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Upper bounds on Nusselt number at finite Prandtl number

Antoine Choffrut, Camilla Nobili and Felix Otto


We study Rayleigh-Bénard convection based on the Boussinesq approximation. We are interested in upper bounds on the Nusselt number $\rm{Nu}$, the upwards heat transport, in terms of the Rayleigh number $\rm{Ra}$, that characterizes the relative strength of the driving mechanism and the Prandtl number $\rm{Pr}$, that characterizes the strength of the inertial effects. We show that, up to logarithmic corrections, the upper bound $\rm{Nu}\lesssim \rm{Ra}^{\frac{1}{3}}$ of Constantin and Doering (1999) persists as long as $\rm{Pr}\gtrsim\rm{Ra}^{\frac{1}{3}}$ and then crosses over to $\rm{Nu}\lesssim\rm{Pr}^{-\frac{1}{2}}\rm{Ra}^{\frac{1}{2}}$. This result improves the one of Wang (2007) by going beyond the perturbative regime $\rm{Pr}\gg \rm{Ra}$.

The proof uses a new way to estimate the transport nonlinearity in the Navier Stokes equations capitalizing on the no-slip boundary condition. It relies on a new Calderón-Zygmund estimate for the non-stationary Stokes equations in $L^1$ with a borderline Muckenhoupt weight.

Dec 19, 2014
Jan 5, 2015
MSC Codes:
35Q30, 35Q35, 76R10, 76E06, 76F99
Rayleigh-B\'enard convection, navier-stokes equations, no-slip boundary condition, finite Prandtl number, Nusselt number, maximal regularity for non-stationary Stokes equat

Related publications

2016 Repository Open Access
Antoine Choffrut, Camilla Nobili and Felix Otto

Upper bounds on Nusselt number at finite Prandtl number

In: Journal of differential equations, 260 (2016) 4, pp. 3860-3880