Axial Symmetric Granular Flow Due to Gravity in a Circular Pipe

Axisymmetric granular flows in vertical cylindrical pipes under action of gravity are studied using mathematical particle–particle models based on the Hertz–Mindlin theory. By and large, in granular flows, the density field and the pressure are unknown scalar functions. A well-known relationship between these fields gives the pressure field a power law of the density. The aim of this paper was to study unsteady, axisymmetric, fully developed granular flow under gravity action in a vertical cylindrical pipe, under the assumptions that the density field is constant and the velocity on the pipe’s wall is time-dependent. Using integral transforms method and appropriate initial-boundary conditions, the analytical solution for axial velocity is determined. The obtained analytical solution is used to determine the steady-state solution (the solution for large values of the time). The properties of the flow in some particular cases of the velocity on the pipe’s surface are analyzed and the transient flow is compared with the stationary one.