# Interdisciplinary Applied Mathematics

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In Figure 4.4 we present the variation of velocity slip along the channel walls normalized    with    centerline    velocity    at    the    channel    inlet.    Due    to the

pressure drop along the channel, the local mean free path increases, resulting in an increase in the local Kn. Also, the density along the channel

X/L    X/L

FIGURE 4.3. Left: Pressure distribution for different pressure ratios along the center of microchannels    for    air    flow.    Right:    Deviation    from    linear    pressure    drop

for П = Pi/Po = 2.58 air flow. Cases 1, 2, and 3 correspond to different channel thickness given in Table 4.1.

FIGURE 4.4. Velocity    slip    variation    on    channel    surface    for    cases    described    in

Table 4.1.

decreases, and thus the average velocity in the channel increases toward downstream to conserve mass. These two effects together increase the slip velocity along the channel walls, as seen in Figure 4.4.

To investigate the compressibility and rarefaction effects further, the de-

FIGURE 4.5. Left: Deviation from linear pressure drop for air and helium flows (h = 0.65 pm,    П =    Pi/Po    =    3.5,    Po    = 1    atm).    Right: Deviation from linear    pres

sure drop for П = Pi/P0 = 2.02, nitrogen flow (h = 1.25im); circles correspond to experimental data of (Pong et al., 1994).

viation from linear pressure distribution for helium and air flows in identical channels with identical inlet and outlet conditions corresponding to pressure ratio П = 3.5 is    given    in Figure    4.5    (left).    Here    we    see    that    un

like the experimental findings, the curvature in the pressure distribution for helium is less pronounced compared to the air microflow. This trend should be expected, since for the same pressure ratio and outlet pressure, the local Mach number for helium flow is smaller than the Mach number for air flow. Also, the rarefaction effects for helium flow are larger than those of the air flow due to the relatively large mean free path of helium molecules compared to air. The finding from the simulation results that rarefaction causes the opposite effect than compressibility is also evident from the analytical expression (equation (4.9)). This is shown in Figure 4.5 (right), where we plot the analytical predictions taking into account first-and second-order Knudsen number effects. The simulation results for nitrogen flow    of Kn    =    0.156    corresponding to the    experiments    of    (Pong    et    al.,

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