Interdisciplinary Applied Mathematics

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mean-square molecular speed. This results in tc « 10~10 seconds for air under standard conditions. This time scale should be compared to a typical scale in the microdomain to determine the validity of thermodynamic equilibrium.

In engineering practice, it is convenient to lump all the molecular effects to space-averaged macroscopic or continuum-based quantities, such as the fluid density, temperature, and velocity. It is important, however, to determine the limitations of these continuum-based descriptions. Specifically, we ask: 1

At what scales will the statistical fluctuations be significant?

It turns out that sampling a volume that contains 10,000 molecules results in 1% statistical fluctuations in the averaged quantities. Based on that, for air at standard conditions the smallest sample volume that will result in 1% statistical variations is about 3.7 x 10~22 m3. If we try to measure the macroscopic gradients (like velocity, density, and temperature) in threedimensional space, one side of our sampling volume will be about 65nm.

A key nondimensional parameter for gas microflows is the Knudsen number, which is defined as the ratio of the mean free path over a characteristic geometric length or a length over which very large variations of a macroscopic quantity may take place. The Knudsen number is related to the Reynolds and Mach numbers as follows:

Kn = у L

Гутт M V YRe’


In complex microgeometries where three-dimensional spatial gradients are expected, definitions of instantaneous macroscopic values and their gradients become problematic for flows with Kn > 1 as the concept of macroscopic property distribution breaks down. However, for microchannels with large aspect ratio (width to height), we can perform spanwise space averaging to define an averaged velocity profile, and thus define the equivalent macroscopic quantities.

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