Interdisciplinary Applied Mathematics

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the numerator of equation (5.9) are absent, and the formula is further simplified; see more details in (Beskok and Karniadakis, 1992; Beskok and Karniadakis, 1994).


The above analytical results can be used to validate computer programs for microfluidic applications. Here we present computations obtained with the spectral element program p,Flow. Comparisons are performed up to Kn = 0.15, and the results are presented in Figure 5.6. The dashed line and the solid line show the drag reduction predicted by the first- and high-order slip flow theory without thermal creep effects, respectively. The triangles correspond to numerical predictions with high-order slip flow theory, and the circles correspond to numerical predictions with high-order slip flow theory including in this case the thermal creep effects (here Ec = 1.0, Re = 1.0, and = 1.0). The differences between the analytical and numerical results are negligible.


The aforementioned simplified analysis can also be used to explain the drag reduction observed in the experiments reported in (Pfahler et al., 1991). For comparison, the experimental results are also plotted in Figure


5.6. The ratio predicted from equation (5.9) for Kn = 0.088 corresponding

FIGURE 5.6. Ratio of Fanning friction coefficients of slip flow to no-slip flow in a pressure-driven channel. (Parameters for thermal creep contribution are Ec = 1.0, Re = 1.0, and = 1.)


to the helium flow (case JP9 in (Pfahler et al., 1991)) is 0.79 in reasonable agreement with    the    measured    value    0.8    to    0.85.    The    nitrogen flow    gives a


slightly greater drag reduction of about 0.86 compared to the theoretical predictions of 0.9 for Kn = 0.04. Our predictions assume accommodation coefficient av = 1 and that compressibility effects in the channels are neglected. Thus, the Knudsen number variation in the channels due to compressibility effects is not taken into account, and the Knudsen number is calculated by taking the arithmetic average of the inlet and outlet Knudsen numbers of the microchannel. Furthermore, isothermal flow conditions are assumed, and thermal creep effects are neglected. For channel thicknesses significantly smaller, corresponding to Kn > 0.1, the experimental results show a strong dependence of the ratio of drag coefficients on the Reynolds number, which is not predicted by the above analysis.

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