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@article{faucris.287616505,
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⟩}≤1/2−cR^{−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⟩}≤1/2−cR^{−4}, with c≈0.0107. This result implies that the Nusselt number Nu, defined as the ratio of the total-to-conductive heat transport, satisfies Nu≲R4. 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.},
author = {Arslan, Ali and Fantuzzi, Giovanni and John, Craske and Wynn, Andrew},
doi = {10.1063/5.0098250},
faupublication = {no},
journal = {Journal of Mathematical Physics},
keywords = {Heat transfer, Mathematical optimization, Auxiliary functions, Calculus of variations, Fluid mechanics, Natural convection},
peerreviewed = {Yes},
title = {{Rigorous} scaling laws for internally heated convection at infinite {Prandtl} number},
volume = {64},
year = {2023}
}