The Equation of Entropy Balance

The local entropy density, j(r, t), was used frequently in Chapter 10. The total entropy S of the fluid contained in a region of volume V is then 

According to the second law of thermodynamics the entropy of a system is not a conserved quantity in an irreversible process. Applying Eq. (10.3.8) to Eq. (13.2.1) then gives 

This is the local equation for the entropy density. It is of fundamental importance in what follows. The quantity a, the local entropy production, is the entropy produced irreversibly per unit time per unit volume and is analogous to the property a a defined prior to Eq. (10.3.4). The quantity Js is the entropy flux and V Js represents the rate of change of the local entropy due to an inflow of entropy from neighboring regions of the fluid. 

In an irreversible process the total entropy production as well as the local entropy production must be positive (a 0). 

Thus we see that according to Eq. (13.2.2) the entropy change of a local fluid element is due to the exchange of entropy with neighboring fluid elements and the internal production of entropy in an irreversible process. 

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There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

The equation of entropy balance in terms of the entropy per unit volume, s, is... 

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