# Interdisciplinary Applied Mathematics

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2.    A    temperature    jump    exists    at    the    driving (top)    plate;    there is    no

temperature jump on the bottom plate, since it is adiabatic.

We also included the variation of velocity in the channel (normalized with the driving plate velocity) in Figure 3.2 (right). It is seen that the lin-

FIGURE 3.2. Couette flow: Density variation (left) and velocity variation (right) across the channel for various values of Mach number M and Knudsen number Kn; L = h is the height of the channel (Re = 5, T,, = 300 K).

ear velocity profile for the incompressible flows is modified for compressible no-slip flows (solid lines); for the range of simulations performed here the deviations from linear profile is small. This is expected, because the compressible no-slip flow friction coefficient for Ыж = 0.863 is only 10% larger than the corresponding incompressible flow. However, the velocity profiles for the compressible slip flow given in Figure 3.2 (right) show significant reduction in the slope of velocity, which explains the large reduction in the friction coefficient.

##### 3.2 Couette Flow: Transition and Free-Molecular Flow Regimes

In this section, we analyze plane Couette flows in the transition and free-molecular flow regimes. Following (Bahukudumbi et al., 2003), we present empirical models for velocity distribution and shear stress, developed using the linearized Boltzmann equation solutions and extensive direct simulation Monte Carlo (DSMC) results.

###### 3.2.1 Velocity Model

We consider rarefied gas flow between two infinite parallel plates, separated by a distance L. The plates are maintained at the same uniform temperature Tw, and they are moving with a uniform velocity of ±U. We investigate steady one-dimensional plane Couette flow induced between the plates subject to the following assumptions:

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