On thermal transpiration and thermomolecular pressure difference

TL;DR

This paper investigates thermal transpiration phenomena in convex domains using the Boltzmann equation, establishing existence of solutions and deriving flux estimates that align with classical thermomolecular pressure difference observations.

Contribution

It provides a rigorous mathematical analysis of thermal transpiration, including existence results and flux estimates, connecting kinetic theory with classical thermomolecular pressure difference.

Findings

Total flux directed toward the hot end.

Flux estimate: $U(x) \,\geq\, C(1 - 1/\sqrt{T_2})$.

Flux order: $\mathcal{O}(1/\kappa)$ under specific pressure-temperature relations.

Abstract

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.