The constitutive relation for the granular flow of rough particles, and its application to the flow down an inclined plane

Abstract

A perturbation expansion of the Boltzmann equation is used to derive constitutive relations for the granular flow of rough spheres in the limit where the energy dissipation in a collision is small compared to the energy of a particle. In the collision model, the post-collisional relative normal velocity at the point of contact is $-e_n$ times the pre-collisional normal velocity, and the post-collisional relative tangential velocity at the point of contact is $-e_t$ times the pre-collisional relative tangential velocity. A perturbation expansion is employed in the limit $(1 - e_n)\,{=}\,\varepsilon^2 \ll 1$, and $(1 - e_t^2) \propto \varepsilon^2 \ll 1$, so that $e_t$ is close to $\pm 1$. In the ‘rough’ particle model, the normal coefficient of restitution $e_n$ is close to 1, and the tangential coefficient of restitution $e_t$ is close to 1. In the ‘partially rough’ particle model, the normal coefficient of restitution $e_n$ is close to 1; and the tangential coefficient of restitution $e_t$ is close to $-1$ if the angle between the relative velocity vector and the line joining the centres of the particles is greater than the ‘roughness angle’ (chosen to be $(\upi/4)$ in the present calculation), and is close to 1 if the angle between the relative velocity vector and the line joining the centres is less than the roughness angle. The conserved variables in this case are mass and momentum; energy is not a conserved variable in the ‘adiabatic limit’ considered here, when the length scale is large compared to the ‘conduction length’. The results for the constitutive relations show that in the Navier–Stokes approximation, the form of the constitutive relation is identical to that for smooth particles, but the coefficient of shear viscosity for rough particles is 10%–50% higher than that for smooth particles. The coefficient of bulk viscosity, which is zero in the dilute limit for smooth particles, is found to be non-zero for rough and partially rough particles, owing to the transport of energy between the translational and rotational modes. In the Burnett approximation, there is an antisymmetric component in the stress tensor for rough and partially rough particles, which is not present for smooth particles.

The constitutive relations are used to analyse the ‘core region’ of a steady granular flow down an inclined plane, where there is a local balance between the production of energy due to the mean shear and the dissipation due to inelastic collisions. It is found that realistic results, such as the decrease in density upon increase in the angle of inclination near close packing, are obtained for the rough and partially rough particle models when the Burnett coefficients are included in the stress tensor, but realistic results are not obtained using the constitutive relations for smooth particles. This shows that the flow dynamics is sensitive to the numerical values of the viscometric coefficients, and provides an indication of the minimal model required to capture the flow dynamics.