# Interdisciplinary Applied Mathematics

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numbers as well as accommodation coefficients (Gampert, 1976). Here, we present simulation results    at    Re = 20.    In    Figure 4.14    (left)    we    plot    the    ve

locity slip along the wall for Kn = 0.03 obtained by using both the first- and high-order slip flow boundary conditions. It is seen that the slip effects are very dominant at the inlet of the channel, and both results predict velocity slip of about 50% of the incoming velocity UTO. However, these effects are reduced well below 10% in the developed region of the flow. It is evident from the plot that the slip velocity experiences a very steep change from

FIGURE 4.14. Developing flow in microchannel: Velocity slip (left) and variation of high-order slip coefficient b as a function of the streamwise location parameter Z (Kn = 0.03, Re = 20).

the inlet to approximately a distance in the streamwise direction equal to the channel height and achieves an asymptotic value corresponding to the fully developed profile approximately four channel heights downstream (for Re = 20 flow). The high-order slip effects reduce the velocity slip compared to the first-order predictions. The value of the high-order slip coefficient (b) is obtained using equation (2.39).

Figure 4.14 (right) shows the variation of high-order slip parameter b as a function of the parameter

c

1

1

1 + l.2x

(where x = x/h, and h is the half channel width) determining the stream-wise location. In Figure 4.15 we plot several streamwise velocity profiles close to inlet for Kn = 0.03 (solid line) and no-slip (dashed line) flows. For various values of C close to the inlet (C < 0.75) the maximum velocity is off centerline until a fully developed profile is established. We also see from the plot that the value of the maximum velocity is always smaller in the case of the slip flow. This behavior of maximum velocities off the centerline cannot be obtained if the convective terms are linearized as in (Sparrow et al., 1962). These results agree with the computations reported in (Gampert, 1976); however, in that work the maximum values were underpredicted by more than a factor of two compared to the present results. The reason for this discrepancy may be due to imposition of the incorrect boundary condition (equation (8) in (Gampert, 1976)) close to the inlet. We also verified the grid independence of our results by performing higher-resolution simulations. The flow at the channel entrance is fully two-dimensional. The cross-flow velocity variation as a function of C is shown in Figure 4.15. The

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