Analytical bounds on the heat transport in internally heated convection

Abstract

We obtain an analytical bound on the mean vertical convective heat flux $⟨wT⟩$ between two parallel boundaries driven by uniform internal heating. We consider two configurations, one with both boundaries held at the same constant temperature, and the other one with a top boundary held at constant temperature and a perfectly insulating bottom boundary. For the first configuration, Arslan et al. (J. Fluid Mech. 919:A15, 2021) recently provided numerical evidence that Rayleigh-number-dependent corrections to the only known rigorous bound $⟨wT⟩\le 1/2$ may be provable if the classical background method is augmented with a minimum principle stating that the fluid's temperature is no smaller than that of the top boundary. Here, we confirm this fact rigorously for both configurations by proving bounds on $⟨wT⟩$ that approach $1/2$ exponentially from below as the Rayleigh number is increased. The key to obtaining these bounds are inner boundary layers in the background fields with a particular inverse-power scaling, which can be controlled in the spectral constraint using Hardy and Rellich inequalities. These allow for qualitative improvements in the analysis not available to standard constructions.

arXiv comment20 pages, 3 figures

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Cited by in corpus (5)

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• Bounds on heat transfer by incompressible flows between balanced sources and sinks
• Rigorous scaling laws for internally heated convection at infinite Prandtl number
• Internally heated convection with rotation: bounds on heat transport
• Bounds on dissipation in three-dimensional planar shear flows: reduction to two-dimensional problems