Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-06-29T15:53:42.953Z Has data issue: false hasContentIssue false
  • English
  • Français

Thermal convection with strongly temperature-dependent viscosity

Published online by Cambridge University Press:  11 April 2006

John R. Booker
John R. Booker
Affiliation:
Geophysics Program, University of Washington, Seattle
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper experimentally investigates the heat transport and structure of convection in a high Prandtl number fluid layer whose viscosity varies by up to a factor of 300 between the boundary temperatures. An appropriate definition of the Rayleigh number R uses the viscosity at the average of the top and bottom boundary temperatures. With rigid boundaries and heating from below, the Nusselt number N normalized with the Nusselt number N0 of a constant-viscosity fluid decreases slightly as the viscosity ratio increases. The drop is 12% at a variation of 300. A slight dependence of N/N0 on R is consistent with a decrease in the exponent in the relation N ∝ Rβ from its constant-viscosity value of 0·281 to 0·25 for R [lsim ] 5 × 104. This may be correlated with a transition from three- to two-dimensional flow. At R ∼ 105 and viscosity variation of 150, the cell structure is still dominated by the horizontal wavelength of the marginally stable state. This is true with both free and rigid upper boundaries. The flow is strongly three-dimensional with a free upper boundary, while it is nearly two-dimensional with a rigid upper boundary.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 76 , Issue 4 , 25 August 1976 , pp. 741 - 754
Copyright
© 1976 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

Booker, J. R. 1972 Large amplitude convection with strongly temperature-dependent viscosity. E.O.S. Trans. Am. Geophys. Un. 53, 520.Google Scholar
Booker, J. R. & Nir, A. 1971 Convection in a temperature variable viscosity fluid and the thermal state of the earth's mantle. E.O.S. Trans. Am. Geophys. Un. 52, 348.Google Scholar
Busse, F. H. 1967 On the stability of two-dimensional convection in a layer heated from below. J. Math. & Phys. 46, 140.Google Scholar
Busse, F. H. & Whitehead, J. A. 1971 Instabilities of convection rolls in a high Prandtl number fluid. J. Fluid Mech. 47, 305.Google Scholar
Carter, N. L. & Ave'Lallemant, H. G. 1970 High temperature flow of Dunite and Peridotite. Bull. Geol. Soc. Am. 81, 2181.Google Scholar
Gordon, R. B. 1971 Observations of crystal plasticity under high pressure with applications to the earth's mantle. J. Geophys. Res. 76, 1248.Google Scholar
Hoard, C. Q., Robertson, C. R. & Acrivos, A. 1970 Experiments on the cellular structure in Bénard convection. Int. J. Heat Mass Transfer. 13, 849.Google Scholar
Houston, M. H. & De Bremaeker, J.-Cl. 1975 Numerical models of convection in the upper mantle. J. Geophys. Res. 80, 742.Google Scholar
Jenssen, O. 1963 Note on the influence of variable viscosity on the critical Rayleigh number. Acta Polytech. Scand. 24, 1.Google Scholar
Koschmieder, E. L. 1974 Bénard convection. Adv. Chem. Phys. 26, 177.Google Scholar
Krishnamurti, R. 1968 Finite amplitude convection with changing mean temperature. Part 2. An experimental test of theory. J. Fluid Mech. 33, 457.Google Scholar
Liang, S. F. 1969 Ph.D. thesis, Stanford University.
Liang, S. F. & Acrivos, A. 1970 Experiments on buoyancy-driven convection in non-Newtonian fluid. Rheol. Acta. 9, 447.Google Scholar
McKenzie, D. P., Roberts, J. M. & Weiss, N. O. 1974 Convection in the earth's mantle: towards a numerical simulation. J. Fluid Mech. 62, 465.Google Scholar
Palm, E. 1960 On the tendency toward hexagonal cells in steady convection. J. Fluid Mech. 8, 183.Google Scholar
Palm, E., Ellingsen, T. & Gjevik, B. 1967 On the occurrence of cellular motion in Bénard convection. J. Fluid Mech. 30, 651.Google Scholar
Rayleigh, C. B. & Kirby, S. H. 1970 Creep in the upper mantle. Am. Min. Special Paper, no. 3, p. 113.Google Scholar
Rossby, H. T. 1969 A study of Bénard convection with and without rotation. J. Fluid Mech. 36, 309.Google Scholar
Silveston, P. L. 1958 Wärmedurchang in Waagerechten Flüssigkeitsschichten. Forsh. Ing. Wes. 24, 59.Google Scholar
Thirlby, R. 1970 Convection in an internally heated layer. J. Fluid Mech. 44, 673.Google Scholar
Torrance, K. E. & Turcotte, D. L. 1971 Thermal convection with large viscosity variations. J. Fluid Mech. 47, 113.Google Scholar
Tozer, D. C. 1967 Towards a theory of mantle convection. In The Earth's Mantle (ed. T. F. Gaskell), p. 327. Academic.
Tritton, D. & Zarraga, M. N. 1967 Convection in horizontal layers with internal heat generation. Experiments. J. Fluid Mech. 30, 21.Google Scholar
Weertman, J. 1970 Creep strength in the earth's mantle. Rev. Geophys. 8, 145.Google Scholar