Bounds on heat flux for Rayleigh-B\'{e}nard convection between Navier-slip fixed-temperature boundaries

TL;DR

This paper derives bounds on heat transfer in 2D Rayleigh-Bénard convection with Navier-slip boundaries, interpolating between free-slip and no-slip conditions as the slip-length varies with Rayleigh number.

Contribution

It establishes new bounds on the Nusselt number for Navier-slip boundary conditions, bridging classical free-slip and no-slip results based on slip-length variation.

Findings

Bounds interpolate between free-slip and no-slip cases

Nusselt number scales as Ra^{5/12} for free-slip

Nusselt number scales as Ra^{1/2} for no-slip

Abstract

We study two-dimensional Rayleigh-B\'{e}nard convection with Navier-slip, fixed temperature boundary conditions and establish bounds on the Nusselt number. As the slip-length varies with Rayleigh number $\rm{Ra}$, this estimate interpolates between the Whitehead-Doering bound by $\rm{Ra}^{\frac{5}{12}}$ for free-slip conditions [13] and the classical Doering-Constantin $\rm{Ra}^{\frac{1}{2}}$ bound [4].