Constitutive Equations of Uniform Fluids

In this Section, we obtain the ultimate form of constitutive equations for model A, B, C, D (2.6)-(2.9) using the constitutive principle of admissibility in the form mentioned at the end of preceding Sect. 2.1 [1, 2, 4-8]. 

Fluid without memory has constitutive equations (2.6) and therefore (see (2.12)) the constitutive equation for the free energy is 

According to the admissibility principle (in the form quoted at the end of the preceding Sect. 2.1) this inequality (2.15) must be valid in any thermokinetic process and therefore also in such a one where T 0, y 0, y are arbitrary constants and f maybe an arbitrary real number. Then coefficient standing next to T in (2.15) must be zero otherwise it is possible to find such a real t that inequality (2.15) is not valid. Here (and often in the following) the Lemma A. 5.1 of Appendix A. 5 is used (with X = f). Therefore, 

Therefore, the free energy is a potential for entropy and pressure, i.e., Gibbs equations are valid 

Model A is in fact the main material model studied in the classical equilibrium thermodynamics. Each state in this model A may be an equilibrium state from postulate S4 in Sect. 1.2 by fixing not only V, T, but also V = 0, t = 0 then not only power but, by (2.1), (2.6)i, also heating should be zero 2 = 0. Its persistence is assured (cf. Rems. 6 in this chapter, 11 in Chap. 1) by stability conditions.  

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