
Generalized Constitutive Laws of Oriented Granular Gas
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In this presentation, we develop a continuum model for the thermomechanics of granular media consisting of rigid oriented particles by extending the state variables to account for the granular state. The state variables to describe oriented granular gas are the granular temperature and the granular orientation in addition to the standard state variables which are density, velocity and thermodynamic temperature. The granular temperature is a nonnegative scalar value that represents the energy of the agitated particles (energy of the translation and rotation velocities fluctuations). The orientation is a symmetric, trace-free and bounded second order tensor representing the particles orientation distribution. The orientation tensor measures the deviation from isotropic orientation distribution, which is taken to be the orientational equilibrium state. The two additional granular state variables are governed by associated balance laws that are the granular energy and orientation which are in addition to the conservation of mass, balances of linear momentum, thermal energy and entropy. Taking the entropy to be quadratic with respect to the orientation tensor, the Gibbs equation is used to identify entropy fluxes (driving to equilibrium) in terms of the forces (deviation from equilibrium) and the state variables. Using the non-negative entropy production requirement, the constitutive laws are constructed for all the fluxes. The constitutive laws for the fluxes, as first order approximation, are constructed by taking the fluxes to be linear combinations of the forces. The phenomenological coefficients matrix is obtained using the Onsager-Casimir reciprocity principles, which is required to be positive semi-definite. The general representation of the phenomenological coefficients matrix includes all the possible coupling, and the phenomenological coefficients are tensor of up to sixth order. This formulation is too general and can only serve as a reference to generalize the constitutive laws if needed. To obtain a meaningful model we must make substantial simplifications by neglecting, what we consider, secondary effects. The developed constitutive laws are compared with models available in the literature for spherical particles as a special case. It is shown that the developed constitutive laws can describe spherical particles, and more importantly, to provide a systematic framework to generalize these laws to oriented particles.