# Rigorous scaling laws for internally heated convection at infinite Prandtl number

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

### Journal

**Book Volume:** 64

**Article Number:** 023101

**DOI:** 10.1063/5.0098250

**Open Access Link:** https://arxiv.org/abs/2205.03175

### 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.

### Authors with CRIS profile

### How to cite

**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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