# Interdisciplinary Applied Mathematics

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In Figure 3.3, velocity profiles for linear Couette flow in the upper half of the channel at three different Knudsen numbers are presented, where к = (0F/2) Kn is the rescaled Knudsen number. The linearized Boltzmann

FIGURE 3.3. Velocity profiles for linear Couette flow in the upper half of the channel at    rescaled    Knudsen    numbers    k =    0.1,1.0,    and    10.0.    The wall    speed

corresponds to M = 0.05.

solutions (triangles) and DSMC (circles) agree quite well, and they both predict essentially linear velocity distribution in the bulk flow region with significant slip effects for increased Kn. Knudsen layers are also visible in this plot, and they become dominant especially in the transition flow regime. The nondimensional results are valid for any monoatomic hard-sphere dilute gas. Figure 3.3 also shows the velocity distribution predicted using Maxwell’s first-order slip model (dashed lines). These analytical results are obtained using Equation (3.7) with Ci = 1.111, and the first-order slip solution is reasonable for Kn < 0.1.

• The velocity profile for incompressible Couette flow remcains essentially linear in the entire Knudsen regime. Therefore, high-order slip conditions that utilize the second- or higher-order derivatives of the velocity field cannot predict the desired slip corrections.

In order to develop an engineering model for these flows, (Bahukudumbi et al., 2003) introduced a modified slip coefficient (Ci) for equation (2.42), and similarly for equation (3.7) in the following form:

Ci = во + в1 tan-1(^2 Kn^3),    (3.8)

where вг (i = 0,1, 2, 3) are empirical constants that are obtained by comparing the slope of the velocity profile, obtained by the linearized Boltzmann solution in (Sone et al., 1990), with that obtained from equation (3.7), using

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