Some Exact Mathematical Solutions for Granular Stock Piles and Granular Flow in Hoppers

Abstract

The flow of granular materials in the presence of gravity in hoppers and the storage of granular materials as a stock pile occur in many industrial situations. The governing ordinary differential equations for two-dimensional wedges and three-dimensional cones are highly nonlinear and there are no known general solutions, apart from that we have given for a special angle of internal friction. Here, we give the overall picture relating to those special cases which give rise to analytical solutions for the two problems of granular flow through hoppers and the stress distributions at the base of stock piles. These equations are fundamental to granular mechanics and previously only some special isolated exact solutions have been known. We list here a number of new exact analytical solutions applying for the two special cases of β= ±1, noting that β= sin φ where φ is the angle of internal friction. The case β= −1 corresponds to a non-physical material, but there are materials such as silica and alumina cake which do indeed exhibit large angles of internal friction and the case β= 1 is by no means unrealistic. However, all the solutions presented are meaningful mathematical solutions of the governing equations and constitute the only known general solutions of these important equations. For certain cases, a full independent numerical solution has been obtained and shown to coincide with the appropriate exact analytical solution.