Interdisciplinary Applied Mathematics

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standard conditions is on the order of 10~10 seconds, and the time step of the DSMC method should be smaller than the mean collision time. Hence, a DSMC algorithm must run at least 1010 time steps for this microchannel prior to settling into a steady-state condition. In practice, due to the simple geometry, settling into a steady state may occur much earlier. However, for a large aspect ratio channel with geometric complications common in microsystems, the above order of magnitude analysis will be valid.

4. Extensive Number of Molecules: If we discretize a rectangular domain of 1 mm x 100 yrm x 1 yrm under standard conditions for Kn = 0.1 flow,    we    will    need at    least    30    cells    per    1 yrm    length    scale. This

results in a total of 2.7 x 109 cells. Each of these cells should contain at least 20 simulated molecules, resulting in a total of 5.4 x 1010 particles. Combined with the number of time step restrictions given above, low-speed microfluidic simulation with DSMC easily goes beyond the capabilities of most computers.

An alternative treatment to overcome the extensive number of simulated molecules and long integration times is utilization of the dynamic similarity of low-pressure rarefied gas flows to gas microflows under atmospheric conditions. The key parameters for the dynamic similarity is the geometric similarity, and matching of the flow Knud-sen, Mach, and Reynolds numbers. Performing actual experiments under dynamically similar conditions may be very difficult. However, parametric studies via numerical simulations are possible. The funda-

mental question to answer for such an approach is whether a specific, unforeseen microscale phenomenon is missed with the dynamic similarity approach. However, all numerical simulations are inherently model-based. Unless microscale-specific models are implemented in the algorithm, we will not be able to obtain more physical information from an atomistic simulation than from a dynamically similar low-pressure simulation.

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