Dense granular flow down an inclined plane: from kinetic theory to granular dynamics

The hydrodynamics of the dense granular flow of rough inelastic particles down an inclined plane is analysed using constitutive relations derived from kinetic theory. The basic equations are the momentum and energy conservation equations, and the granular energy conservation equation contains a term which represents the dissipation of energy due to inelastic collisions. A fundamental length scale in the flow is the ‘conduction length’ $\delta=(d/(1-e_n)^{1/2})$, which is the length over which the rate of conduction of energy is comparable to the rate of dissipation. Here, d is the particle diameter and $e_n$ is the normal coefficient of restitution. For a thick granular layer with height $h\gg\delta$, the flow in the bulk is analysed using an asymptotic analysis in the small parameter $\delta/h$. In the leading approximation, the rate of conduction of energy is small compared to the rates of production and dissipation, and there is a balance between the rate of production due to mean shear and the rate of dissipation due to inelastic collisions. A direct consequence of this is that the volume fraction in the bulk is a constant in the leading approximation. The first correction due to the conduction of energy is determined using asymptotic analysis, and is found to be $O(\delta/h)^2$ smaller than the leading-order volume fraction. The numerical value of this correction is found to be negligible for systems of practical interest, resulting in a lack of variation of volume fraction with height in the bulk