Constitutive equations for granular flow with uniform mean shear and   spin fields
  Numerical simulations of two-dimensional granular flows under uniform shear and external body torque were performed in order to extract the constitutive equations for the system. The outcome of the numerical simulations is analyzed on the basis of the micropolar fluid model. Uniform mean shear field and mean spin field, which is not subordinate to the vorticity field, are realized in the simulations. The estimates of stresses based on kinetic theory by Lun [Lun, J. Fluid Mech., 1991, 233, 539] are in good agreement with the simulation results for a low area fraction $\nu=0.1$ but the agreement becomes weaker as the area fraction gets higher. However, the estimates in the kinetic theory can be fitted to the simulation results up to $\nu=0.7$ by renormalizing the coefficient of roughness. For a relatively dense granular flow ($\nu=0.8$), the simulation results are also compared with Kanatani's theory [Kanatani, Int. J. Eng. Sci., 1979, 17, 419]. It is found that the dissipation function and its decomposition into the constitutive equations in Kanatani's theory are not consistent with the simulation results. 
K. Takechi, K. Yoshida, T. Arimitsu
cond-mat.stat-mech cond-mat.soft