# Interdisciplinary Applied Mathematics

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In order to identify the limitations and accuracy of an incompressible model, we compare the results of the incompressible model, with the compressible model for the same Reynolds and Knudsen numbers. We present the variation    of    mass flowrate    in    the    channel    versus    the drag    on    the top

channel wall, both normalized with their no-slip incompressible counterparts in Figure 3.27. Drag reduction due to rarefaction effects is seen. For example, for Kn = 0.128 flow, about 30% drag reduction is observed compared to the no-slip case. Both models predict a reduction in mass flowrate for slip flow. This is due to the reduction in the volumetric flowrate as seen in Figure 3.26. The mass flowrate predicted by the incompressible model

FIGURE 3.26. Streamwise velocity distribution along the center of the grooved channel. y = 0 corresponds to the lower boundary of the grooves, and y = corresponds to the top wall.

 Kn=0.0 Д A Kn=0.042 Д ▲ Kn=0.086 Kn=0.128 A Incompressible A Compressible Д ▲

0.8 1

Drag/Dragns

FIGURE 3.27. Variation of mass flowrate versus drag force for the grooved channel, normalized with values of the corresponding incompressible no-slip model.

is less than the mass flowrate predicted by the compressible model. This is due to the inability of the incompressible model to predict the variations in fluid density.

The temperature contours for no-slip and slip (Kn = 0.086) flows are

Kn=0.0    Kn=0.086

U

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