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Bounds on heat transport for convection driven by internal heating

Published online by Cambridge University Press:  26 May 2021

Ali Arslan Open the ORCID record for Ali Arslan [Opens in a new window] ,
Giovanni Fantuzzi Open the ORCID record for Giovanni Fantuzzi [Opens in a new window] ,
John Craske Open the ORCID record for John Craske [Opens in a new window]  and
Andrew Wynn Open the ORCID record for Andrew Wynn [Opens in a new window]
Ali Arslan*
Affiliation:
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
Giovanni Fantuzzi
Affiliation:
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
John Craske
Affiliation:
Department of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, UK
Andrew Wynn
Affiliation:
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: a.arslan18@imperial.ac.uk
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Abstract

The mean vertical convective heat transport $\langle wT \rangle$ between isothermal plates driven by uniform internal heating is investigated by means of rigorous bounds. These are obtained as a function of the Rayleigh number R by constructing feasible solutions to a convex variational problem, derived using a formulation of the classical background method in terms of a quadratic auxiliary function. When the fluid's temperature relative to the boundaries is allowed to be positive or negative, numerical solution of the variational problem shows that best previous bound $\langle wT \rangle \leqslant 1/2$ (Goluskin & Spiegel, Phys. Lett. A, vol. 377, issue 1–2, 2012, pp. 83–92) can only be improved up to finite R. Indeed, we demonstrate analytically that $\langle wT \rangle \leqslant 2^{-21/5} {\textit {R}}^{1/5}$ and therefore prove that $\langle wT\rangle < 1/2$ for ${\textit {R}} < 65\,536$. However, if the minimum principle for temperature is invoked, which asserts that internal temperature is at least as large as the temperature of the isothermal boundaries, then numerically optimised bounds are strictly smaller than $1/2$ until at least ${\textit {R}}=3.4\times 10^{5}$. While the computational results suggest that the best bound on $\langle wT\rangle$ approaches $1/2$ asymptotically from below as ${\textit {R}}\rightarrow \infty$, we prove that typical analytical constructions cannot be used to prove this conjecture.

JFM classification

Mathematical Foundations: Variational methods

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JFM Papers
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Journal of Fluid Mechanics , Volume 919 , 25 July 2021 , A15
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© The Author(s), 2021. Published by Cambridge University Press

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