# The Knudsen Paradox in Micro-Channel Poiseuille Flows with a Symmetric Particle
## 3 DSMC Framework: \(dsmcFoam+\)  
The direct simulation Monte-Carlo (DSMC), developed by [PERSON] (1992), is a stochastic particle-based technique for modeling real gases in which the molecular diameter is much smaller than the mean free path (\(\lambda\)). The primary objective of this method is to decouple molecular motion (modeled deterministically) and inter-molecular collision (modeled probabilistically) over small time intervals. The time interval over which the solution is sought is subdivided into smaller sub-intervals over which the particle motion and collisions are decoupled. Consequently, the trajectories are computed over a time interval which is lesser than the mean collision time. Further, DSMC relies on discrete parcels, which each contain a collection of molecules, to represent the total molecular number density (\(N\)) of the real gas. Thus, this method is computationally cheaper than a fully deterministic method like molecular dynamics (MD) and is ideally suited for studying rarefied gas dynamics [PERSON] (1992). We use the open-source solver \(dsmcFoam+\)(K  
in Figure 1) is introduced in the center of the channel to obstruct the rarefied flow at varying degrees. This is given by particle blockage ratios (or \(H/h\)) 8, 4 and 2, which represent progressively increasing degree of channel obstruction. The surface and inlet temperatures are set to 300 K, while the inlet pressure is fixed based on the desired \(Kn\) (in our study, this varies from 5000 pa to 255,000 Pa). Further, the inlet to outlet pressure ratio is maintained at 3.0 for all the cases run (synonymous with the range prescribed in similar studies [34; 35; 37; 41]). The time step size used in computing the DSMC parcel motion was determined smaller than the local mean collision time. The values of mass flow rate at the inlet and outlet are monitored until these are equivalent. The relevant macroscopic properties (e.g., temperature, density and velocity) are sampled after this steady state has been attained. The sampling is performed within each computational cell over a sufficiently large time step and over multiple realizations (to minimize statistical scattering) using relationships given by [PERSON] et al. [42]. These simulation conditions are summarized in Table 1.  
A typical DSMC simulation is usually constrained by the following discretization criteria: (i) the computational cell size must be smaller than the local mean free path (if possible collision partners are restricted to a parcel's current cell); (ii) the number of parcels per cell must be large enough to preserve collision statistics; and (iii) the simulation time step must be chosen so that the parcels only cross a fraction of the average cell length during each time step. Conditions (ii) and (iii) are satisfied by employing a time-step much smaller than the mean collision time, while (i) is met by utilizing a spatial discretization that has already been validated exhaustively for the current setup (cf. [34; 35; 36]). All the simulations were carried out serially on a single node with 20 cores using an Intel(r) Xeon(r) (E5-2650) processor (up to 3.0 GHz). The simulations contained around 200,000 DSMC parcels in the case at a pressure of 255,000 Pa (each parcel has a weighting factor of 10). Note that the parcel count reduces as the pressure (and consequently the mass flow of the gas) in the micro-channel reduces. Hence, to minimize the statistical scatter in the collected data, two realizations of each DSMC simulation were run for a total of 100,000 sampling intervals. The results presented in this paper are ensemble averaged over these two realizations (the scatter was noticeably reduced). Furthermore, the velocity extracted from the simulation (shown in Figure 2) is displayed both using the scattered raw data as well as a polynomial fitted using a weighted least squares estimate [43], so as to facilitate a simplified visualization of the data. Each DSMC simulation needed a physical time of 100 h.