On Knudsen-minimum effect and temperature bimodality in a dilute granular Poiseuille flow

Abstract

The numerical simulation of gravity-driven flow of smooth inelastic hard disks through a channel, dubbed 'granular' Poiseuille flow, is conducted using event-driven techniques. We find that the variation of the mass-flow rate (Q) with Knudsen number (Kn) can be non-monotonic in the elastic limit (i.e. the restitution coefficient e(n) -> 1) in channels with very smooth walls. The Knudsen-minimum effect (i.e. the minimum flow rate occurring at Kn similar to 0(1) for the Poiseuille flow of a molecular gas) is found to be absent in a granular gas with e(n) < 0.99, irrespective of the value of the wall roughness. Another rarefaction phenomenon, the bimodality of the temperature profile, with a local minimum (T-min) at the channel centerline and two symmetric maxima (T-max) away from the centerline, is also studied. We show that the inelastic dissipation is responsible for the onset of temperature bimodality (i.e. the 'excess' temperature, Delta T = (T-max/T-min - 1) not equal 0) near the continuum limit (Kn similar to 0), but the rarefaction being its origin (as in the molecular gas) holds beyond Kn similar to O(0.1). The dependence of the excess temperature Delta T on the restitution coefficient is compared with the predictions of a kinetic model, with reasonable agreement in the appropriate limit. The competition between dissipation and rarefaction seems to be responsible for the observed dependence of both the mass-flow rate and the temperature bimodality on Kn and e(n) in this flow. The validity of the Navier-Stokes-order hydrodynamics for granular Poiseuille flow is discussed with reference to the prediction of bimodal temperature profiles and related surrogates.