Bounds on heat transport for convection driven by internal heating
File(s)
Accepted version
Author(s)
Arslan, Ali
Fantuzzi, Giovanni
Craske, John
Wynn, Andrew
Type
Journal Article
Abstract
The mean vertical heat transport ⟨wT⟩ in convection 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 quadratic auxiliary functions. 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 ⟨wT⟩≤1/2 can only be improved up to finite R. Indeed, we demonstrate analytically that ⟨wT⟩≤2−21/5R1/5 and therefore prove that ⟨wT⟩<1/2 for R<65536. 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 R=3.4×105. While the computational results suggest that the best bound on ⟨wT⟩ approaches 1/2 asymptotically from below as R→∞, we prove that typical analytical constructions cannot be used to prove this conjecture.
Date Issued
2021-07
Date Acceptance
2021-04-16
Citation
Journal of Fluid Mechanics, 2021, 919, pp.1-34
ISSN
0022-1120
Publisher
Cambridge University Press
Start Page
1
End Page
34
Journal / Book Title
Journal of Fluid Mechanics
Volume
919
Copyright Statement
© The Author(s), 2021. Published by Cambridge University Press. This article has been published in a revised form in Journal of Fluid Mechanics https://doi.org/10.1017/jfm.2021.360. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works.
Sponsor
Engineering and Physical Sciences Research Council
Identifier
https://arxiv.org/abs/2102.06458
Grant Number
EP/L016230/1
Subjects
physics.flu-dyn
physics.flu-dyn
01 Mathematical Sciences
09 Engineering
Fluids & Plasmas
Publication Status
Published
Date Publish Online
2021-05-26