Interdisciplinary Applied Mathematics

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above equation provides an implicit relation for P; the pressure distribution for a first-order boundary condition is obtained explicitly by neglecting the second-order terms (O(Kn2)) in equation (4.9) .

The formula for the flowrate has been tested directly using experimental data reported in (Arkilic et al., 1997), as well as simulation results obtained using the program p,Flow (see Section 14.1). In Figure 4.1 we plot the normalized flowrate



versus the pressure ratio П. Three cases are included, corresponding to no-slip compressible air flow (lower curve; equation (4.7)), rarefied air flow (middle curve; equation (4.6)), and rarefied helium flow (upper curve; equation (4.6)). The open circles correspond to microchannel experimental data in (Arkilic et al., 1997), and solid circles denote the corresponding numerical simulation results of rarefied helium flow. It is seen that there is a significant mass flowrate increase, especially for helium flow due to rarefaction (Kn = 0.165), and that in this range there are no significant deviations between the formula and the simulations; the latter correspond to solutions of full compressible Navier-Stokes equations with rarefaction effects expressed via equation (2.26).

The differences between simulation, experiments, and analysis are evaluated more accurately by computing the ratio of slip mass flowrate to the corresponding no-slip flowrate predicted by equation (4.8) as a function of pressure ratio in Figure 4.2. Microchannel helium flow experiments in (Arkilic et al., 1997), show a maximum of 10% deviation from the first-order theoretical curve. The deviations are more significant especially for low pressure ratio cases. The gain in the mass flowrate due to slip effects is reduced as the pressure ratio is increased. This is a trend expected by equation (4.8) as well. However, due to significant scatter in the experimental data, it is difficult to determine the slope of this decrease. Comparison of the high-order formula with the first-order formula shows about 8% deviations    for    small    pressure    ratios;    the    deviations    are reduced    for    higher

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