Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-19T10:47:30.520Z Has data issue: false hasContentIssue false
  • English
  • Français

Rolls versus squares in thermal convection of fluids with temperature-dependent viscosity

Published online by Cambridge University Press:  21 April 2006

D. R. Jenkins
D. R. Jenkins
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK Present address: CSIRO Division of Mathematics and Statistics, PO Box 218, Lindfield, NSW 2070, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider finite-amplitude thermal convection, in a horizontal fluid layer. The viscosity of the fluid is dependent upon its temperature. Using a weakly nonlinear expansion procedure, we examine the stability of two-dimensional roll and three-dimensional square planforms, in order to determine which should be preferred in convection experiments. The analysis shows that the roll planform is preferred for low values of the ratio of the viscosities at the top and bottom boundaries, but the square planform is preferred for larger values of the ratio. At still larger values, subcritical convection is predicted. We also include the effects of boundaries having finite thermal conductivity, which enables favourable comparison to be made with experimental studies. A discrepancy between the present work and a previous study of this problem (Busse & Frick 1985) is discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 178 , May 1987 , pp. 491 - 506
Copyright
© 1987 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

Bernoff, A. J. 1985 Sidewall stabilisation of convection. Preprint, University of Cambridge.
Booker, J. R. 1976 Thermal convection with strongly temperature-dependent viscosity. J. Fluid Mech. 76, 741–754.Google Scholar
Booker, J. R. & Stengel, K. C. 1978 Further thoughts on convective heat transport in a variable-viscosity fluid. J. Fluid Mech. 86, 289–291.Google Scholar
Busse, F. H. 1967 The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 30, 625–629.Google Scholar
Busse, F. H. & Frick, H. 1985 Square-pattern convection in fluids with strongly temperaturedependent viscosity. J. Fluid Mech. 150, 451–465.Google Scholar
Chapman, C. J. & Proctor, M. R. E. 1980 Nonlinear Rayleigh-Bénard convection between poorly conducting boundaries. J. Fluid Mech. 101, 759–782.Google Scholar
Depassier, M. C. & Spiegel, E. A. 1982 Convection with heat flux prescribed on the boundaries of the system. I. The effect of temperature dependence of material properties. Geophys. Astrophys. Fluid Dyn. 21, 167–188.Google Scholar
Gertsberg, V. L. & Sivashinsky, G. I. 1981 Large cells in nonlinear Rayleigh-Bénard convection. Prog. Theor. Phys. 66, 1219–1229.Google Scholar
Hoard, C. Q., Robertson, C. R. & Acrivos, A. 1970 Experiments on the cellular structure in Bénard convection. Intl J. Heat Mass Transfer 13, 849–855.Google Scholar
Jenkins, D. R. & Proctor, M. R. E. 1984 The transition from roll to square-cell solutions in Rayleigh-Bénard convection. J. Fluid Mech. 139, 461–471.Google Scholar
Jenkins, D. R. 1987 Interpretation of shadowgraph patterns in Rayleigh-Bénard convection. J. Fluid Mech. (submitted).Google Scholar
Le Gal, P., Pocheau, A. & Croquette, V. 1985 Square versus roll pattern at convective threshold. Phys. Rev. Lett. 54, 2501–2504.Google Scholar
Mckenzie, D., Watts, A., Parsons, B. & Roufosse, M. 1980 Planform of mantle convection beneath the Pacific Ocean. Nature 288, 422–445.Google Scholar
Oliver, D. S. 1980 Bénard convection with strongly temperature dependent viscosity. Ph.D. thesis, University of Washington.
Oliver, D. S. & Booker, J. R. 1983 Planform of convection with strongly temperature dependent viscosity. Geophys. Astrophys. Fluid Dyn. 27, 73–85.Google Scholar
Palm, E. 1960 On the tendency towards hexagonal cells in steady convection. J. Fluid Mech. 8, 183–192.Google Scholar
Palm, E. & Oiann, H. 1964 Contribution to the theory of cellular thermal convection. J. Fluid Mech. 19, 353–365.Google Scholar
Peltier, W. R. 1985 Mantle convection and viscoelasticity. Ann. Rev. Fluid Mech. 17, 561–608.Google Scholar
Proctor, M. R. E. 1981 Planform selection by finite-amplitude thermal convection between poorly conducting slabs. J. Fluid Mech. 113, 469–485.Google Scholar
Segel L. A. & Stuart, J. T. 1962 On the question of the preferred mode in cellular thermal convection. J. Fluid Mech. 13, 289–306.Google Scholar
Somerscales, E. F. C. & Dougherty T. S. 1970 Observed flow patterns at the initiation of convection in a horizontal liquid layer heated from below. J. Fluid Mech. 42, 755–768.Google Scholar
Stengel, K. C., Oliver, D. S. & Booker, J. R. 1982 Onset of convection in a variable-viscosity fluid. J. Fluid Mech. 120, 411–431.Google Scholar
Stuart, J. T. 1964 On the cellular patterns in thermal convection. J. Fluid Mech. 18, 481–496.Google Scholar
Swift, J. W. 1984 Bifurcation and symmetry in convection. Ph.D. dissertation, University of California, Berkeley.
Turcotte, D. L. & Oxburgh, E. R. 1972 Mantle convection and the new global tectonics. Ann. Rev. Fluid Mech. 4, 33–68.Google Scholar
White, D. B. 1982 Experiments with convection in a variable viscosity fluid. Ph.D. thesis, University of Cambridge.