A comparison of the solutions of some proposed equations of motion of granular materials for fully developed flow down inclined planes

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Abstract
In the past few years kinetic theory has been used to derive equations of motion for rapidly shearing granular materials, and there have been empirical extensions of these to take into account stress transmitted by sustained sliding and rolling contacts between particles. The equations are complicated and solutions have been generated only for very simple flows. In this paper three forms for the equations of motion are considered; one representing interaction by collisions only, one which is a high-density asymptotic form of this, and a third which includes terms representing the ‘frictional’ stresses associated with the sustained contacts referred to above. Solutions are found for fully developed flow under gravity down an inclined plane, and it is shown that the relation between the flow rate and the depth of the flowing layer predicted by the first two sets of equations is not in accord with observations. The third form appears to eliminate much of the discrepancy, but its predictions have not been explored over the whole parameter space. It is emphasized that the form of the solutions should be studied over a wide range of operating conditions in order to assess the usefulness of proposed equations.