Extended kinetic theory for granular flow over and within an inclined erodible bed

We employ kinetic theory, extended to incorporate the influence of velocity correlations, friction and particle stiffness, and a model for rate-independent, elastic components of the stresses at volume fractions larger than a critical value, in an attempt to reproduce the results of discrete-element numerical simulations of steady, fully developed, dissipative, collisional shearing flows over and within inclined, erodible, fragile beds. The flows take place between vertical, frictional sidewalls at different separations with sufficient total particle flux so that differently inclined, erodible beds result. Numerical solutions of the spanwise-averaged differential equations of the theory and the associated boundary conditions are seen to be capable of reproducing profiles of stresses, solid volume fraction, average velocity and the strength of the particle velocity fluctuations, both in the rapid collisional flow above the bed and in the slower creeping flow within the bed. The indication is that extended kinetic theory has the unique ability to faithfully describe steady, inhomogeneous, granular shearing flows, ranging from dilute to extremely dense, using balances of momentum and energy and employing boundary conditions that are associated with the balances, with a small number of physically determined, microscopic parameters