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Nonlinear Rayleigh–Bénard convection between poorly conducting boundaries

Published online by Cambridge University Press:  19 April 2006

C. J. Chapman  and
M. R. E. Proctor
C. J. Chapman
Affiliation:
School of Mathematics, University of Bristol
M. R. E. Proctor
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
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Abstract

The convective instability of a layer of fluid heated from below is studied on the assumption that the flux of heat through the boundaries is unaffected by the motion in the layer. It is shown that when the heat flux is above the critical value for the onset of convection, motion takes place on a horizontal scale much greater than the layer depth. Following Childress & Spiegel (1980) the disparity of scales is exploited in an expansion scheme that results in a nonlinear evolution equation for the leading-order temperature perturbation. This equation which does not depend on the vertical co-ordinate, is solved analytically where possible and numerically where necessary; most attention is concentrated on solutions representing two-dimensional rolls. It is found that for any given heat flux a continuum of steady solutions is possible for all wave numbers smaller than a given cut off. Stability analysis reveals, however, that each mode is unstable to one of longer wavelength than itself, so that any long box will eventually contain a single roll, even though the most rapidly growing mode on linear theory has much shorter wavelength.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 101 , Issue 4 , 29 December 1980 , pp. 759 - 782
Copyright
© 1980 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. Washington: National Bureau of Standards.
Backus, G. E. 1958 A class of self-sustaining dissipative spherical dynamos. Ann. Phys. 4, 372–447.Google Scholar
Busse, F. H. 1978 Non-linear properties of thermal convection. Rep. Prog. Phys. 41, 1929–1937.Google Scholar
Busse, F. H. & Riahi, N. 1980 Nonlinear convection in a layer with nearly insulating boundaries J. Fluid Mech. 96, 243–256.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability, Clarendon.
Chapman, C. J. 1978 Bénard convection with constant heat flux boundaries. Woods Hole Summer Fellowship Lectures, vol. 2, 1–17.Google Scholar
Chapman, C. J. 1980 Bénard convection between insulating boundaries. Ph.D. thesis, Bristol University.
Chapman, C. J., Childress, S. & Proctor, M. R. E. 1980 Long wavelength thermal convection between non-conducting boundaries. Earth & Planet. Sci. Letters (in press).Google Scholar
Childress, S., Levandowsky, M. & Spiegel, E. A. 1975 Pattern formation in a suspension of swimming micro-organisms: equations and stability theory. J. Fluid Mech. 63, 591–613.Google Scholar
Childress, S. & Spiegel, E. A. 1980 Pattern formation in a suspension of swimming microorganisms: nonlinear aspects. (To be published.)
Hurle, D. T. J., Jakeman, E. & Pike, E. R. 1967 On the solution of the Bénard problem with boundaries of finite conductivity. Proc. Roy. Soc. A 296, 469–475.Google Scholar
Jakeman, E. 1968 Convective instability in fluids of high thermal diffusivity. Phys. Fluids 11, 10–14.Google Scholar
Jeffreys, H. 1926 The stability of a layer heated from below. Phil. Mag. 2, 833–844.Google Scholar
Joseph, D. D. 1976 Stability of Fluid Motions (2 vols), Springer.
Malkus, W. V. R. & Veronis, G. 1958 Finite-amplitude cellular convection. J. Fluid Mech. 4, 225–260.Google Scholar
Minorsky, N. 1974 Nonlinear Oscillations, Krieger.
Moore, D. R. & Weiss, N. O. 1973 Two-dimensional Rayleigh—Bénard convection. J. Fluid Mech. 58, 289–312.Google Scholar
Newell, A. C. & Whitehead, J. A. 1969 Finite bandwidth finite-amplitude convection. J. Fluid Mech. 38, 279–303.Google Scholar
Nield, D. A. 1967 The thermohaline Rayleigh—Jeffreys problem. J. Fluid Mech. 29, 545–553.Google Scholar
Nield, D. A. 1975 The onset of transient convective instability. J. Fluid Mech. 71, 441–454.Google Scholar
Rayleigh, Lord 1916 On convective currents in a horizontal layer of fluid when the higher temperature is on the under side. Phil. Mag. 32 (6), 529–546.Google Scholar
Schlüter, A., Lortz, D. & Busse, F. 1965 On the stability of steady finite-amplitude convection. J. Fluid Mech. 23, 129–144.Google Scholar
Serrin, J. 1959 On the stability of viscous fluid motions. Arch. Rat. Mech. Anal. 3, 1–13.Google Scholar
Sparrow, E. M., Goldstein, R. J. & Jonsson, V. H. 1964 Thermal instability in a horizontal fluid laver: effect of boundary conditions and nonlinear temperature profile. J. Fluid Mech. 18, 513–528.Google Scholar