# Interdisciplinary Applied Mathematics

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Rarefaction effects become more important as the Knudsen number increases and thus pressure drop, shear stress, heat flux, and corresponding mass flowrate cannot be predicted from flow and heat transfer models based on the continuum hypothesis. On the other hand, models based on kinetic gas    theory    concepts    are    not    appropriate    either,    except    in the    very    high

Knudsen number regime corresponding to near vacuum conditions or very small clearances. The appropriate flow and heat transfer models depend on the range of the Knudsen number. A classification of the different flow regimes is given in (Schaaf and Chambre, 1961):

for Kn < 10~2 the fluid can be considered as a continuum, while

for Kn > 0(10) it is considered a free-molecular flow.

A rarefied gas can be considered neither an absolutely continuous medium nor a free-molecular flow in the Knudsen number range between 10~2 and 10. In that region, a further classification is needed, i.e.,

slip flow (10~2 < Kn < 0.1), and

transition flow (0.1 < Kn < 10).

This classification is based on empirical information and thus the limits between the different flow regimes may depend on the problem geometry.

L (microns)

FIGURE 1.11. Limits of approximations in modeling gas microflows. L (vertical axis) corresponds    to    the    characteristic    length    and n/n0    is    the    number    density

normalized with corresponding atmospheric conditions. The lines that define the various Knudsen number regimes are based on air at isothermal conditions at T = 273 K. Statistical fluctuations are significant below the line L/S = 20.

n/n0

This separation in different regimes is plotted in Figure 1.11, where we define the various flow regions as a function of the characteristic length scale L in microns, and also the number density. In addition, we have included a line that corresponds to L/S = 20, below which statistical fluctuations are present; this line corresponds to 1% fluctuations in macroscopic measurements.

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