Hydrodynamics, wall-slip, and normal-stress differences in rarefied granular Poiseuille flow.

Abstract

Hydrodynamic fields, macroscopic boundary conditions, and non-Newtonian rheology of the acceleration-driven Poiseuille flow of a dilute granular gas are probed using "direct simulation Monte Carlo" method for a range of Knudsen numbers (Kn, the ratio between the mean free path and the macroscopic length), spanning the rarefied regime of slip and transitional flows. It is shown that the "dissipation-induced clustering" (for 1-e_{n}>0, where e_{n} is the restitution coefficient), leading to inhomogeneous density profiles along the transverse direction, competes with "rarefaction-induced declustering" (for Kn>0) phenomenon, leaving seemingly "anomalous" footprints on several hydrodynamic and rheological quantities; one example is the well-known rarefaction-induced temperature bimodality, which could also result from inelastic dissipation that dominates in the continuum limit (Kn→0) as found recently [Alam et al., J. Fluid Mech. 782, 99 (2015)JFLSA70022-112010.1017/jfm.2015.523]. The simulation data on the slip velocity and the temperature slip are contrasted with well-established boundary conditions for molecular gases. A modified Maxwell-Navier-type boundary condition is found to hold in granular Poiseuille flow, with the velocity slip length following a power-law relation with Knudsen number Kn^{δ}, with δ≈0.95, for Kn≤0.1. Transverse profiles of both first [N_{1}(y)] and second [N_{2}(y)] normal stress differences seem to correlate well with respective density profiles at small Kn; their centerline values [N_{1}(0) and N_{2}(0)] can be of "odd" sign with respect to their counterparts in molecular gases. The phase diagrams are constructed in the (Kn,1-e_{n}) plane that demarcates the regions of influence of inelasticity and rarefaction, which compete with each other resulting in the sign change of both N_{1}(0) and N_{2}(0). The results on normal stress differences are rationalized via a comparison with a Burnett-order theory [Sela and Goldhirsch, J. Fluid Mech. 361, 41 (1998)JFLSA70022-112010.1017/S0022112098008660], which is able to predict their correct behavior at small values of the Knudsen number. Lastly, the Knudsen paradox and its dependence on inelasticity are analyzed and contrasted with related recent works.