DOI: 10.1017/s0022112067001028. The thermohaline Rayleigh-Jeffreys problem

The thermohaline Rayleigh-Jeffreys problem

10.1017/s0022112067001028

Crossref journal-article

Cambridge University Press (CUP)

Journal of Fluid Mechanics (56)

Abstract

The onset of convection induced by thermal and solute concentration gradients, in a horizontal layer of a viscous fluid, is studied by means of linear stability analysis. A Fourier series method is used to obtain the eigenvalue equation, which involves a thermal Rayleigh numberRand an analogous solute Rayleigh numberS, for a general set of boundary conditions. Numerical solutions are obtained for selected cases. Both oscillatory and monotonic instability are considered, but only the latter is treated in detail. The former can occur when a strongly stabilizing solvent gradient is opposed by a destablizing thermal gradient. When the same boundary equations are required to be satisfied by the temperature and concentration perturbations, the monotonic stability boundary curve in the (R, S)-plane is a straight line. Otherwise this curve is concave towards the origin. For certain combinations of boundary conditions the critical value ofRdoes not depend onS(for some range ofS) or vice versa. This situation pertains when the critical horizontal wave-number is zero.A general discussion of the possibility and significance of convection at ‘zero’ wave-number (single convection cell) is presented in an appendix.

Bibliography

Nield, D. A. (1967). The thermohaline Rayleigh-Jeffreys problem. Journal of Fluid Mechanics, 29(3), 545â558.

Authors 1

D. A. Nield (first)

References 30 Referenced 191

Tabor, H. & Matz, R. 1965 Solar Energy 9,177–82.
Veronis, G. 1965 J. Marine Res. 23,1–17.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability .Oxford:Clarendon Press.
Weiss, N. O. 1964 Phil. Trans. Roy. Soc A 256,99–147.
Stern, M. E. 1960 Tellus 12,172–5.
Nield, D. A. 1964 J. Fluid Mech. 19,341–52.
Shirtcliffe, T. G. L. 1964 J. Geophys. Res. 69,5257–68.
Sani, R. 1965 Amer. Inst. Chem. Engrs J. 11,971–80.
Gershuni, G. Z. & Zhukhovitskii, E. M. 1963 Prikl. Mat. Mekh. 27,784–93. English translation in J. Appl. Math. Mech. 27, 1197–204.
Hoare, R. A. 1966 Nature, Lond. 210,787–9.
Weinberger, H. 1964 Solar Energy 8,45–56.
Turner, J. S. & Stommel, H. 1964 Proc. Natn. Acad. Sci. 52,49–53.
Pearson, J. R. A. 1958 J. Fluid Mech. 4,489–500.
Nield, D. A. 1966b Extended Theory of the Bénard Convection Problem. Ph.D. Thesis,Univ. of Auckland.
Benard, H. 1900 C.R. Acad. Sci., Paris 130,1004–7 and1065–8.
Nield, D. A. 1965 Proc. 2nd Australasian Conf. Hydraulics and Fluid Mech., University of Auckland , C45–56.
Vertgeim, B. A. 1955 Prikl. Mat. Mekh. 19,745–50.
Sparrow, E. M. , Goldstein, R. J. & Jonsson, V. K. 1964 J. Fluid Mech. 18,513–28.
Bénard, H. 1901 Ann. Chim. Phys. 23,62–144.
Jeffreys, H. 1926 Phil. Mag. 2,833–44.
Hurle, D. T. J. , Jakeman, E. & Pike, E. R. 1967 Proc. Roy. Soc A296,469–75.
Reid, W. H. & Harris, D. L. 1958 Phys. Fluids 1,102–10.
Walin, G. 1964 Tellus 16,389–393.
Jeffreys, H. 1928 Proc. Roy. Soc A118,195–208.
Nield, D. A. 1966a Z.A.M.P. 17,226–32.
Lieber, P. & Rintel, L. 1963 Convective instability of a horizontal layer of fluid with maintained concentration of diffusive substance and temperature on the boundaries. Rept. MD-6, Inst. Engrg Res., Univ. of Calif., Berkeley .
Yih, C.-S. 1965 Dynamics of Nonhomogeneous Fluids .New York:Macmillan.
Rayleigh, Lord 1916 Phil. Mag. 32,529–46. (10.1080/14786441608635602)
Weinberger, H. 1962 Stability Criteria for Liquid Systems with Temperature and Concentration Gradients . Publ. Nat. Phys. Lab. Israel.
Shirtcliffe, T. G. L. 1967 Nature, Lond. 213,489–90.

Dates

Type	When
Created	19 years, 4 months ago (March 28, 2006, 8:22 a.m.)
Deposited	6 years, 4 months ago (April 17, 2019, 1:16 p.m.)
Indexed	3 weeks ago (Aug. 6, 2025, 9:13 a.m.)
Issued	57 years, 11 months ago (Sept. 6, 1967)
Published	57 years, 11 months ago (Sept. 6, 1967)
Published Online	19 years, 4 months ago (March 28, 2006)
Published Print	57 years, 11 months ago (Sept. 6, 1967)

Funders 0

None

BibTeX

@article{Nield_1967, title={The thermohaline Rayleigh-Jeffreys problem}, volume={29}, ISSN={1469-7645}, url={http://dx.doi.org/10.1017/s0022112067001028}, DOI={10.1017/s0022112067001028}, number={3}, journal={Journal of Fluid Mechanics}, publisher={Cambridge University Press (CUP)}, author={Nield, D. A.}, year={1967}, month=sep, pages={545–558} }

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