Plane simple shear of smooth inelastic circular disks: the anisotropy of the second moment in the dilute and dense limits

We consider a plane, steady, homogeneous flow of circular disks. The disks are identical, smooth, and inelastic. We adopt the assumption of molecular chaos and introduce an anisotropic Maxwellian velocity distribution function based on the full second moment of the velocity fluctuations. In the limits of dilute and dense flows, we determine approximate analytic solutions of the balance law for the second moment that result in stresses whose qualitative behaviour and magnitudes are in good agreement with numerical simulations.