Thermodynamic Analysis II Residual Inequality, Thermodynamic Equilibrium, Isotropic Expansion

Abstract

After the full exploitation of the LIU identities in the preceding chapter, we draw in this chapter some (but not all) inferences which follow from the condition that the entropy production density assumes its minimum value in thermodynamic equilibrium. This requirement implies that ∂π^ρη/∂n _I | _E = 0, where π^ρη is the entropy production density and n _I are those independent variables which vanish in equilibrium. The evaluation of this condition first requires π^ρη to be expressed in an appropriate form. Choosing for n _I , in turn, the variables v _α , $\dot \theta$, ∇ θ and D _α , which are the constituent velocities, the time rates of change of the temperature, the temperature gradient and constituent stretchings, allows evaluation of the equilibrium representations of the constituent interaction forces, entropy, heat flux vector and constituent stress tensors, which exhibit a clear structure of their dependences on (i) a thermodynamic potential (HELMHOLTZ-like free energy) and thermodynamic, configuration and saturation pressures, (ii) extra entropy flux, (iii) frictional effects via their production terms and (iv) interaction rate densities of constituent mass and volume fractions. It becomes very clear how the various equilibrium terms are affected if simplifying assumptions are made about the functional dependencies of the above mentioned production terms.

The remainder of the chapter deals with quasi-linear expansions of the constitutive relations for the interaction rate densities of mass, volume fraction and for the extra entropy flux vector about a state in which these quantities are linear in the thermodynamic non-equilibrium variables and in the constituent mass and volume fraction gradients. The formulae for the constituent interaction forces, entropy, equilibrium heat flux and constituent stresses are slightly simplified thereby.