We use the principles of non-equilibrium thermodynamics to rigorously formulate the transport equations for granular systems consisting of oriented particles. The state variables are taken to be the density, velocity, thermal temperature, granular temperature (particles agitation) and the orientation tensor. The evolution of the state variables is governed by the associated balance laws in terms of fluxes. The contributions of the granular agitation energy and orientation to entropy are introduced into the Gibbs equation. The balance of entropy is used to identify the entropy production as the product of thermodynamics forces and fluxes. Using classical linear non-equilibrium thermodynamics the fluxes are considered to be linear in the thermodynamic forces. The Onsager–Casimir reciprocal relations and the representation theorem of isotropic tensors provide further restrictions that simplify the formulation. The non-negative entropy production requirement is satisfied by restricting the matrix of phenomenological coefficients to be positive semidefinite. Similarly the boundary conditions are constructed. The transport coefficients are then determined by comparison with available results from the granular kinetic theory of spherical particles and other available results for oriented particles. It is shown that not only these results are well captured, but also the formulation provides a framework for further generalization. The significant contribution of this work is the rigorous formulation of a physically admissible generalization to granular gases of oriented particles which reveals the role of the orientation in the transport equations and identifies couplings that might otherwise be omitted.
Amereh et al. (Mon,) studied this question.