Rigid bounds on heat transport by a fluid between slippery boundaries

Rigorous bounds on heat transport are derived for thermal convection between stress-free horizontal plates. For three-dimensional Rayleigh–Bénard convection at infinite Prandtl number ( Pr ), the Nusselt number ( Nu ) is bounded according to Nu≤ 0. 28764Ra5/ 12 where Ra is the standard Rayleigh number. For convection driven by a uniform steady internal heat source between isothermal boundaries, the spatially and temporally averaged (non-dimensional) temperature is bounded from below by T ≥ 0. 6910R^- 5/ 17 in three dimensions at infinite Pr and by T ≥ 0. 8473R^- 5/ 17 in two dimensions at arbitrary Pr , where R is the heat Rayleigh number proportional to the injected flux.