  The hydrodynamic conservation equations and constitutive relations for a
binary granular mixture composed of smooth, nearly elastic, hard spheres with
non-equipartition energies and different mean velocities are derived. This
research is aimed to build three-dimensional kinetic theory to characterize the
behaviors of two species of particles suffering different forces. The standard
Enskog method is employed assuming a Maxwell velocity distribution for each
species of particles. The collision components of the stress tensor and the
other parameters are calculated from the zeroth- and first-order approximation.
Our results demonstrate that three factors, namely the ratios between two
granular masses, temperatures and mean velocities all play important roles in
the stress-strain relation of the binary mixture. The collision frequency and
the solid viscosity escalate with increasing of two granular temperatures and
are maximized when both of two granular temperatures reach maximums. The first
part of the energy source varies greatly with the mean velocities of spheres of
two species, and further, it reaches maximum at the maximum of relative
velocity between two mean velocities of spheres of two species.
