A rigorous bound on the vertical transport of heat in Rayleigh-Bénard convection at infinite Prandtl number with mixed thermal boundary conditions

Abstract

A rigorous upper bound on the Nusselt number is derived for infinite Prandtl number Rayleigh-Bénard convection for a fluid constrained between no-slip, mixed thermal vertical boundaries. The result suggests that the thermal boundary condition does not affect the qualitative nature of the heat transport. The bound is obtained with the use of a nonlinear, stably stratified background temperature profile in the bulk, notwithstanding the lack of boundary control of the temperature due to the Robin boundary conditions.