## 3 Simulation Results  
For combined heat conduction, Couette flow, and simple pressure gradient channel flow, the values given in Table 1 were used. During the simulations, different wall speeds, wall temperatures, and pressure gradients were examined. All the values used are dimensionless. The \(V_{x}\) value, which expresses the wall speed, varies between the value at which the wall has no speed (\(V_{x}=0.0\)) and the value at which the wall speed is \(V_{x}=5.0\). The speed of sound Is calculated as 2.89, and therefore \(V_{x}=5.0\) can be considered to be approximately 1.7 Mach. The pressure gradient varies between 1.0 and 0.5. There is no pressure gradient in the case wherein \(PW=1.0\). In the case of \(PW=0.5\), the pressure gradient is at its maximum point, and this means that half of the molecules approaching the wall are reflected to the other side, and the other half are reflected via specular reflection. \(T_{u}\), which expresses the upper wall temperature, was chosen to be between 4.0 and 22.0, and the lower wall temperature was set as \(T_{l}=10\) and kept constant in all the simulations. The simulations were conducted for a set of various numbers of molecules \((N)\) and aspect ratios \((AR)\), as presented in Table 2. The different wall speeds, wall temperatures, and pressure gradients given in Table 1 were applied to these data sets. In order to converge toward the steady state profiles, a final long-time averaging procedure was computed between 198 and 200 cpp. It was verified that the steady-state conditions were well established by comparing these data with previous cpp values. The local temperature depends on local molecular velocities:  
\[3 kT\equiv\langle\,m|\stackrel{{\rightarrow}}{{c}}-\stackrel{{ \rightarrow}}{{u}}|^{2}\rangle=m\langle\,|\stackrel{{ \rightarrow}}{{c}}-\stackrel{{\rightarrow}}{{u}}|^{2}\rangle \tag{1}\]  
where \(\langle\,\rangle\) is an averaging operator, \(k\) is the Boltzmann constant, \(T\) is absolute temperature, \(m\) is molecular mass, \(\stackrel{{\rightarrow}}{{c}}\) is molecular velocity, and \(\stackrel{{\rightarrow}}{{u}}\) is the local velocity vector averaged over the molecules in the vicinity. In our problem, since there are only pressure gradient and velocity boundary conditions acting in the \(x\)-direction, we expected a non-zero velocity component only in the \(x\)-direction:  
\[\langle\,c_{x}\,\rangle=|\stackrel{{\rightarrow}}{{u}}|=V_{x}. \tag{2}\]  
Substituting Equation (2) into Equation (1) yields  
\[3 kT=m\langle\,|\stackrel{{\rightarrow}}{{c}}-\stackrel{{ \rightarrow}}{{u}}|^{2}\rangle=m(\langle\,c^{2}\,\rangle-V_{x}^{2}). \tag{3}\]  
Kinetic energy, on the other hand, is defined as  
\[E\equiv\frac{1}{2}mc^{2}. \tag{4}\]  
By using Equations (3) and (4), for temperature, the relationship  
\[3 kT=2\langle E\rangle-\langle m\rangle V_{x}^{2} \tag{5}\]  
can be deduced. \(n\) denotes number density; pressure is expressed by the state equation:  
\[p=nkT \tag{6}\]  
As an example, number density plots for when \(V_{x}=0.0\), \(AR=1.0\), and \(N=204800\) are presented in Figure 2. A single simulation took 20 h on a Xeon Power Station HP-Z840 in a single processor run. The first column consisted of zero pressure gradient cases \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameters** & \multicolumn{6}{c}{**Values**} \\ \hline \(V_{x}\) & 0.0 & 1.0 & 2.0 & 3.0 & 4.0 & 5.0 & \\ \(PW\) & 1.0 & 0.9 & 0.8 & 0.7 & 0.6 & 0.5 & \\ \(T_{u}\) & 4.0 & 7.0 & 10.0 & 13.0 & 16.0 & 19.0 & 22.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Sets of Values**} \\ \hline
**Parameters** & **Set 1** & **Set 2** & **Set 3** & **Set 4** & **Set 5** \\ \hline \(AR\) & 1.0 & 2.0 & 2.0 & 4.0 & 4.0 \\ \(N\) & 204,800 & 25,600 & 51,200 & 204,800 & 51,200 \\ \(Kn\) & 0.0345 & 0.0689 & 0.1378 & 0.0345 & 0.0689 \\ \(L_{y}/d\) & 280 & 140 & 70 & 280 & 140 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The data sets. (\(PW=1.0\)).