We have decided to discontinue the publication of preprints on our preprint server end of 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
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.