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On thermal transpiration and thermomolecular pressure difference

Kai-Li Wang, I-Kun Chen

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

On thermal transpiration and thermomolecular pressure difference

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 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 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 . This result is consistent with Knudsen's finding of thermomolecular pressure difference in 1909.
Paper Structure (17 sections, 33 theorems, 255 equations)

This paper contains 17 sections, 33 theorems, 255 equations.

Table of Contents

  1. Introduction
  2. Introduction
  3. Organization of the article
  4. Preliminaries
  5. Estimates for $\nu$ and $K$
  6. Proposition of $L$
  7. Proposition of $S_\Omega$ and $J$
  8. Estimate of $Q_+$ and $Q_-$
  9. $L^2$ estimate
  10. $L^{\infty}$ estimate
  11. The existence of solution to \ref{['3.1']}
  12. the proof of Theorem \ref{['theorem 1.1']}
  13. Application to Thermal transpiration and thermomolecular pressure difference
  14. the model of the transpiration : the proof of Theorem \ref{['theorem 1.2']} and Theorem \ref{['corollary 1.1']}
  15. Appendix A: The $H^2$ estimate of Laplace equation with Robin boundary condition
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}^3$ be a bounded convex domain with $C^2$ boundary. Suppose that $\Omega$ is unsealed with respect to $\partial\Omega_1$ and satisfies the assumption assumption of boundary. Assume that $-3<\gamma\leq 1$ and that eq 1.2 holds. Then, for any $0< \alpha <1/4$ and $\kappa_0\ and then the equation eq 1.1 has a unique mild solution for $\kappa\geq \kappa_0$. Furthermore, H

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Remark 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • ...and 54 more