# The Knudsen Paradox in Micro-Channel Poiseuille Flows with a Symmetric Particle
As the pressure crosses this threshold (\(\lambda<a\)), drift transport increases (towards the Poiseuille form) leading to an increase in \(Q\). The [PERSON] paradox has also been confirmed through numerical assessments, first by [PERSON] and co-workers [5; 8; 9] and later by others (e.g., [10; 11; 12]). These authors solved the steady Boltzmann equation with a BGK (Bhatnagar-Gross-Krook) [13] collision term. For a detailed review of these approaches, the reader is referred to the book by [PERSON] [14]. More recently, these numerical methods have been further extended and validated to handle a wide range of rarefied flows in a variety of geometries, as described in the reviews by [PERSON] [15], [PERSON] [16], [PERSON] [17] and [PERSON] [18]. In general, the reported methods are at the following levels of abstraction: (i) micro-scale resolved techniques that directly model molecular scale effects (such as inter-molecular and molecule-wall interactions); (ii) meso-scale resolved techniques (using the Boltzmann equation) that solve the molecular distribution function; and (iii) macro-scale (continuum) resolved techniques that incorporate velocity slip and temperature jump boundary conditions into the Navier-Stokes equations to model molecular phenomena. The continuum based methods are valid up to the transition regime [19], while the meso- and micro-scale methods can in principle be applied across the entire \(Kn\) range [6]. Thus, these two alternatives are preferred while evaluating the Knudsen paradox. Consequently, the lattice Boltzmann method (LBM) [20; 21] and the direct simulation Monte-Carlo (DSMC) method [22; 23] have gained prominence in rarefied micro-channel flow studies. In general, research on single-phase rarefied Poiseuille flows are extensive (cf. [16; 17; 18]). However, related work on their multiphase counterparts (where a solid particle and a rarefied gas co-exist in the micro-channel) are relatively scarce. Currently, these reported multiphase studies are mostly limited to assessing stationary flow past a particle (e.g., [24; 25; 26; 27]), indicative of a knowledge gap. In fact, investigations on the effect of an additional particulate phase on the Knudsen paradox noted in micro-channel Poiseuille flows (in the transition regime) have not been previously reported. An additional motive to study such rarefied particulate Poiseuille flows stems from the recent advancements in nanofabrication techniques. Novel nanofabricated reactor systems are increasingly used to study nanoscale reactive flows [28]. Such systems are often designed to mimic naturally occurring reactor geometries, such as individual pores inside a porous medium. Particles can either be dispersed (as in a gas-solid flow) or stationary (as in typical heterogeneous catalysis applications). A requirement for the existence of a Knudsen minimum is that the duct exhibits free paths for molecular motion that are much longer than the mean radius, a configuration which is typically not readily encountered in natural porous media [4]. However, due to restrictionsimposed by nanofabrication as well as increasing demand for advanced experimental equipment, artificial model pores may well fulfill these requirements and thus exhibit Knudsen minima effects. Furthermore, the spatial variability in the heterogeneous topology of natural porous media has been shown to reduce the relevance of rarefaction effects on the total flow rate in the slip flow regime [29], which could imply that effects observed in systems with regular pore patterns may differ from the systems they are meant to represent. As model pore manufacturing methods are swiftly advancing, possibilities to experimentally investigate reactive rarefied gas-solid flows in model pores will soon become available. It is then of great interest to understand the differences in flow and transport characteristics of artificially designed systems to their intended natural counterparts.