On thermal transpiration and thermomolecular pressure difference

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 $\mathcal{O}(\frac{1}{\kappa})$. This result is consistent with Knudsen's finding of thermomolecular pressure difference in 1909.