Bounds on heat flux for Rayleigh-B\'{e}nard convection between Navier-slip fixed-temperature boundaries

We study two-dimensional Rayleigh-B'{e}nard convection with Navier-slip, fixed temperature boundary conditions and establish bounds on the Nusselt number. As the slip-length varies with Rayleigh number $\mathrm{R}\mathrm{a}$, this estimate interpolates between the Whitehead-Doering bound by ${\mathrm{R}\mathrm{a}}^{\frac{5}{12}}$ for free-slip conditions [13] and the classical Doering-Constantin ${\mathrm{R}\mathrm{a}}^{\frac{1}{2}}$ bound [4].