Energy flux, collisional contribution

If we choose as the summational invariants m, mC and mC, we obtain the Cauchy set of conservation equations named the continuity (2.217), the equation of motion (2.223), and the equation of energy (2.230). The only difference in the final result is that the pressure tensor, p, and the heat flux vector, q, are made up of two parts (i.e., a kinetic and a collisional contribution) ... 

Overlaps reduce the frequency of collisions while chattering replaces strong binary collisions with numerous weak ones. In a range of volume fractions between 0.49 and 0.60 and coefficients of restitution greater than 0.60, the influence of the correlations on the fluxes of momentum and energy is compensated for by nonlocal transport associated with the correlated motion. Consequently, overlaps and chattering first influence the collisional rate of dissipation. For denser and/or more inelastic flows, anisotropic and rate-independent contributions to the pressure and shear stress, associated with chains or clusters that eventually span the flow, are anticipated to develop (Goldman and Swinney 2006, Hatano et al. 2007, Schroter et al. 2007, Chialvo et al. 2012). 

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