关于热输运和热分子压力差
On thermal transpiration and thermomolecular pressure difference
AI总结：
本文研究了有界凸域内的热输运现象，利用截断势的稳态玻尔兹曼方程，证明了在特定边界条件下的解存在性，并推导了总流量估计及热分子压力差的结论。
摘要：
本文演示了在有界凸域内的热输运现象。我们使用具有截断势的稳态玻尔兹曼方程。对于边界条件，我们将边界分为漫反射区域和入射区域。我们建立了在加权$ L^\infty $空间中解存在的存在性。此外，我们考虑了一个中间有漫反射边界条件、两端有入射边界条件的凸域。我们首先考虑两端压力相同但温度不同的麦克斯韦分布。我们证明总流量$ U(x) $指向热端。此外，我们推导了总流量的估计：$$ U(x)\geq C\left(1-\frac{1}{\sqrt{T_2}}\right) $$。此外，我们表明当两端的压力和温度满足关系$$ \frac{P_1}{P_2}=\sqrt{\frac{T_1}{T_2}} $$，解的总流量为阶$$ \mathcal{O}(\frac{1}κ) $$。这一结果与Knudsen于1909年发现的热分子压力差结论一致。
英文摘要：
In this article, we demonstrate the phenomenon of thermal transpiration in a bounded convex domain. We employ the stationary Boltzmann equation with a cutoff potential. For boundary condition, we partition the boundary into diffuse reflection and incoming regions. We establish the existence of solution in a weighted $L^\infty$ space. Furthermore, we consider a convex domain with diffuse reflection boundary condition in the middle and incoming boundary condition at the two ends. We first consider Maxwellians with the same pressure but different temperatures at the two ends. We prove that the total flux $U(x)$ is directed toward the hot end. Furthermore, we derive an estimate for the total flux: \begin{align} U(x)\geq C\left(1-\frac{1}{\sqrt{T_2}}\right). \end{align} In addition, we show that when the pressures and temperatures on the two ends satisfy the relation \begin{align} \frac{P_1}{P_2}=\sqrt{\frac{T_1}{T_2}}, \end{align} the total flux of the solution is of order $\mathcal{O}(\frac{1}κ)$. This result is consistent with Knudsen's finding of thermomolecular pressure difference in 1909.
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