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New bounds on the vertical heat transport for Bénard–Marangoni convection at infinite Prandtl number

Published online by Cambridge University Press:  27 December 2019

Giovanni Fantuzzi Open the ORCID record for Giovanni Fantuzzi [Opens in a new window] ,
Camilla Nobili  and
Andrew Wynn
Giovanni Fantuzzi*
Affiliation:
Department of Aeronautics, Imperial College London, LondonSW7 2AZ, UK
Camilla Nobili
Affiliation:
Department of Mathematics, University of Hamburg, 20146Hamburg, Germany
Andrew Wynn
Affiliation:
Department of Aeronautics, Imperial College London, LondonSW7 2AZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

We prove a new rigorous upper bound on the vertical heat transport for Bénard–Marangoni  convection of a two- or three-dimensional fluid layer with infinite Prandtl number. Precisely, for Marangoni number $Ma\gg 1$ the Nusselt number $Nu$ is bounded asymptotically by $Nu\leqslant \text{const.}\times Ma^{2/7}(\ln Ma)^{-1/7}$. Key to our proof are a background temperature field with a hyperbolic profile near the fluid’s surface and new estimates for the coupling between temperature and vertical velocity.

JFM classification

Mathematical Foundations: Variational methods Convection: Marangoni convection
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 885 , 25 February 2020 , R4
Copyright
© The Author(s), 2019. Published by Cambridge University Press

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