Second Law of Thermodynamics in a Thermomechanical Continuum Eulerian Description

4 Second Law of Thermodynamics in a Thermomechanical Continuum Eulerian Description 

Assume that, corresponding to the entropy S of a body, there exists an entropy density function s, measured per unit mass of a continuum, such that 

As mentioned in Appendix D.4.2, the entropy change dS of a system consists of two parts in the form dS = dS +dS, where dS is due to an energy exchange with the surroundings and dS is due to an internal irreversible process, so that the Second Law of Thermodynamics can be written as dS 0. 

This can be applied to the continuum as understood from the definition of s (D.72). That is, the change of entropy ds is divided into two parts as ds = ds +ds, where ds is a result of energy exchange, and ds is the internal entropy production. For the reversible process ds = 0 and for the irreversible process ds 0. Thus the Second Law of Thermodynamics for the continuum can be written as 

Note that CIT may be able to come to the same conclusion however, it is important that we do not require the assumption of a local-equilibrium. Since the Second 

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