Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T00:53:48.243Z Has data issue: false hasContentIssue false

Finite amplitude convection in a self-gravitating fluid sphere containing heat sources

Published online by Cambridge University Press:  24 October 2008

P. Baldwin
Affiliation:
Rutherford College of Technology, Newcastle upon Tyne

Abstract

The non-linear equations governing the convective flow of heat, beyond marginal stability, in a fluid sphere containing a uniform distribution of heat sources, and cooled at its surface, are expressed in the Boussinesq approximation as a perturbation of the steady-state conduction solution. A steady-state axisymmetric poloidal flow is assumed, and the equations for an approximate solution, involving the first spherical harmonic only, are obtained using a special case of the general evolution criterion of Glansdorff and Prigogine. The solution is found numerically using an iterative procedure, the results showing good agreement with those obtained by Stuart's ‘shape assumption’ for values of the Rayleigh number just above the critical value for the onset of marginal stability.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1967

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Chandrasekhar, S.Hydrodynamic and hydromagnetic stability (Oxford University Press, 1961).Google Scholar
(2)Heiskanen, W. A. and Vening-Meistesz, F. A.The earth and its gravity field (McGraw Hill, 1958).Google Scholar
(3)Jeffreys, H. and Bland, M. E. M.The instability of a fluid sphere heated within. Monthly Not. Roy. Astr. Soc. Geophys. Suppl. 3 (1951), 148158.Google Scholar
(4)Chandrasekhar, S.The thermal instability of a fluid sphere heated within. Phil. Mag. (7), 43 (1952), 13171329.CrossRefGoogle Scholar
(5)Backus, G. E.On the application of eigenfunction expansions to the problem of thermal instability of a fluid sphere heated within. Phil. Mag. (7), 46 (1955), 13101327.CrossRefGoogle Scholar
(6)Jeffreys, H.Problems of thermal instability in a sphere. Monthly Not. Roy. Astr. Soc. Geophys. Suppl. 6 (1952), 272277.Google Scholar
(7)Chandrasekhar, S.The onset of convection by thermal instability in spherical shells. Phil. Mag. (7), 44 (1953), 233241.CrossRefGoogle Scholar
(8)Roberts, P. H.Convection in a self-gravitating fluid sphere. Mathematika 12 (1965), 128137.CrossRefGoogle Scholar
(9)Roberts, P. H. On non-linear Bénard convection. Non-equilibrium thermo-dynamics, variational techniques and stability, edited by Donnelly, R. J., Herman, R. and Prigogine, I., pp. 125157 (The University of Chicago Press, 1966).Google Scholar
(10)Glansdorff, P. and Prigogine, I.On a general evolution criterion in macroscopic physics. Physica (30), 2 (1964), 351374.Google Scholar
(11)Lamb, H.On the oscillations of a viscous spheroid. Proc. London Math. Soc. 13 (1881), 5166.CrossRefGoogle Scholar
(12)Backus, G. E.A class of self-sustaining dissipative spherical dynamos. Ann. Physics 4 (1958), 372447.Google Scholar
(13)Stuart, J. T.On the non-linear mechanics of hydrodynamic stability. J. Fluid Mech. 4 (1958), 121.Google Scholar
(14)Roberts, P. H. On the relationship between Galerkin's method and the local potential method. Non-equilibrium thermo-dynamics, variational techniques and stability, edited by Donnelly, R. J., Herman, R. and Prigogine, I., pp. 299302 (The University of Chicago Press, 1966).Google Scholar
(15)Glansdorff, P.Mechanical flow processes and variational methods based on fluctuation theory, pp. 4554.Google Scholar
(16)Filon, L. N. G.On a quadratic formula for trigonometric integrals. Proc. Roy. Soc. Edinburgh 49 (1928), 3847.Google Scholar
(17)Wright, K.Chebyshev collocation methods for ordinary differential equations. Comput. J. (6), 4 (1964), 358365.CrossRefGoogle Scholar
(18)Stewartson, K. Appendix to a paper by Roberts (9). Non-equilibrium thermo-dynamics, variational techniques and stability, edited by Donnelly, R. J., Herman, R. and Prigogine, I., pp. 158162 (The University of Chicago Press, 1966).Google Scholar