Arslan A, Fantuzzi G, John C, Wynn A (2023)
Publication Language: English
Publication Status: In review
Publication Type: Journal article, Original article
Future Publication Type: Journal article
Publication year: 2023
Book Volume: 64
Article Number: 023101
DOI: 10.1063/5.0098250
Open Access Link: https://arxiv.org/abs/2205.03175
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.
APA:
Arslan, A., Fantuzzi, G., John, C., & Wynn, A. (2023). Rigorous scaling laws for internally heated convection at infinite Prandtl number. Journal of Mathematical Physics, 64. https://dx.doi.org/10.1063/5.0098250
MLA:
Arslan, Ali, et al. "Rigorous scaling laws for internally heated convection at infinite Prandtl number." Journal of Mathematical Physics 64 (2023).
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