## 3 Simulation Results
In this case, a periodic boundary condition was applied, wherein all molecules approaching the wall pass to the other side. The lower wall was maintained at an initial medium temperature (\(T_{l}=10\)), while the upper wall temperature (\(T_{u}\)) was kept at various values during the simulations. Since \(T_{u}=10\) corresponds to the same wall temperatures for the upper and lower walls, the third-row, first-column plot represents a case where only the wall velocities are in effect. Figure 3 shows the density plots for when \(V_{x}=5.0\), \(AR=1.0\), and \(N=204800\). In Figures 2 and 3, the first column only shows the effects of velocities and upper-wall temperatures. As the upper wall cools, the number of molecules close to the wall decreases. When \(T_{u}=4\), the upper wall is considerably colder than the lower wall. The number of molecules increases in the cooling region. This result is clearly depicted in Figure 3, where the speed of the upper wall is high. In the case of \(PW=1.0\), there is no pressure gradient; therefore, the molecules are distributed more homogeneously than when there is a pressure gradient. As the pressure gradient grew stronger, molecules concentrated near  
Figure 2: The number density plots for when \(V_{x}=0.0\), \(AR=1.0\), and \(N=204800\). the cold wall. When \(T_{u}=10\), the lower wall and upper wall temperatures were equal, and the graph is symmetrical in the case where there is no pressure gradient. The 2-D density plots presented in Figures 2 and 3 show variations that agree with the corresponding temperature profiles. Density increases near the cold wall and decreases near the hot wall. The asymmetry of higher-density regions around the corners of cooler walls is caused by the pressure gradient that supports the movement of walls in the forward direction. This is particularly important for the column located on the rightmost side, which has the most intense pressure gradient. The pressure difference was considered in the simulation results to examine how the system responded to the pressure difference when a selectively permeable periodic boundary condition was applied. It was observed that the pressure gradients in all the  
Figure 3: The number density plots for when \(V_{x}=5.0\), \(AR=1.0\), and \(N=204800\). simulations in this study could be divided into five selected intervals, whose value ranges were taken as approximate values. Table 3 presents these observed pressure gradients. The range with the lowest pressure difference is called dPdx1, and the range with the highest-pressure gradient is called dPdx5. In Figure 4a-c, variations in average pressure in the computational domain with \(T_{u}\) are depicted for various pairs of wall velocities. A legend corresponding to the dPdx values and wall speeds is given in Figure 4d. Seemingly, in compliance with Equation (6), the average pressure increases almost linearly with \(T_{u}\). Note that \(T_{u}\) affects the average temperature in the whole region. This behavior is also affected by Couette flow wall velocities simply because faster-moving walls impart more energy to the system, causing a pressure increase. There are bands of velocity ranges arising from the existence of separate cases with different values of Knudsen numbers and aspect ratios. Figure 4: The average pressures in the computational domain for different \(Kn\) values: (**a**) \(Kn=0.034\); (**b**) \(Kn=0.069\); (**c**) \(Kn=0.137\); and (**d**) legend. \begin{table}
\begin{tabular}{c c} \hline \hline
**Assigned Name of Pressure Gradient** & **Corresponding Interval** \\ \hline dPdx1 & \(-\frac{dP}{dx}<0.0008\) \\ dPdx2 & \(0.0020>-\frac{dP}{dx}>0.0008\) \\ dPdx3 & \(0.0032>-\frac{dP}{dx}>0.0020\) \\ dPdx4 & \(0.0043>-\frac{dP}{dx}>0.0032\) \\ dPdx5 & \(-\frac{dP}{dx}>0.0043\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Nomenclature of pressure gradient intervals.