary convection the equations describing the eigenvalue problem for thermohaline convection problems are identical to equations describing the eigenvalue problem for classical Bénard convection problem. As a consequence, the values of the critical Rayleigh numbers for the onset of stationary convection in thermohaline convection problems are obtained. Also, necessary conditions for the validity of principle of exchange of stabilities for thermohaline convection problems are derived using variational principle.

Keywords

Thermohaline Convection, Stationary Convection, Eigenvalue Problem, Principle of Exchange of Stabilities, Rayleigh Number

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	H. Bénard, “Les Tourbillons Cellulaires dans une Napple Liquide,” Revue Generale des Sciences Pures at Appliqués, Vol. 11, 1900, pp. 1261-1271.
[2]	L. Rayleigh, “On the Convective Current in a Horizontal Layer of Fluid When the Higher Temperature is on the Upper Side,” Philosophical Magazine, Vol. 32, No. 3, pp. 529-543.
[3]	J. S. Turner, “Multicomponent Convection,” Annual Review of Fluid Mechanics, Vol. 17, No. 1, 1985, pp. 11-44.
[4]	A. Brandt and H. J. S. Fernando, “Double Diffusive Convection,” American Geophysical Union, Washington, DC, 1996.
[5]	M. E. Stern, “The Salt Fountain and Thermohaline Convection,” Tellus, Vol. 12, No. 2, 1960, pp. 172-175.
[6]	G. Veronis, “On Finite Amplitude Instability in Thermohaline Convection,” Journal of Marine Research, Vol. 23, 1965, pp. 1-17.
[7]	J. R. Gupta, J. S. Dhiman and J. Thakur, “Thermohaline Convection of Veronis and Stern Types Revisited,” Journal of Mathematical Analysis and Applications, Vol. 264, No. 2, 2001, pp. 398-407.
[8]	S. Chandrasekhar, “Hydrodynamic and Hydromagnetic Stability,” Oxford University Press, Amen House, London, 1961.
[9]	E. Knobloch, “Convection in Binary Fluids,” Physics of Fluids, Vol. 23, No. 9, 1980, pp. 1918-1920.