Infinite Prandtl number convection with Navier-slip boundary conditions
Christian Seis
TL;DR
The paper analyzes infinite Prandtl number Rayleigh--Bénard convection with Navier-slip boundary conditions, deriving upper bounds for the Nusselt number $Nu$ that interpolate between the classical no-slip and free-slip scalings as a function of the slip length $\sigma$ and Rayleigh number $Ra$. By a strategic rescaling that eliminates $Ra$ and localizing the heat flux near the boundaries, the authors reduce the problem to a fourth-order elliptic equation for the vertical velocity and develop a boundary-layer framework to control velocity corrections. They establish two main bounds: (i) $Nu \lesssim 1+\sigma^{1/2}$ for moderate to large $H$ with $\sigma\lesssim H$, capturing the no-slip/Navier-slip regime, and (ii) $Nu \lesssim H^{1/4}$ in the free-slip regime, with a complete interpolation between regimes suggesting $Nu \lesssim Ra^{1/3}$ when $\sigma \lesssim Ra^{-1/3}$, $Nu \lesssim (\sigma Ra)^{1/2}$ for $Ra^{-1/3}\lesssim \sigma \lesssim Ra^{-1/6}$, and $Nu \lesssim Ra^{5/12}$ for $\sigma \gtrsim Ra^{-1/6}$. This provides a rigorous, physically relevant bridge between boundary conditions and heat transport, and introduces a novel intermediate scaling in the Navier-slip setting.
Abstract
We are concerned with infinite Prandtl number Rayleigh--Bénard convection with Navier-slip boundary conditions. The goal of this work is to estimate the average upward heat flux measured by the nondimensional Nusselt number $Nu$ in terms of the Rayleigh number $Ra$, which is a nondimensional quantity measuring the imposed temperature gradient. We derive bounds on the Nusselt number that coincide for relatively small slip lengths with the optimal Nusselt number scaling for no-slip boundaries, $Nu\lesssim Ra^{1/3}$; for relatively large slip lengths, we recover scaling estimates for free-slip boundaries, $Nu\lesssim Ra^{5/12}$.