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

TL;DR

This paper derives new upper bounds for the Nusselt number in 2D Rayleigh-Bénard convection with Navier-slip boundaries, revealing five possible scaling laws depending on flow parameters and improving previous bounds.

Contribution

It introduces a novel approach combining localization and interpolation bounds, providing shorter proofs and refined estimates for convection with slip boundaries.

Findings

Five distinct scaling laws for Nusselt number depending on parameters.

Shorter proof of existing free-slip boundary results.

Improved bounds for slippery boundary convection.

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 $\textit{Nu} \sim \textit{Pr}^{-\frac{1}{6}} (L_s^{-\frac{1}{3}}\textit{Ra})^{\frac{1}{2}}$