Interdisciplinary Applied Mathematics

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The slip model (2.43) gives good agreement with DSMC data and the linearized Boltzmann solutions for the nondimensional velocity profile, but it does not predict correctly the flowrate. This is expected, since the Navier-Stokes equations are invalid in this regime. In fact, the dynamic viscosity, which defines the diffusion of momentum due to the intermolecular collisions, must be modified to account for the increased rarefaction effects. The kinetic theory description for dynamic viscosity requires


Mo ~ Avp,


where v is the mean thermal speed. Using mean free path A in this relation is valid as long as intermolecular collisions are the dominant part of


momentum transport in the fluid (i.e., Kn ^ 1). However, for increased rarefaction, the intermolecular collisions are reduced significantly, and in the free-molecular flow regime, only the collisions of the molecules with the walls should be considered. Therefore, in free-molecular channel flow the diffusion coefficient should be based on characteristic length scale (channel height) and thus p « hvp (Polard and Present, 1948). Since the diffusion coefficient is based on A in slip or continuum flow regimes and h in the free-molecular flow regime, we propose to model the variation of diffusion coefficient with the following hybrid formula:


1



pvA



1


1 + Kn


which can be simplified to


p(Kn)



po



1 «


1 +Kn ’



(4.20)


where p0 is the dynamic viscosity of the gas at a specified temperature and p is the generalized diffusion coefficient. The variable diffusion coefficient model presented above is based on a simple analysis. Another point of view is to consider the ratio of intermolecular collisions of the molecules (fg) to the total number of collisions per unit time (i.e., sum of intermolecular and wall collisions fg + fw). Following (Thompson and Owens, 1975), the frequency of wall collisions in a channel section is (with width w, height h, and length dx)

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