Fully developed flow of coarse granular materials through a vertical channel

Abstract

Some recent studies have proposed a simple manner of combining the frictional contributions to the stress with the kinetic theory based models for granular flow. In this paper, kinetic and frictional-kinetic models are used to study the gravity flow of granular materials through a vertical channel. The results show that as the mean density ν ^̅ increases, the velocity profile predicted by the kinetic model shows a region of plug flow near the centre, and a shear layer adjacent to the wall. At high densities (ν ^̅ ≈ 0.64 ), the thickness of the shear layer is a few particle diameters. The centerline velocity and the mass flow rate exhibit maxima as ν ^̅ increases. The maxima are due to the opposing effects of the weight of the material and the shear viscosity. For the parameter values used here, the grain temperature profile exhibits a maximum near the wall and, correspondingly, the density profile exhibits a minimum. Predictions of the frictional-kinetic model are qualitatively similar, except that (i) the profiles are discontinuous at the interface between the plug and the shear layer, and (ii) at high densities, the thickness of the shear layer is much less than in the kinetic case. When mean densities are matched with measured values, velocity profiles (scaled by the centerline velocities) predicted by both models are found to be in fair agreement with the experimental data. In contrast, the absolute centerline velocities are higher than the reported values by a factor of 10-50. This discrepancy cannot be attributed to air-drag, as calculations show that the latter has a negligible effect on the velocity profiles. When either the mass flow rates or the centerline velocities are matched with the experimental data, the kinetic profiles fit the data fairly well. This is a puzzling result, as at high densities, many of the assumptions of the kinetic model are unlikely to hold. Though the frictional-kinetic model is expected to be more realistic, the shear layer is much thinner than that observed. Both models underestimate the shear stresses at the channel walls by a factor of 1.5-3, and overestimate the normal stresses by a similar factor.