标题： 无限普朗特数内部加热对流的严格尺度定律 显示英文标题

标题： Rigorous scaling laws for internally heated convection at infinite Prandtl number

摘要： 新的界限被证明了关于平均垂直对流热输送，$\overline{\langle wT \rangle}$，对于无限普朗特数极限下的均匀内部加热（IH）对流。对于在等温边界之间的水平周期层中的流体，我们证明了$\overline{\langle wT \rangle} \leq \frac12 - c R^{-2}$，其中$R$是一个无量纲的“通量”雷诺数，用于量化内部加热的强度，以及$c = 216$。然后，$\overline{\langle wT \rangle} = 0$对应于仅由传导引起的垂直热输送，而$\overline{\langle wT \rangle} > 0$表示由于对流运动导致的向上垂直热输送的增强。 如果下边界是热绝缘体，则我们得到$\overline{\langle wT \rangle} \leq \frac12 - c R^{-4}$，其中$c\approx 0.0107$。这个结果意味着努塞尔数$Nu$，定义为总传热与导热传热的比值，满足$Nu \lesssim R^{4}$。这两个界通过将背景方法与流体温度的最小原理相结合，并利用 Hardy--Rellich 不等式来利用垂直速度和温度之间的联系而获得。在两种情况下，对$R$的幂律依赖性改进了之前已知的最佳界限，尽管这些界限在无限和有限普朗特数下都有效，但它们随着$R$指数级接近统一界限。 显示英文摘要

摘要： New bounds are proven on the mean vertical convective heat transport, $\overline{\langle wT \rangle}$, for uniform internally heated (IH) convection in the limit of infinite Prandtl number. For fluid in a horizontally-periodic layer between isothermal boundaries, we show that $\overline{\langle wT \rangle} \leq \frac12 - c R^{-2}$, where $R$ is a nondimensional `flux' Rayleigh number quantifying the strength of internal heating and $c = 216$. Then, $\overline{\langle wT \rangle} = 0$ corresponds to vertical heat transport by conduction alone, while $\overline{\langle wT \rangle} > 0$ represents the enhancement of vertical heat transport upwards due to convective motion. If, instead, the lower boundary is a thermal insulator, then we obtain $\overline{\langle wT \rangle} \leq \frac12 - c R^{-4}$, with $c\approx 0.0107$. This result implies that the Nusselt number $Nu$, defined as the ratio of the total-to-conductive heat transport, satisfies $Nu \lesssim R^{4}$. Both bounds are obtained by combining the background method with a minimum principle for the fluid's temperature and with Hardy--Rellich inequalities to exploit the link between the vertical velocity and temperature. In both cases, power-law dependence on $R$ improves the previously best-known bounds, which, although valid at both infinite and finite Prandtl numbers, approach the uniform bound exponentially with $R$.