A new analytic solution of the Navier-Stokes equations for microchannel flows

Abstract

In this article we present a new analytic solution of the Navier-Stokes equations for microchannel flows. The solution is based on the concept of the continuum approach using the Chapman-Enskog method, but built upon the proposal to introduce a hyperbolic tangent function of Kn number in the power series of the distribution function and slip boundary condition. The physics behind the mathematical modification are discussed. With the slip boundary condition accurate to O(tanh(Kn)), the solution of the Navier-Stokes equations is extended successfully to the transition flow regime. The analytic solutions are compared with results of DSMC in both slip flow and transition flow regimes. Satisfactory agreements on the velocity profiles and pressure distributions have been achieved. The extension of the upper Knudsen number limits of continuum approach is significant in molecular gas dynamics.