# The Knudsen Paradox in Micro-Channel Poiseuille Flows with a Symmetric Particle
## 5 Results and Discussion
### Hydrodynamics of a Stationary Particle in the Transition Regime  
The presence of a stationary and inert particle in a 2D planar micro-channel Poiseuille flow of a rarefied gas in the transition regime alters the inherent response of the system. This is because of the additional momentum sink in the flow due to the particle. We investigate the hydrodynamic consequences of such a flow obstruction by comparing the velocity in the micro-channel with a benchmark case without the particle. Such an assessment has a two-fold objective: (i) deviations from the ideal micro-channel Poiseuille flow due to the presence of the particle can be identified; and (ii) the hydrodynamic basis for the observed Knudsen paradox can be deduced. We first discuss the impacts on the channel velocity profile (along the height) and follow this with a brief discussion on the compressibility of such particulate flows. Note that all these hydrodynamic assessments are done under the conditions defined in Table 1 and at an outlet pressure \(P_{o}\) of 1 bar (\(Kn=0.1\)). This \(Kn\) is chosen as it represents the unique hydrodynamics in the transition regime, where the Knudsen minimum has been observed in micro-channel Poiseuille flows [5; 41]. The 2D planar micro-channel Poiseuille flow benchmark case without the particle is characterized by a parabolic velocity profile. Consequently, when the flow encounters a stationary particle at the center of the channel, its response is markedly altered. Figure 2 shows the variation of the co-axial velocity along the \(y\)-direction (\(U_{x}(y)\)). As expected, the presence of the particle decelerates the overall flow, with the maximum velocity reduced due to the blockage. It follows that, at the center of the channel, the particle partitions the flow into two narrower channels each developing its own parabolic velocity profile for \(U_{x}(y)\), with a maximum velocity lower than the benchmark case. Note that velocities at the wall boundaries are always nonzero due to a net slip-effect in the transition regime. The non-uniformities noted in the reported velocity profiles are attributed to the statistical scatter from the DSMC data. This scatter is more visible at the lowest blockage ratios as the largest particle drastically alters the local length scale of the system, thereby decreasing the probability of molecular collisions in this region, which in turn increases the need for sampling to converge the statistics. Nevertheless, the results obtained for the lowest blockage ratio are deemed reasonable enough as they follow the expected trend despite the scatter. These observations are further confirmed by the streamlines of the velocity along a \(x-y\) plane, shown in terms of the Mach number (\(Ma=u/c\), where \(u\) is the local fluid velocity and \(c\) is the speed of sound) in Figure 3. The flow accelerates as it passes the particle, partly due to the pressure decrease but mostly due to continuity. Further, a symmetric low-velocity region is developed before and after the particle. The channeling effect created by the presence of the particle has a consequent impact on the Knudsen paradox, as will be explained later. Furthermore, the mach number (\(Ma\)) is also used to assess compressibility effects in a flow system [44], where such effects are generally deemed negligible for \(Ma<0.3\). This criterion may however be violated in micro-channel Poiseuille flows. Figure 3 shows \(Ma\) in the micro-channel for all cases with and without the particle at the conditions reported in Table 1. The benchmark case shows that such Poiseuille flows are sub-sonic in nature with \(Ma<0.8\). The highest \(Ma\) is observed at the flow outlet as expected. The additional momentum sink, due to the particle, leads to a lower flow velocity of the rarefied gas at the outlet and as the particle size increases, there is a corresponding decrease in the developed \(Ma\).