Infinite Prandtl number convection with Navier-slip boundary conditions

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, $Nu\lesssim R{a}^{1/3}$; for relatively large slip lengths, we recover scaling estimates for free-slip boundaries, $Nu\lesssim R{a}^{5/12}$.

Final version. To appear in Comm. Math. Phys