# Interdisciplinary Applied Mathematics

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M.

0    0.2    0.4    0.6    0.8    1

FIGURE 3.1. Variation of skin friction as a function of M and Kn for Couette flow. (Re = 5,TX = 300 K). Lines and symbols denote analytical and numerical results, respectively. Compressible no-slip results denoted by L&R are from (Liepmann and Roshko, 1957).

Figure 3.1 that significant deviations from incompressible flows (with either slip or no-slip) are obtained for MTO > 0.3. In particular, we investigated a case in which the bottom plate is kept at Tw = 350 K while the top plate is kept at Тж = 300 K. The friction coefficient of this case is also given in    Figure    3.1. The results are    shown    by    solid    and    open triangles    for    the

no-slip and the slip cases, respectively. The trend is different from the adiabatic bottom plate case. The no-slip results show small variation of Cf as a function of M, while for slip flows Cf is reduced significantly as Kn is increased.

The density variation across the channel for compressible no-slip as well as slip flows is shown in Figure 3.2 (left) for the case with adiabatic bottom wall. Here,    we normalized    the    density variation    by    the    top    plate    density    of

the no-slip case (solid line). The no-slip cases exhibit large density variations    for    relatively    large    values    of    MTO. Since    the    pressure    is    constant,

density variation across the channel is due to the drastic change in temperature, which is attributed to viscous heating. However, density variations are reduced in slip flows. There are two reasons for this behavior:

1. The shear stress is reduced due to slip, reducing the viscous heating effects, i.e., work done by viscous stresses in the energy equation.

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