The analytical and numerical solutions for gaseous slip flows in micro‐channels

Abstract

The present work studies analytically as well as numerically gaseous flow in micro‐channels. The working fluids are nitrogen and helium. The proposed model assumes the fluid is a continuum, but employs a slip boundary condition on the channel wall. Although slip flow in micro‐channels can be investigated by solving numerically the compressible Navier‐Stokes equations, as was done previously by several investigators, the hyperbolic‐parabolic character of the equations makes it very inefficient. The results of the present work show that they can be predicted accurately by solving the compressible boundary‐layer equations. The parabolic character of the boundary‐layer equations renders the present method a very efficient and accurate tool in studying slip flows. The results also demonstrate that diffusion is the dominant mechanism in momentum and energy transfers in micro‐channel flows. The convective terms in the boundary‐layer equations can be neglected when compared with the diffusive terms. This reduces the governing equations to a simple parabolic equation which can be solved analytically. Both analytical and numerical solutions compare quite well with experimental data. The slip boundary condition is the result of rarefaction, which is due to the incomplete momentum and energy exchanges between gas molecules and the walls. Our results show that the slip condition has decisive effects on the velocity and mass flow rate of the flow and has to be taken into account.