Scaling laws for Rayleigh-B\'enard convection between Navier-slip boundaries

Abstract

We consider the two-dimensional Rayeigh-B\'enard convection problem between Navier-slip fixed-temperature boundary conditions and present a new upper bound for the Nusselt number. The result, based on a localization principle for the Nusselt number and an interpolation bound, exploits the regularity of the flow. On one hand our method yields a shorter proof of the celebrated result in Whitehead & Doering (2011) in the case of free-slip boundary conditions. On the other hand, its combination with a new, refined estimate for the pressure gives a substantial improvement of the interpolation bounds in Drivas et al. (2022) for slippery boundaries. A rich description of the scaling behaviour arises from our result: depending on the magnitude of the Prandtl number and slip-length, our upper bounds indicate five possible scaling laws: $\textit{Nu} \sim (L_s^{-1}\textit{Ra})^{\frac{1}{3}}$, $\textit{Nu} \sim (L_s^{-\frac{2}{5}}\textit{Ra})^{\frac{5}{13}}$, $\textit{Nu} \sim \textit{Ra}^{\frac{5}{12}}$, $\textit{Nu} \sim \textit{Pr}^{-\frac{1}{6}} (L_s^{-\frac{4}{3}}\textit{Ra})^{\frac{1}{2}}$ and $\textit{Nu} \sim \textit{Pr}^{-\frac{1}{6}} (L_s^{-\frac{1}{3}}\textit{Ra})^{\frac{1}{2}}$