Rigorous scaling laws for internally heated convection at infinite Prandtl number

نویسندگان

چکیده

New bounds are proven on the mean vertical convective heat transport, $\overline{\langle wT \rangle}$, for uniform internally heated (IH) convection in limit of infinite Prandtl number. For fluid a horizontally-periodic layer between isothermal boundaries, we show that \rangle} \leq \frac12 - c R^{-2}$, where $R$ is nondimensional `flux' Rayleigh number quantifying strength internal heating and $c = 216$. Then, 0$ corresponds to transport by conduction alone, while > represents enhancement upwards due motion. If, instead, lower boundary thermal insulator, then obtain R^{-4}$, with $c\approx 0.0107$. This result implies Nusselt $Nu$, defined as ratio total-to-conductive satisfies $Nu \lesssim R^{4}$. Both obtained combining background method minimum principle fluid's temperature Hardy--Rellich inequalities exploit link velocity temperature. In both cases, power-law dependence improves previously best-known bounds, which, although valid at finite numbers, approach bound exponentially $R$.

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ژورنال

عنوان ژورنال: Journal of Mathematical Physics

سال: 2023

ISSN: ['0022-2488', '1527-2427', '1089-7658']

DOI: https://doi.org/10.1063/5.0098250