# Interdisciplinary Applied Mathematics

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The fundamental assumption in this analysis is the dynamic similarity of microflows with rarefied flows encountered in a low-pressure environment.

This assumption is justified theoretically based on the analysis of the Boltzmann equation for small Reynolds number and Knudsen number (Sone, 2002). Based on this assumption, gas microflows are simulated subject to slip boundary conditions. In order to demonstrate the effects of rarefaction and slip in microgeometries we compare these findings with the continuum no-slip solutions whenever possible. In gas microflows we encounter four important effects:

rarefaction,

compressibility,

viscous heating, and

thermal creep.

In particular, we investigate the competing effects of compressibility and rarefaction, which result in a nonlinear pressure distribution in microchannels in the slip flow regime. Curvature in pressure distribution is due to the compressibility effects, and it increases with increased inlet to outlet pressure ratios across the channels. The effect of rarefaction is to reduce the curvature in the pressure distribution. This finding is consistent with the fact that the pressure distribution becomes linear as the free-molecular flow regime is reached (Kennard, 1938). The viscous heating effects are due to the work done by viscous stresses (dissipation), and they are important for microflows, especially in creating temperature gradients within the domain even for isothermal surfaces. The thermal creep (transpiration) phenomenon is a rarefaction effect. For a rarefied gas flow it is possible to start the flow with tangential temperature gradients along the channel surface. In such a case the gas molecules start creeping from cold toward hot (Kennard, 1938; Kruger et al., 1970). Thermal creep effects can be important in causing variation of pressure along microchannels in the presence of tangential temperature gradients (Fukui and Kaneko, 1988). This mechanism is also significant for transport through porous media in atmospheric conditions (Loeb, 1961; Vargo and Muntz, 1996). Other temperature-induced flows are studied in Section 5.2.

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