Bounds on Rayleigh-B\'enard convection with imperfectly conducting plates

We investigate the influence of the thermal properties of the boundaries in turbulent Rayleigh-B\'enard convection on analytical upper bounds on convective heat transport. We model imperfectly conducting bounding plates in two ways: using idealized mixed thermal boundary conditions of constant Biot number $\eta$, continuously interpolating between the previously studied fixed temperature ($\eta = 0$) and fixed flux ($\eta = \infty$) cases; and by explicitly coupling the evolution equations in the fluid in the Boussinesq approximation through temperature and flux continuity to identical upper and lower conducting plates. In both cases, we systematically formulate a bounding principle and obtain explicit upper bounds on the Nusselt number $Nu$ in terms of the usual Rayleigh number $Ra$ measuring the average temperature drop across the fluid layer, using the ``background method'' developed by Doering and Constantin. In the presence of plates, we find that the bounds depend on $\sigma = d/\lambda$, where $d$ is the ratio of plate to fluid thickness and $\lambda$ is the conductivity ratio, and that the bounding problem may be mapped onto that for Biot number $\eta = \sigma$. In particular, for each $\sigma > 0$, for sufficiently large $Ra$ (depending on $\sigma$) we show that $Nu \leq c(\sigma) R^{1/3} \leq C Ra^{1/2}$, where $C$ is a $\sigma$-independent constant, and where the control parameter $R$ is a Rayleigh number defined in terms of the full temperature drop across the entire plate-fluid-plate system. In the $Ra \to \infty$ limit, the usual fixed temperature assumption is a singular limit of the general bounding problem, while fixed flux conditions appear most relevant to the asymptotic $Nu$--$Ra$ scaling even for highly conducting plates.