The maximal heat-transfer exponent at finite Prandtl number
On $\Omega = (\mathbb{R}/\mathbb{Z}) \times (0,1)$ consider the two-dimensional Boussinesq system at Prandtl number one, $$\partial_t u + u \cdot \nabla u + \nabla p = \Delta u + \mathrm{Ra}\, T e_2, \qquad \operatorname{div} u = 0, \qquad \partial_t T + u \cdot \nabla T = \Delta T,$$ with horizontal periodicity, $u = 0$ on the walls, $T(x,0) = 1$ and $T(x,1) = 0$. For $\mathrm{Ra} > 0$ let $\mathcal{N}(\mathrm{Ra})$ be the supremum, over smooth compatible initial data with $0 \le T_0 \le 1$, of $$\limsup_{\tau \to \infty} \frac{1}{\tau} \int_0^\tau \int_\Omega |\nabla T(t,x)|^2 \, dx \, dt,$$ which in these units is the Nusselt number of the solution, the ratio of its heat flux to the conductive one. The conductive state $u = 0$, $T = 1 - x_2$ gives $\mathcal{N}(\mathrm{Ra}) \ge 1$. Rigorous upper bounds, among them Doering and Constantin's background method, give $\mathrm{Nu} \lesssim \mathrm{Ra}^{1/2}$ for convection between no-slip walls at finite Prandtl number. In the infinite-Prandtl-number limit in three dimensions, Chanillo and Malchiodi proved $\mathrm{Nu} \le C\,\mathrm{Ra}^{1/3}$, the scaling predicted by Malkus and by Howard.
Problem. Determine $$\beta_* := \limsup_{\mathrm{Ra} \to \infty} \frac{\log \mathcal{N}(\mathrm{Ra})}{\log \mathrm{Ra}},$$ the optimal power-law exponent for actual solutions with these fixed boundary conditions at finite Prandtl number.
Source: Stated by AIM — Open Applied Problems, after the question of the high-Rayleigh-number scaling of the Nusselt number in Rayleigh–Bénard convection, for which Malkus (1954) predicted the exponent 1/3; see Ding and Kerswell (2020) and Ding, Wen and Li (2026).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Boussinesq equations, Nusselt numbers, Rayleigh–Bénard convection
References
@article{malkus1954,
author = {Malkus, W. V. R.},
title = {The heat transport and spectrum of thermal turbulence},
journal = {Proc. R. Soc. Lond. Ser. A},
volume = {225},
number = {1161},
year = {1954},
pages = {196--212},
doi = {10.1098/rspa.1954.0197},
}
@article{ding2020,
author = {Ding, Zijing and Kerswell, Rich R.},
title = {Exhausting the background approach for bounding the heat transport in {R}ayleigh--{B}{\'e}nard convection},
journal = {J. Fluid Mech.},
volume = {889},
year = {2020},
pages = {A33},
doi = {10.1017/jfm.2020.41},
eprint = {1906.03376},
archivePrefix = {arXiv},
}
@misc{ding2026,
author = {Ding, Zijing and Wen, Baole and Li, Hui},
title = {Optimal heat transport at the edge of energy stability},
year = {2026},
eprint = {2606.14138},
archivePrefix = {arXiv},
}
@article{doering1996,
author = {Doering, Charles R. and Constantin, Peter},
title = {Variational bounds on energy dissipation in incompressible flows. {III}. {C}onvection},
journal = {Phys. Rev. E},
volume = {53},
number = {6},
year = {1996},
pages = {5957--5981},
doi = {10.1103/PhysRevE.53.5957},
}
@article{chanillo2025,
author = {Chanillo, Sagun and Malchiodi, Andrea},
title = {Sharp bounds on the {N}usselt number in {R}ayleigh--{B}{\'e}nard convection and a bilinear estimate by {C}oifman--{M}eyer},
journal = {Invent. Math.},
volume = {240},
number = {2},
year = {2025},
pages = {633--660},
doi = {10.1007/s00222-025-01326-z},
eprint = {2001.01662},
archivePrefix = {arXiv},
}
@misc{aim-finite-prandtl-heat-scaling,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The maximal heat-transfer exponent at finite Prandtl number}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/161-finite-prandtl-heat-scaling.md},
note = {Entry 161 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00865, "The maximal heat-transfer exponent at finite Prandtl number", clopen.solutions.