Bounds on Rayleigh-Benard convection with general thermal boundary   conditions. Part 1. Fixed Biot number boundaries
  We investigate the influence of the thermal properties of the boundaries in turbulent Rayleigh-Benard convection on analytical bounds on convective heat transport. Using the Doering-Constantin background flow method, we systematically formulate a bounding principle on the Nusselt-Rayleigh number relationship for general mixed thermal boundary conditions of constant Biot number \eta which continuously interpolates between the previously studied fixed temperature ($\eta = 0$) and fixed flux ($\eta = \infty$) cases, and derive explicit asymptotic and rigorous bounds. Introducing a control parameter R as a measure of the driving which is in general different from the usual Rayleigh number Ra, we find that for each $\eta > 0$, as R increases the bound on the Nusselt number Nu approaches that for the fixed flux problem. Specifically, for $0 < \eta \leq \infty$ and for sufficiently large R ($R > R_s = O(\eta^{-2})$ for small \eta) the Nusselt number is bounded as $Nu \leq c(\eta) R^{1/3} \leq C Ra^{1/2}$, where C is an \eta-independent constant. In the $R \to \infty$ limit, the usual fixed temperature assumption is thus a singular limit of this general bounding problem. 
Ralf W. Wittenberg