## 1 Introduction
As the Knudsen number increases, the behavior of gas shifts towards rarefied flow, where the mean free path becomes similar to or greater than the characteristic length scale. MD simulations are more useful than Navier-Stokes equations for compressible flows because they can offer precise insights into the behavior of compressible gaseous continua at the molecular level. Conducting this investigation at the molecular level enables a more thorough comprehension of compressibility, which is essential for accurately capturing the dynamics of compressible flows [21]. In MD simulations, Couette flow and Poiseuille flow are often used to study the flow of fluids at the molecular level [22; 23; 24; 25; 26; 27]. In these simulations, the behavior of individual particles in response to external forces such as shear forces and pressure gradients is investigated. Couette flow is a fundamental concept in fluid dynamics that characterizes the motion of a fluid between two parallel plates, where one plate or two plates move at a constant velocity. This flow structure can be defined by the production of shear stress and the creation of velocity gradients within the fluid. Another approach used in solving physical problems, Poiseuille flow is characterized by a pressure difference (\(\Delta\)P) along the length of a tube, and the flow is driven by this pressure gradient. It refers to the steady, laminar flow of an incompressible fluid through a pipe or tube. When considering the assumption of incompressibility, there is a simple analytical solution to the problem, and this solution has been examined using various methods [28]. Most MD simulations documented in the literature focus on flows induced by an accelerating body force rather than flows induced by pressure [29; 30; 31; 32]. These limitations arise from the use of periodic boundary conditions that cannot create a pressure gradient within the simulation domain. In many studies, new techniques have been developed to imitate flows caused by pressure differences. [PERSON] & [PERSON] (1990) positioned two rigid walls at entry and exit points and then exerted external forces upon them ([PERSON] & [PERSON], 1990). [PERSON] et al. (1998) used an imaginary membrane that allows the unidirectional movement of molecules while reflecting molecules entering from the opposite direction with elastic reflection and certain probability ([PERSON] & [PERSON], 1990). [PERSON] et al. dealt with periodic entry and exit velocity conditions, where the square velocity difference between molecules entering and exiting the simulation area is kept constant ([PERSON] & [PERSON], 1990). In the present study, MD simulations of Couette and Poiseuille flows with the addition of wall temperature effects for a monatomic gas were conducted by utilizing a multi-cell hard-sphere method. Combinations of the coupled effects of wall speeds, pressure gradients, and wall temperatures were investigated to better understand the implications of various parameters. Subsonic, transonic, and supersonic speeds in nanochannels were achieved. In order to illustrate Poiseuille flow, we used a density differentiation algorithm that, for the sake of normalization, eliminates the dependency of variation in molecular mass on imposed pressure gradients. This approach also allowed us to avoid the application of a special thermostatic treatment, which was the case for many Poiseuille flow studies in the literature ([PERSON] & [PERSON], 1990; [PERSON], 1990; [PERSON], 1990). A membrane with a periodic boundary condition was employed, permitting unidirectional movement of molecules while elastically reflecting molecules approaching from the opposite direction with a specific probability. While we used low spacing ratios in the simulations, the results can be extended to rarefied gas flows of similar Knudsen numbers by using proper scaling ([PERSON] & [PERSON], 1990). The simulations in this study focus on \(0.03<Kn<0.14\) and \(M<1.7\) for Knudsen and Mach numbers, respectively.