熱滲透現象與熱分子壓力差之研究

Abstract

本文研究了有界凸域中的熱滲透（thermal transpiration）現象。我們採用帶截斷勢能的波茲曼方程。在邊界條件的設定上，我們將邊界劃分為漫反射區域與入射區域。我們在加權的 L^\infty 空間中建立了解的存在性。 此外，我們考慮一個凸域，其中中間部分施加漫反射邊界條件，而兩端則施加入射邊界條件。我們首先考慮兩端具有相同壓力但不同溫度的 Maxwell 分佈。左端溫度設為 1，右端溫度設為 T_2>1. 我們證明總通量 U(x) 的方向指向高溫端，並進一步推導出總通量的估計： U(x)>= C(1-\frac{1}{\sqrt{T_2}}). 另外，我們指出當兩端的壓力 P_1,P_2 與溫度 T_1,T_2 滿足關係 \frac{P_1}{P_2}=\sqrt{\frac{T_1}{T_2}}, 時，解的總通量為 O(\frac{1}{\kappa}) 的量級。此結果與 Knudsen 於 1909 年提出的熱分子壓差（thermomolecular pressure difference）現象一致。

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, $T_1=1$ and $T_2>1$. We prove that the total flux $U(x)$ is directed toward the hot end. Furthermore, we derive an estimate for the total flux: U(x)>= C(1-\frac{1}{\sqrt{T_2}}). In addition, we show that when the pressures P_1,P_2 and temperatures T_1,T_2 on the two ends satisfy the relation \frac{P_1}{P_2}=\sqrt{\frac{T_1}{T_2}} the total flux of the solution is of order O(\frac{1}{\kappa}). This result is consistent with Knudsen's finding of thermomolecular pressure difference in 1909.