Bounds on buoyancy driven flows with Navier-slip conditions on rough boundaries

We consider two-dimensional Rayleigh-B\'enard convection with Navier-slip and fixed temperature boundary conditions at the two horizontal rough walls described by the height function $h$. We prove rigorous upper bounds on the Nusselt number $\text{Nu}$ which capture the dependence on the curvature of the boundary $\kappa$ and the (non-constant) friction coefficient $\alpha$ explicitly. If $h\in W^{2,\infty}$ and $\kappa$ satisfies a smallness condition with respect to $\alpha$, we find $$\text{Nu}\lesssim \text{Ra}^{\frac{1}{2}}+\|\kappa\|_{\infty}\,,$$ where $\text{Ra}$ is the Rayleigh number, which agrees with the predicted Spiegel-Kraichnan scaling when $\kappa=0$. This bound is obtained via local regularity estimates in a small strip at the boundary. When $h\in W^{3,\infty}$, the functions $\kappa$ and $\alpha$ are sufficiently small in $L^{\infty}$ and the Prandtl number $\Pr$ is sufficiently large, we prove upper bounds using the background field method, which interpolate between $\text{Ra}^{\frac{1}{2}}$ and $\text{Ra}^{\frac{5}{12}}$ with non-trivial dependence on $\alpha$ and $\kappa$. These bounds agree with the result in Drivas et al (2022 Phil. Trans. R. Soc. A 380 20210025) for flat boundaries and constant friction coefficient. Furthermore, in the regime $\Pr\geq \text{Ra}^{\frac 57}$, we improve the $\text{Ra}^{\frac 12}$-upper bound, showing $$\text{Nu}\lesssim_{\alpha,\kappa}\text{Ra}^{\frac{3}{7}}\,,$$ where $\lesssim_{\alpha,\kappa}$ hides an additional dependency of the implicit constant on $\alpha$ and $\kappa$.