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Exact energy stability of Bénard–Marangoni convection at infinite Prandtl number

Published online by Cambridge University Press:  01 June 2017

Giovanni Fantuzzi Open the ORCID record for Giovanni Fantuzzi [Opens in a new window]  and
Andrew Wynn
Giovanni Fantuzzi*
Affiliation:
Department of Aeronautics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
Andrew Wynn
Affiliation:
Department of Aeronautics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

Using the energy method we investigate the stability of pure conduction in Pearson’s model for Bénard–Marangoni convection in a layer of fluid at infinite Prandtl number. Upon extending the space of admissible perturbations to the conductive state, we find an exact solution to the energy stability variational problem for a range of thermal boundary conditions describing perfectly conducting, imperfectly conducting, and insulating boundaries. Our analysis extends and improves previous results, and shows that with the energy method global stability can be proven up to the linear instability threshold only when the top and bottom boundaries of the fluid layer are insulating. Contrary to the well-known Rayleigh–Bénard convection set-up, therefore, energy stability theory does not exclude the possibility of subcritical instabilities against finite-amplitude perturbations.

JFM classification

Convection: Marangoni convection Instability: Nonlinear instability Mathematical Foundations: Variational methods
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 822 , 10 July 2017 , R1
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Copyright
© 2017 Cambridge University Press 

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