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Time-Dependent Pattern Formation for Two-Layer Convection

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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 article is a review of double-layer convection in which the pattern formation arises due to a competition between the bulk motions in each fluid (see Figure 1). An instability takes place when the temperature difference between the upper and lower walls reaches a threshold value, and the response of the two-layer system depends on the properties of the constituent fluids. Our motivation is the search for patterns formed in non-equilibrium fluid dynamical systems which exhibit time-dependence at or near the onset of a pattern. Such a time-dependent state is predicted for the two-layer Rayleigh-Benard system and is accessible experimentally as well as theoretically. We expect to see oscillatory and spatio-temporal chaotic behavior. The presence of the interface and the coupling between the fluids in this model problem may provide an understanding of generic behaviors in related applications, such as the modeling of the earth’s mantle as a two-layer convecting system [3, 11], and liquid encapsulated crystal growth [19].

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Authors and Affiliations

  1. Dept. of Mathematics, Virginia Tech, 460 McBryde Hall, Blacksburg, VA, 24061-0123, USA

    Y. Renardy & C. G. Stoltz

Authors
  1. Y. Renardy
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  2. C. G. Stoltz
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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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Renardy, Y., Stoltz, C.G. (1999). Time-Dependent Pattern Formation for Two-Layer Convection. 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_16

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  • DOI: https://doi.org/10.1007/978-1-4612-1558-5_16

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7192-5

  • Online ISBN: 978-1-4612-1558-5

  • eBook Packages: Springer Book Archive

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