Disentangling the role of athermal walls on the Knudsen paradox in molecular and granular gases.

Abstract

The nature of particle-wall interactions is shown to have a profound impact on the well-known "Knudsen paradox" [or the "Knudsen minimum" effect, which refers to the decrease of the mass-flow rate of a gas with increasing Knudsen number Kn, reaching a minimum at Kn∼O(1) and increasing logarithmically with Kn as Kn→∞] in the acceleration-driven Poiseuille flow of rarefied gases. The nonmonotonic variation of the flow rate with Kn occurs even in a granular or dissipative gas in contact with thermal walls. The latter result is in contradiction with recent work [Alam et al., J. Fluid Mech. 782, 99 (2015)JFLSA70022-112010.1017/jfm.2015.523] that revealed the absence of the Knudsen minimum in granular Poiseuille flow for which the flow rate was found to decrease at large values of Kn. The above conundrum is resolved by distinguishing between "thermal" and "athermal" walls, and it is shown that, for both molecular and granular gases, the momentum transfer to athermal walls is much different than that to thermal walls which is directly responsible for the anomalous flow-rate variation with Kn in the rarefied regime. In the continuum limit of Kn→0, the athermal walls are shown to be closely related to "no-flux" ("adiabatic") walls for which the Knudsen minimum does not exist either. A possible characterization of athermal walls in terms of (1) an effective specularity coefficient for the slip velocity and (2) a flux-type boundary condition for granular temperature is suggested based on simulation results.