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

We consider two-dimensional Rayleigh–Bénard 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 Nu which capture the dependence on the curvature of the boundary κ and the (non-constant) friction coefficient α explicitly. If h ∈ W 2 , ∞ and κ satisfies a smallness condition with respect to α , we find Nu ≲ Ra 1 2 + ∥ κ ∥ ∞ , where Ra is the Rayleigh number, which agrees with the predicted Spiegel–Kraichnan scaling when κ = 0. This bound is obtained via local regularity estimates in a small strip at the boundary. When h ∈ W 3 , ∞ , the functions κ and α are sufficiently small in L ∞ and the Prandtl number Pr is sufficiently large, we prove upper bounds using the background field method, which interpolate between Ra 1 2 and Ra 5 12 with non-trivial dependence on α and κ . 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 ⩾ Ra 5 7 , we improve the Ra 1 2 -upper bound, showing Nu ≲ α , κ Ra 3 7 , where ≲ α , κ hides an additional dependency of the implicit constant on α and κ .