## 2 Methodology and Model
### Computational Model  
In this study, the MD simulation method was used to process every molecule and molecular interaction. No real collisions are missed, and no fictitious events are imposed. Serving as the computational domain, a 3-dimensional region (box) was filled with molecules of a compressible fluid, as shown in Figure 1. The monatomic gas molecules are modelled as hard spheres undergoing elastic intermolecular collisions. \(L_{y}\) is a characteristic length in the problem in the \(y\)-direction. A molecule-wall interaction model called diffuse reflection is used so that any molecule reaching \(\pm y\) walls is reflected back to the computational region with a velocity generated based on the wall characteristics, namely, temperature and velocity ([PERSON], 1990). In the \(z\)-direction, a complete periodicity condition is applied in which any molecule that reaches a boundary in that direction is instantly transferred to the other boundary, retaining its original velocity. These periodic boundary conditions in the \(z\) direction allow for simulations of systems that extend infinitely along the \(z\)-axis, providing a more accurate representation of particle behavior without the need for an excessively large simulation box. This is particularly useful in studying systems with periodic or long-range effects, allowing researchers to observe behavior over an extended spatial domain without the computational burden of simulating an infinitely large system. Although there are 2 DComputational Fluid Dynamics (CFD) solutions abounded in the literature when it comes to MD simulations this simplification cannot be applied simply because reducing molecular degree of freedom changes behavior of the gas (for example, speed of sound, heat capacity, specific heat ratios, viscosity, etc.). An algorithm is used for this periodicity condition, which prevents the violation of the primary principle of Event-Driven MD Simulations, which stipulates that any two molecules cannot overlap at any time during the simulation. In order to increase computational speed, the multi-Cell Method is used [41]. In this approach, the whole computational region is divided into cells. This cell division improves computational speed simply because distant molecules are less likely to collide with each other. Nevertheless, molecules can cross multiple cells in 3-dimensional space, and any distant pair of molecules can still interact; no real collision is lost. Any molecule may undergo three major kinds of events: intermolecular collision, cell switching (a pseudo-event), and an outer boundary event (as explained above). Intermolecular collisions cause changes in the velocity components of the colliding pair based on their incoming velocities, masses, sizes, and relative positions through momentum and energy conservation with the assumption of perfectly elastic interaction. Cell switching, as a virtual event, does not really affect the velocity or computational position of a molecule. It merely changes the cell the molecule is assigned to, allowing a new set of molecules in the cellular neighborhood to gain candidacy for becoming a future partner of an interaction. Every molecule has a time variable that is updated after each real event it is involved in while relocating, but this is not the case in cell switching. This approach greatly reduces the incremental errors caused by molecular relocations. The simulation starts with equally spaced molecules with the same _rms_ speeds but arbitrarily different velocity components. The earliest event is calculated and carried out, and then the simulation moves on to the next event. This process repeats in order to generate the simulation. It is worth noting that in a few collisions per particle (cpp), the overall molecular behavior spontaneously adjusts itself to the Maxwellian speed distribution. The whole process is event-driven, and any molecule's position changes only when it is involved in a real event [42; 43; 44].