We compute steady planar incompressible flows and wall shapes that maximize the rate of heat transfer ( $Nu$ ) between hot and cold walls, for a given rate of viscous dissipation by the flow ( $Pe^2$ ), with no-slip boundary conditions at the walls. In the case of no flow, we show theoretically that the optimal walls are flat and horizontal, at the minimum separation distance. We use a decoupled approximation to show that flat walls remain optimal up to a critical non-zero flow magnitude. Beyond this value, our computed optimal flows and wall shapes converge to a set of forms that are invariant except for a $Pe^{-1/3}$ scaling of horizontal lengths. The corresponding rate of heat transfer $Nu \sim Pe^{2/3}$ . We show that these scalings result from flows at the interface between the diffusion-dominated and convection-dominated regimes. We also show that the separation distance of the walls remains at its minimum value at large $Pe$ .

中文翻译：

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稳定平面对流的最佳壁形状和流动

我们计算稳定的平面不可压缩流动和壁形状，以最大化传热速率（ $Nu$ ）在热壁和冷壁之间，对于给定的流动粘性耗散率（ $Pe^2$ ），壁上具有无滑移边界条件。在没有流动的情况下，我们从理论上证明，最佳的墙壁是平坦且水平的，且间隔距离最小。我们使用解耦近似来表明，平坦的壁在达到临界非零流量大小时仍然保持最佳状态。超过这个值，我们计算出的最佳流动和壁形状收敛到一组不变的形式，除了 $Pe^{-1/3}$ 水平长度的缩放。相应的传热速率 $Nu \sim Pe^{2/3}$ 。我们表明，这些尺度是由扩散主导和对流主导状态之间的界面处的流动引起的。我们还表明，墙壁的间距在很大程度上保持在最小值 $Pe$ 。