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∈W2,∞ and κ satisfies a smallness condition with respect to α, we find Nu≲Ra12+∥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∈W3,∞ , 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 Ra12 and Ra512 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⩾Ra57 , we improve the Ra12 -upper bound, showing Nu≲α,κRa37, where ≲α,κ hides an additional dependency of the implicit constant on α and κ.

中文翻译：

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粗糙边界上具有纳维滑移条件的浮力驱动流的边界

我们考虑在高度函数描述的两个水平粗糙壁处具有纳维滑移和固定温度边界条件的二维瑞利-贝纳德对流H。我们证明了努塞尔数的严格上限 努 捕获对边界曲率的依赖性κ和（非常数）摩擦系数α明确地。如果 Hε瓦2,无穷大 和κ满足较小条件α， 我们发现 努≲拉12+∥κ∥无穷大, 在哪里 拉 是瑞利数，它与预测的 Spiegel-Kraichnan 标度一致，当κ = 0。此界限是通过边界处小条带中的局部规律性估计获得的。什么时候 Hε瓦3,无穷大 , 函数κ和α足够小 L无穷大 和普朗特数 普罗 足够大，我们使用背景场​​方法证明上限，该方法在 拉12 和 拉512 具有不平凡的依赖α和κ。这些界限与 Drivas 中的结果一致等人（2022年菲尔. 跨。R.苏克。 A 38020210025）用于平坦边界和恒定摩擦系数。此外，在政权中 普罗⩾拉57 ，我们改进了 拉12 - 上限，显示 努≲α,κ拉37, 在哪里 ≲α,κ 隐藏了隐式常量的额外依赖α和κ。