Kinetic theory models for granular mixtures with unequal granular temperature. Derivation of analytical constitutive equations

We consider kinetic theory of granular mixtures in which each solid phase is described by its own granular temperature. The particle velocities are assumed to follow a Mawellian distribution with zero-order approximation of the Boltzmann equation. The literature proposes a number of approximate solutions to the constitutive equations resulting from particle collisions. Here, we derive and show that analytical solutions exist, and reveal that some of the proposed approximate solutions are inaccurate whereas others might give good approximations. We also review the balance laws and constitutive relations resulting from the mono- and poly-disperse kinetic granular flow model frameworks. Such a review has been done here because the model framework on kinetic theory of mono-disperse granular flow systems has sometimes been adopted in models for granular mixtures. The consequence of such a simplification is that the kinetic energy loss in a collision between two particles of different species is equally shared among them.