Analytical modeling of rarefied Poiseuille flow in microchannels

Abstract

The prediction of rarefied gas flow in the transition regime remains an obstacle due to its complexity. The moment method is commonly used in kinetic theory of gases to obtain approximate solutions in this regime, though substantial effort is still needed either numerically or mathematically. In this article Grad’s 13-moment method was first applied to the derivation of a new governing equation for Poiseuille flow in microchannels and novel boundary conditions were introduced. For validation purposes, Poiseuille flow in micro channels was investigated. The flow rate and velocity profiles were obtained analytically. The results were compared with the linearized Boltzmann equation and available experimental data. The error is within 10% in both slip and transition regimes. This shows a substantial improvement over first-, and second-order slip models where the errors can be as high as several tens or even hundreds of percent. The major attraction of the current approach is that it allows a simple analytical equation to explain the effects of various forces on the velocity profiles, flow rates, and pressure distributions.