# The Knudsen Paradox in Micro-Channel Poiseuille Flows with a Symmetric Particle
## 5 Results and Discussion
### Effect of the Particle on the Knudsen Paradox  
We assess the Knudsen paradox in the 2D setup (described in Figure 1) by varying the inlet pressure (\(P_{in}\)) and maintaining the inlet to outlet pressure ratio as 3. As detailed above, the normalized mass flow rate in the channel \(Q_{norm}\) (see Equation (1)) is extracted from the DSMC simulations across the entire \(Kn\) range, i.e., at different \(\delta_{m}\) (which is in turn related to \(Kn\) as shown in Equation (2)). Figure 4 shows the Knudsen minima estimated from the DSMC simulations along with inherent uncertainties in the sampled data (through error bars). In this figure, the solid lines represent cases without a particle, while the dashed lines represent the cases with one. The comparisons between the benchmark case (without a particle) and the results given by [PERSON] et al. [5; 8] are reasonable, given the statistical spread of the sampled data. The agreement is good at high \(\delta_{m}\) (or low \(Kn\)) values with a deviation noted at the lower ones. The result in this region is more sensitive to the scatter in the DSMC data (indicated by the wider error bars in this region) and is also known to be affected by the lateral boundary condition in pseudo-3D systems [45]. However, these obtained results clearly depict the Knudsen paradox (within the limits of certainty of the simulation setup) and are synonymous with those reported by [PERSON] et al. [34] using the same solver. Moreover, the minima is noted at \(\delta_{m}=0.97\) (which is at around \(Kn\approx 1\), i.e., in the transition regime), as reported by others (cf. [5; 8; 41]). The predicted Knudsen minima for all the cases simulated are summarized in Table 2.  
When a particle is introduced in the channel, it serves as an additional momentum sink in the system and sections of high fluid velocity are created in the regions between the particle and wall (see Figure 3). The Knudsen minimum may have shifted towards higher Knudsen numbers (see Table 2), although our data does not provide enough statistical confidence to confirm this. Nevertheless, the noted trends could be due to a shift towards relatively more ballistic molecules (that do not collide with each other) at shorter geometrical distances--a trend similar to that observed when going from a channel flow to a duct flow [45] or with flows in a bent channel [46]. In other words, the presence of the particle reduces the governing local length scales of the system, with a decreased probability of molecular interactions along the diffusion paths of the gas molecules. Moreover, the momentum sink effect causes the total mass flow rate in the channel to be reduced. Such a reduction in \(Q_{norm}\) is noticeable in Figure 4, with lower mass flows noted as the particle size increases, thereby effectively blocking the channel to a higher degree. It can also be noted that the relative changes in \(Q_{norm}\) with increasing particle size in the transition regime diminishes, with the curve tending to flatten out. This occurs as beyond the continuum regime the frictional losses become relatively less important due to velocity slip. Figure 3: Contours of Mach number (\(Ma\)), overlaid with velocity streamlines (colored by \(Ma\)), along an \(x-y\) plane for the benchmark case (see Table 1) and the DSMC simulations at blockage ratios \(H/h\) of 8, 4 and 2, respectively. Moreover, as the particle becomes larger, there is an increased chance that the Knudsen paradox totally disappears, which means that \(Q_{norm}\) monotonically decreases without any increase at higher \(Kn\) (or lower \(\delta_{m}\)) similar to observations made in bent channels by [PERSON] et al. [46]. Synonymous with the inferences drawn from the bent channel studies, we also attribute the observed behavior to the apparent increase in the local hydrodynamic resistances due to the particle.