无限普朗特数下Bénard-Marangoni对流换热的边界条件
Bounds on heat transfer for Bénard-Marangoni convection at infinite Prandtl number
摘要：
利用Nusselt数$Nu$的上界作为Marangoni数$Ma$的函数，研究了具有无限普朗特数的流体层的Bénard-Marangoni对流中的垂直换热.最近Hagstrom和Doering利用温度场的背景方法证明了$ Nu\leq 0.838\,Ma^{2/7}$。在这项工作中，我们扩展了以前的背景方法分析，包括平衡参数，并推导出一个变分原理的约束$Nu$，表示在一个缩放的背景场，产生一个更好的约束比Hagstrom和Doering的制定在一个给定的$Ma$。使用分段线性，单调递减的配置文件，然后我们表明，$Nu \leq 0.803\,Ma^{2/7}$，降低了4.2%的前因子。然而，我们也证明了平衡参数的优化并不影响渐近缩放的最佳界实现与哈格斯特罗姆和Doering的原始配方。随后，我们利用凸优化优化$Nu$上的所有容许的背景字段，以及在两个较小的家庭的配置文件约束的单调性和凸性。结果表明，当背景场在近地表有非单调边界层时，$Nu \leq O(Ma^{2/7}(\ln Ma)^{-1/2})$是最优的，而在单调背景场类中，指数为2/7的幂律界是最优的。进一步分析我们的上限原理揭示了非单调性的作用，以及它如何在严格的数学论证中被利用。
英文摘要：
The vertical heat transfer in Bénard-Marangoni convection of a fluid layer with infinite Prandtl number is studied by means of upper bounds on the Nusselt number $Nu$ as a function of the Marangoni number $Ma$. Using the background method for the temperature field, it has recently been proven by Hagstrom & Doering that $ Nu\leq 0.838\,Ma^{2/7}$. In this work we extend previous background method analysis to include balance parameters and derive a variational principle for the bound on $Nu$, expressed in terms of a scaled background field, that yields a better bound than Hagstrom & Doering's formulation at a given $Ma$. Using a piecewise-linear, monotonically decreasing profile we then show that $Nu \leq 0.803\,Ma^{2/7}$, lowering the previous prefactor by 4.2%. However, we also demonstrate that optimisation of the balance parameters does not affect the asymptotic scaling of the optimal bound achievable with Hagstrom & Doering's original formulation. We subsequently utilise convex optimisation to optimise the bound on $Nu$ over all admissible background fields, as well as over two smaller families of profiles constrained by monotonicity and convexity. The results show that $Nu \leq O(Ma^{2/7}(\ln Ma)^{-1/2})$ when the background field has a non-monotonic boundary layer near the surface, while a power-law bound with exponent 2/7 is optimal within the class of monotonic background fields. Further analysis of our upper-bounding principle reveals the role of non-monotonicity, and how it may be exploited in a rigorous mathematical argument.
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