Perturbation analysis on gas flow in a straight microchannel

Abstract

In the present paper, a steady subsonic gas flow either in a circular micropipe or in a planar microchannel driven by pressure within the slip flow regime is studied theoretically by using a perturbation expansion method to solve compressible Navier-Stokes equations. The isothermal flow assumption used in previous theoretical studies is given up. High-order boundary conditions of velocity slip and temperature jump are adopted at the wall. The set of dimensionless governing equations with two small similarity parameters, namely, the ratio of height to length ε, and the Knudsen number Kn, is approximated successively by using the perturbation expansions. The various cases such as ε≪Kn2, ε∼Kn2, and ε∼Kn1.5 are studied in detail. Explicit analytical solutions for pressure, density, velocity, temperature, and mass flow rate are obtained up to order of O(Kn2). It is shown that the solution formulas for long channels (ε≪Kn2) in lower order are in exact agreement with previous theoretical results. In particular, it is proved that the isothermal flow assumption is indeed reliable for relatively lower-order expansions. However, for higher-order expansions, the flow cannot be considered as isothermal, and the higher-order temperature correction is also given. The present high-order perturbation solutions can be applied even to a relatively shorter channel, and the results agree very well with those by the direct simulation Monte Carlo approach.