# Interdisciplinary Applied Mathematics

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l + Pr^Mc Re

2

(3.4)

Cf0 = 2

where MTO is the Mach number based on the upper plate velocity and temperature, and Re is the Reynolds number based on the channel height.

We performed a series of simulations with the program pFlow, which is based on high-order discretization (see Section 14.1 for details) to explore the compressibility effects in shear-driven flows. One set of simulation results corresponds to top plate temperature Тж = 300 K and Reynolds number Re =5. The simulations are performed using nine spectral elements with sixth-order polynomial expansions per direction in each element. The Mach number MTO is specified by varying the driving velocity of the top plate UTO. Correspondingly, rarefaction effects are specified through the Knudsen number, since

TV    O’ Moo

##### Klb— = f2 TiW    (35)

The variation of friction coefficient as a function of Mach number and corresponding Knudsen number is shown in Figure 3.1. The friction coefficient of no-slip compressible flow increases quadratically, in agreement with equation    (3.4),    well    above    the    constant    value    of    the    corresponding

incompressible flow. The no-slip compressible flow simulations match the theoretical results exactly. For rarefied flows, slip effects change the friction coefficient significantly. Compressible slip flow results are denoted by open circles in Figure 3.1. It is seen that the compressible slip flow results correspond to small deviations from the incompressible slip flow results obtained from

In linear Couette flow the pressure is constant, and therefore compressibility effects are due to the temperature changes only. As MTO increases, the temperature difference between the two plates gets larger due to viscous heating. Thus, compressibility effects become significant. It is seen in

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