## 4 Discussion  
In this study, MD simulations of Couette and Poiseuille flows with the addition of wall-temperature effects for a monatomic gas were conducted by using a multi-cell hard-sphere method. Simulations of various parameters, combinations of the coupled effects of wall speeds, pressure gradients, and wall temperatures were executed. Subsonic, transonic, and supersonic speeds in nanochannels are achieved. The periodicity boundary condition in the streamwise direction was modified by developing an acceptance-rejection scheme so that a density difference between upstream and downstream regions formed and yielded a pressure-gradient-driven flow.  
In total, the results of 1260 simulation cases were evaluated. The temperature and velocity slips show dependencies not only with respect to the temperature and velocity values but also the magnitudes of the pressure gradient. The pressure gradient also causes nonlinearities in the velocity profiles, which is in compliance with basic flow theory.  
In the simulations, the upper and lower walls move in opposite directions at the specified speeds. Therefore, according to Equation (5), we predicted that higher temperatures would occur in the central region. Since the flow in the middle region is slower, a parabolic behavior was observed in the temperature profile. As \(PW\) decreases and accordingly the pressure gradient increases, the profiles lose their parabolic properties: a stronger pressure gradient results in a lower peak temperature in the profile. As the pressure gradient becomes stronger, the zero-velocity layer moves toward the wall moving in the opposite direction of the flow.  
The molecular spacing ratio in this study is 3.5, which corresponds to a rather high pressure. Additionally, the number of molecules is very low for experimental examination. The application of pressure and temperature gradients complicates the problem of experimental examination even more. Therefore, similar experimental data were not found in the literature. For simple Couette flow, Couette flow with a pressure gradient, and simple Poiseuille flow with a very low Knudsen number (\(Kn=0.034\)), our results match the analytical continuum parabolic results for velocity and temperature profiles, which are explained in Figures 6 and 7 in the Simulation Results Section.  
It is understood that with the currently available computational capabilities, MD simulations of gases can successfully handle hundreds of thousands of molecules with various combinations of wall temperatures, wall velocities, and pressure gradients in nanochannels of high and low Knudsen numbers.  
E.P.O.: Formal analysis, Visualization, Investigation, and Writing; I.K.: Methodology, Software, Visualization, Writing, and Review and Editing. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
Data Availability Statement: Data are contained within the article.  
The authors thank TUSAS Engine Industries Inc. (TEI) for granting access to the computational resources developed at Gebze Technical University High-Performance Computing Laboratory.  
The authors declare no conflicts of interest.