Scaling laws for Rayleigh-Bénard convection between Navier-slip boundaries

Abstract

We consider the two-dimensional Rayeigh-Bénard 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: $\text{Nu}\sim (L_s^{-1}\text{Ra}{)}^{\frac{1}{3}}$, $\text{Nu}\sim (L_s^{-\frac{2}{5}}\text{Ra}{)}^{\frac{5}{13}}$, $\text{Nu}\sim {\text{Ra}}^{\frac{5}{12}}$, $\text{Nu}\sim {\text{Pr}}^{-\frac{1}{6}}(L_s^{-\frac{4}{3}}\text{Ra}{)}^{\frac{1}{2}}$ and $\text{Nu}\sim {\text{Pr}}^{-\frac{1}{6}}(L_s^{-\frac{1}{3}}\text{Ra}{)}^{\frac{1}{2}}$

arXiv comment18 pages, 1 figure, 1 table

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