Interdisciplinary Applied Mathematics

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In many fluid-mechanical applications an analogy between different ge-

ometric scales and dynamic conditions can be obtained by invoking the concept of dynamic similarity. This enables us to determine the performance of a fluidic device by experimenting on a scaled prototype under similar physical conditions, characterized by a set of nondimensional parameters, such as the Reynolds, Mach, Prandtl, and Knudsen numbers. It is therefore appropriate to pose this question for gas microflows:

Are the low-pressure rarefied gas flows dynamically similar to the gas microflows?

The answer to this question depends on the onset state of statistical fluctuations and also on wall surface effects. For example, at standard conditions for air,    the    value of    the    Knudsen    number    Kn    =1 is    obtained    at    about

a 65 nm length scale. For smaller length scales, corresponding to higher Knudsen number regimes, the average macroscopic quantities cannot be defined. However, for low-pressure flows, for example, at 100 Pa and 270 K, the 1% statistical scatter limit sets in at about L « 0.65pm, since 8 ~ p-1/3. However, at this low-pressure condition, Kn = 1 corresponds to the characteristic length of about 65pm. This length scale is two orders of magnitude    larger than the    one    at the    onset    of    the    statistical    scatter    at

these conditions. Therefore, macroscopic property distributions can be defined without any significant statistical fluctuations. Hence, for dynamic similarity approaches for gas microflows to be valid, the onset of statistical scatter should be carefully considered. Also, Figure 1.11 shows a dense gas region where the Kn = 0.1 line crosses the 1% statistical scatter line. For dense gas flows in this region, the Navier-Stokes equations are valid, but the    results    show    large    statistical    deviations    due    to the    onset of    the

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