# The Knudsen Paradox in Micro-Channel Poiseuille Flows with a Symmetric Particle
## 6 Conclusions  
In this paper, we present a DSMC investigation of the effect of a stationary particle on the Knudsen paradox observed in micro-channel Poiseuille flows. Such an investigation has previously not been reported in the literature. The objective with this study was to assess the effects from representing natural porous media through nanofabricated model pores in rarefied gas-solid systems and heterogeneous catalysis applications. The open source solver \(dsmcFoam+\) was used to carry out the relevant studies. We show that the presence of a particle markedly alters the response of the rarefied flow system as it serves as \begin{table}
\begin{tabular}{c c c} \hline \hline
**Case** & \(\delta_{m}\) & \(Kn\) \\ \hline [PERSON] et al. [5, 8, 9] & 0.77 & 1.16 \\ Benchmark (without particle) & 0.97 & 0.92 \\ \(H/h=8\) & 0.38 & 2.32 \\ \(H/h=4\) & 0.36 & 2.46 \\ \(H/h=2\) & 0.17 & 5.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Knudsen minimum in 2D planar micro-channel Poiseuille flows with a stationary particle in the middle. Note that the reported minima is subject to the variations in the sampled data shown in Figure 4.  
an additional momentum sink in the flow. Further, as it becomes larger and subsequently blocks a larger portion of the duct (represented as the blockage ratios \(H/h\)), the mass-flow rate at the same nominal pressure difference decreases. Despite the noticeable scatter in the reported data, the general trends (in \(Q_{norm}\)) observed agree with conventional rarefied flow dynamics in micro-channels and parallelisms can be drawn from our study and that of [PERSON] et al. [46] on micro-channels with bends. In both assessments, an increase in the local hydrodynamic resistances in the flow leads to noticeable deviations from the behavior derived for straight micro-channel Poiseuille flows by [PERSON] et al. [5]. Consequently, with the presence of a particle, the Knudsen minimum is shifted towards higher Knudsen numbers with the curve flattening out towards the free-molecular flow regime. This effect is more pronounced as the particle becomes large and it results from a shift towards relatively more ballistic molecular motion at shorter geometrical distances. These observations have significant implications for the design of model pore systems for reactive rarefied gas-solid flows, as well as investigations of heterogeneous catalysis using stationary catalyst particles. Most notably, the local mass diffusivity, which determines the overall reaction rate at mass-transfer limited conditions, will depend on the local flow configuration, which may thus be indirectly assessed from the response of the global mass-flow rate as a function of absolute pressure. Moreover, attention should be paid to the possibility for flow oscillations to manifest in nano-scale reactor systems with rarefied reactive gas-solid flow. In conclusion, the present work addresses a perceived knowledge gap in rarefied flows along with a relevant challenge while modeling reactive gas-solid flows in confined geometries at the nano-scale, where simultaneous handling of local and non-local transport mechanisms over the particle surfaces must be realized. The observed trends can be extended towards asymmetric particles as well, since the presence of any particle in the flow increases the overall hydrodynamic resistances in it. Thus, we underline the need to account for these complex transition regime effects while assessing such rarefied gas-particle systems, particularly in the relevant micro-fluidic applications.