Granular flow through a vertical axisymmetric pipe

We employ dense kinetic theory extended to include velocity correlations, contact duration, and the presence of a particle bed for steady, fully developed gravitational flow of grains in a vertical pipe with a circular cross section and bumpy boundaries. We determine the variation of average flow properties with both pipe radii and total flow rate. There are two solutions associated with high and low pressures for a given pipe radius and total flow rate.