A new analytical solution of microchannel gas flow

Abstract

A two-dimensional, isothermal, compressible and subsonic gas flow in a microchannel with slowly varying cross-section is analytically investigated. The microchannel gas flow is described by the Burnett momentum equation that comprises the shear stress tensor of the second order. The application of the Burnett equation allows the introduction of the second-order boundary condition for the gas velocity slip on microchannel walls. The model solution is based on perturbation analysis. The analytical solution procedure incorporates the exact physical relation between Mach, Reynolds and Knudsen numbers, which enables the evaluation of the order of all terms within the balance equations and the boundary conditions and provides high accuracy of the model. It is shown that the Burnet equation has the same form as the Navier-Stokes equation for low Mach number flows in channels with slowly varying cross sections. The obtained analytical results are in excellent agreement with available measured data and numerical solutions of the Boltzmann equation. The presented analytical model is new since previous solutions are limited to microchannels with uniform cross sections, while here the presented solution is obtained for microchannels with varying cross sections. The presented model is a useful tool for the design and analysis of gas flows in microchannels, and it provides accurate benchmark results for the validation of the numerically obtained results.