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MiS Preprint
116/2014
Upper bounds on Nusselt number at finite Prandtl number
Antoine Choffrut, Camilla Nobili and Felix Otto
Abstract
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.
