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Rayleigh-Bénard Convection with Rotation at Small Prandtl Numbers

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Pattern Formation in Continuous and Coupled Systems

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 115))

Abstract

This paper reviews past results from and future prospects for experimental studies of Rayleigh-Bénard convection with rotation about a vertical axis. At dimensionless rotation rates 0 ≤ Ω ≤ 20 and for Prandtl numbers σ ≃ 1, Küppers-Lortz-unstable patterns offered a unique opportunity to study spatio-temporal chaos immediately above a supercritical bifurcation where weakly-nonlinear theories in the form of Ginzburg-Landau (GL) or Swift-Hohenberg (SH) equations can be expected to be valid. However, the dependence of the time and length scales of the chaotic state on ε ≡ ΔT/ΔT C - 1 was found to be different from the expected dependence based on the structure of GL equations. For Ω ≳ 70 and 0.7 ≲ σ ≲ 5 patterns were found to be cellular near onset with local four-fold coordination. They differ from the theoretically expected Küppers-Lortz-unstable state. Stable as well as intermittent defect-free rotating square lattices exist in this parameter range.

Smaller Prandtl numbers ( 0.16 ≲ σ ≲ 0.7) can only be reached in mixtures of gases. These fluids are expected to offer rich future opportunities for the study of a line of tricritical bifurcations, of supercritical Hopf bifurcations to standing waves, of a line of codimension-two points, and of a codimension-three point.

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Author information

Authors and Affiliations

  1. Department of Physics and Center for Nonlinear Science, University of California, Santa Barbara, CA, 93106, USA

    Guenter Ahlers & Kapil M.S. Bajaj

Authors
  1. Guenter Ahlers
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  2. Kapil M.S. Bajaj
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, TX, 77204-4792, USA

    Martin Golubitsky

  2. Department of Chemical Engineering, University of Houston, Houston, TX, 77204-4792, USA

    Dan Luss

  3. Theoretical and Applied Mechanics, Cornell University, Ithaca, NY, 14853, USA

    Steven H. Strogatz

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Ahlers, G., Bajaj, K.M. (1999). Rayleigh-Bénard Convection with Rotation at Small Prandtl Numbers. In: Golubitsky, M., Luss, D., Strogatz, S.H. (eds) Pattern Formation in Continuous and Coupled Systems. The IMA Volumes in Mathematics and its Applications, vol 115. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1558-5_1

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  • DOI: https://doi.org/10.1007/978-1-4612-1558-5_1

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7192-5

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