Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-29T12:31:18.306Z Has data issue: false hasContentIssue false
  • English
  • Français

The asymptotic stability of a bounded rotating fluid heated from below: conductive basic state

Published online by Cambridge University Press:  29 March 2006

G. M. Homsy  and
J. L. Hudson
G. M. Homsy
Affiliation:
Department of Chemistry and Chemical Engineering, University of Illinois, Urbana Present address: Department of Chemical Engineering, Stanford University.
J. L. Hudson
Affiliation:
Department of Chemistry and Chemical Engineering, University of Illinois, Urbana
Get access Rights & Permissions [Opens in a new window]

Abstract

The asymptotic stability of a rapidly rotating, horizontally bounded fluid, heated from below, is treated using boundary-layer methods. It is shown that the rotational constraint is so strong as to preclude instabilities, if the interior regions of the fluid are considered to be inviscid. The correct formulation allows this constraint to be broken by introducing horizontal diffusive effects into the interior, while vertical diffusion is confined to Ekman layers on the horizontal surfaces; no vertical layers exist. Moreover, the mechanism of instability is (to the lowest order) associated with energy conversions entirely within the interior region. The present formulation elucidates the role of the Ekman layers in producing high-order corrections to the limiting critical Rayleigh number, and the asymptotic results are extended to include higher-order terms. The effect of rigid side walls on the critical Rayleigh number, and on the azimuthal wavenumber, is considered. Except for very tall cylinders, the critical Rayleigh number is unaffected by the presence of side walls; the results for different azimuthal modes of convection are inconclusive, but indicate that no great error occurs if disturbances are assumed axisymmetric.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 45 , Issue 2 , 30 January 1971 , pp. 353 - 373
Copyright
© 1971 Cambridge University Press

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

Barcilon, V. 1964 Role of the Ekman layers in the stability of the symmetric regime obtained in a rotating fluid annulus J. Atmos. Sci. 21, 291.Google Scholar
Barcilon, V. 1967 On the motion due to sources and sinks distributed along the vertical boundary of a rotating fluid J. Fluid Mech. 27, 551.Google Scholar
Barcilon, V. & Pedlosky, J. 1967a Linear theory of rotating stratified fluid motions J. Fluid Mech. 29, 1.Google Scholar
Barcilon, V. & Pedlosky, J. 1967b A unified linear theory of homogeneous and stratified rotating fluids J. Fluid Mech. 29, 609.Google Scholar
Barcilon, V. & Pedlosky, J. 1967c On the steady motions produced by a stable stratification in a rapidly rotating fluid J. Fluid Mech. 29, 673.Google Scholar
Chandrasekhar, S. 1953 The instability of a layer of fluid heated below and subject to Coriolis forces. Proc. Roy. Soc. A 217, 306.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press.
Chandrasekhar, S. & Elbert, D. D. 1955 The instability of a layer of fluid heated below and subject to Coriolis forces II. Proc. Roy. Soc. A 231, 198.Google Scholar
Davis, S. H. 1967 Convection in a box: linear theory. J. Fluid Mech. 30, 465.Google Scholar
Finlayson, B. A. 1968 The Galerkin method applied to convective instability problems J. Fluid Mech. 33, 201.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Homsy, G. M. 1969 Ph.D. Thesis, University of Illinois, Urbana.
Homsy, G. M. & Hudson, J. L. 1969 Centrifugally driven thermal convection in a rotating cylinder J. Fluid Mech. 35, 33.Google Scholar
Koschmieder, E. L. 1966 On convection on a uniformly heated plane. Beitr. Phys. Atmos. 39, 1.Google Scholar
Koschmieder, E. L. 1967a On convection on a uniformly heated rotating plane Beitr. Phys. Atmos. 40, 216.Google Scholar
Koschmieder, E. L. 1967b On convection under an air surface J. Fluid Mech. 30, 9.Google Scholar
Koschmieder, E. L. 1968 Convection on a non-uniformly heated, rotating plane J. Fluid Mech. 33, 515.Google Scholar
Liang, S. F., Vidal, A. & Acrivos, A. 1969 Buoyancy-driven convection in cylindrical geometries J. Fluid Mech. 36, 239.Google Scholar
Müller, U. 1965 Untersuchungen an rotationssymmetrischen Zellularkonvektionsströmungen. I. Stationäre Strömungsfelder in rotierenden Flüssigkeitsschichten Beitr. Phys. Atmos. 38, 1.Google Scholar
Niiler, P. P. & Bisshopp, F. E. 1965 On the influence of Coriolos force on the onset of convection J. Fluid Mech. 22, 753.Google Scholar
Veronis, G. 1966 Motions at subcritical values of the Rayleigh number in a rotating fluid J. Fluid Mech. 24, 545.Google Scholar
Veronis, G. 1968 Large amplitude Bénard convection in a rotating fluid J. Fluid Mech. 31, 113.Google Scholar