Rigorous scaling laws for internally heated convection at infinite Prandtl number

Abstract

New bounds are proven on the mean vertical convective heat transport, $\overline{⟨wT⟩}$, for uniform internally heated (IH) convection in the limit of infinite Prandtl number. For fluid in a horizontally-periodic layer between isothermal boundaries, we show that $\overline{⟨wT⟩}\le \frac{1}{2}-c{R}^{-2}$, where $R$ is a nondimensional `flux' Rayleigh number quantifying the strength of internal heating and $c=216$. Then, $\overline{⟨wT⟩}=0$ corresponds to vertical heat transport by conduction alone, while $\overline{⟨wT⟩}>0$ represents the enhancement of vertical heat transport upwards due to convective motion. If, instead, the lower boundary is a thermal insulator, then we obtain $\overline{⟨wT⟩}\le \frac{1}{2}-c{R}^{-4}$, with $c\approx 0.0107$. This result implies that the Nusselt number $Nu$, defined as the ratio of the total-to-conductive heat transport, satisfies $Nu\lesssim R^4$. Both bounds are obtained by combining the background method with a minimum principle for the fluid's temperature and with Hardy--Rellich inequalities to exploit the link between the vertical velocity and temperature. In both cases, power-law dependence on $R$ improves the previously best-known bounds, which, although valid at both infinite and finite Prandtl numbers, approach the uniform bound exponentially with $R$.

28 pages, 5 figures

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Cited by in corpus (2)

• Bounds on heat transfer by incompressible flows between balanced sources and sinks
• Internally heated convection with rotation: bounds on heat transport