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@unpublished{Arslan:2022,
author = {Arslan, A and Fantuzzi, G and Craske, J and Wynn, A},
publisher = {arXiv},
title = {Rigorous scaling laws for internally heated convection at infinite Prandtl number},
url = {https://arxiv.org/abs/2205.03175},
year = {2022}
}

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TY  - UNPB
AB  - New bounds are proven on the mean vertical convective heat transport, 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 wT¯¯¯¯¯¯¯¯¯¯¯≤12−cR−2, where R is a nondimensional `flux' Rayleigh number quantifying the strength of internal heating and c=216. Then, wT¯¯¯¯¯¯¯¯¯¯¯=0 corresponds to vertical heat transport by conduction alone, while 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 wT¯¯¯¯¯¯¯¯¯¯¯≤12−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 NuR4. 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.
AU  - Arslan,A
AU  - Fantuzzi,G
AU  - Craske,J
AU  - Wynn,A
PB  - arXiv
PY  - 2022///
TI  - Rigorous scaling laws for internally heated convection at infinite Prandtl number
UR  - https://arxiv.org/abs/2205.03175
UR  - http://hdl.handle.net/10044/1/97430
ER  -

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