Gravity flow of granular materials in contracting cylinders and tapered tubes

This paper presents two applications of the double-shearing theory for flow of granular materials under gravitational forces for axially symmetric flow. In the first the material is contained in a vertical circular cylinder which compresses the material by contracting radially. Exact solutions for the stress and velocity fields are derived, under the boundary conditions that the cylinder wall is either `perfectly rough' or subject to Coulomb friction. In the second problem the material flows under gravity through a tube with circular cross-section of radius that decreases slowly with depth. For this problem an approximate solution is derived that is accurate to first order in the slope of the tube wall relative to the vertical. It is also shown that an alternative theory, in which it is postulated that the principal axes of the stress and rate-of-strain tensors are coincident, leads to the prediction of unacceptable singularities in the flow field in the interior of the material.