Treatments of Micro-channel Flows Revisited: Continuum Versus Rarified Gas Considerations

Abstract

There are numerous treatments of micro-channel flows available that point out the breakdown of the Navier–Stokes equations under molecular flow conditions when continuum conditions should still apply. The wrong conclusion regarding the validity of the Navier–Stokes equations comes about from the missing mass diffusion terms in the continuity equation. This term also affects the Navier–Stokes equations and yields improved theoretical results for flows with strong pressure and temperature gradients. Rarified gas treatments are often applied, although the fluid mechanics conditions are still such, that continuum equations should work. In the paper, the missing mass diffusion term is added, and it is shown that those extended fluid mechanics equation (EFME) allows micro-channel flows to be treated in the so-called slip regime. Hence, the continuum approach for flow treatment holds in micro-channel flows, in the flow regimes where modeling of wall interactions is applied these days. The paper also describes the treatments of micro-channel flows in the slip-flow regime by the rarified gas flow treatment method of Shapiro and Seleznev, and the results are compared with the corresponding results of the EFME. Good agreement was obtained, but differences exist regarding the wall interactions, which are explained, and suggested to use both methods to obtain a deeper insight into molecule–wall interactions in micro-channel flows. The EFME claim that large pressure and temperature gradients are the reasons for the differences between experimental and theoretical results of micro-channel flows. Such differences also exist in other fluid flows with strong property gradients, e.g., in shock waves. Flows of this kind are also treated in this paper in a way to show that the EFME have a wide range of applications.