## 3 Simulation Results
In Figure 5, the behavior of the relationship between pressure and upper wall temperature is further examined by means of linearly curve fitting the data presented in Figure 4a-c in terms of their Knudsen numbers. In this figure, the effect of Knudsen number and wall velocities on the average pressure is depicted. As the Knudsen number or wall velocities increased, the average pressure increased. As shown in Figure 5, wall velocity and wall temperature have a linear effect on the average pressure. This linearity is directly affected when the pressure gradient is included in the system. The velocity profiles when \(AR=1.0\) and \(N=204800\) are given in Figure 6. In the first column, the velocity profiles are almost linear because the Knudsen number is low (\(Kn=0.034\)) and there is a periodic boundary condition with \(PW=1.0\). As the pressure gradient increases, the nonlinearity in the velocity profiles gains significance. The effect of the pressure gradient on the velocity profiles is greater than that of wall temperature. Note that the velocity slips on the walls are not symmetrical when a pressure gradient exists. For example, in the extreme case when \(PW=0.5\), the velocity slip is greater on the lower wall. The effect of the wall temperatures on the velocity slips seems to be insignificant. The upper and lower walls move in opposite directions with given velocities. This is why, as per Equation (5), we expected higher temperatures in the middle region. This corresponds to a lower density in the same region because of Equation (6), which is clearly depicted in the plot. This parabolic velocity profile and corresponding temperature profile are shown in Figures 6 and 7, respectively. In Figures 6 and 7, due to the pressure gradient supporting forward-moving walls, the behaviors of higher-density regions accumulated near the corners of colder walls are not symmetrical. This is especially significant for the rightmost column with the strongest pressure gradient. In Figure 7, the temperature profiles when \(AR=1.0\) and \(N=204800\) are given. When \(T_{u}=10\), the lower wall and upper wall temperatures are equal, and the graph is symmetrical in the case where \(PW=1.0\). For \(T_{u}\
eq T_{l}\), the profiles are not symmetrical. This figure shows that the temperature slips increase with the increasing wall velocities. Pressure gradients affect these slips as well. Stronger pressure gradients cause greater temperature slips on the lower walls. This is due to the fact that because of the higher pressure at the left wall, some of the molecules traveling upstream are reflected back, while the others are instantly beamed to the right wall with their existing velocity. Figure 5: The average pressure in the computational domain for various \(Kn\) numbers and Couette velocities. Figure 6: The velocity profiles when \(AR=1.0\) and \(N=204800\). In Figures 6 and 7, the upper and lower walls move in opposite directions at the specified velocities. Therefore, according to Equation (5), we anticipated higher temperatures in the central area. In the temperature profile, a parabolic behavior can be observed because the flow in the middle region is slower, as shown in Figure 6. This behavior identically matches the low-Knudsen-number analytical solution [48]. As \(PW\) decreases and, correspondingly, the pressure gradient increases, the profiles in the right column of Figure 7 lose their parabolic nature: a stronger pressure gradient causes a lower peak temperature in the profile. Note that wall velocities strongly define the behavior of the velocity profile near the  
Figure 7: The temperature profiles when \(AR=1.0\) and \(N=204\),800.  
walls. The existence of a pressure gradient causes higher velocities around the centerline, as predicted by Poiseuille flow theory. Combined effects of pressure gradient, wall velocities, and temperatures govern different nonlinear temperature and velocity profiles. All these effects cause density variation that plays a role in the region as well.