Fluid with Linear Transport Properties

Our main goal is to apply the admissibility principle to fluids [39, 53, 75-78], namely to nonsimple fluid (3.127) (the special cases of simple (3.129) and thermoelastic (3.130) fluids will be discussed at the end of this section but the most important are fluids with linear transport properties contained in Sects. 3.7 and 3.8). In nonsimple fluid (3.127) it is sufficient to use the field of velocity (instead of motion), cf. (3.14)i, (3.15)i. Therefore we define the thermokinetic process in fluids as the fields of (instead of (3.114))... 

In this paragraph we specialize the results for the nonsimple fluid (3.171)-(3180) on the linear dependence in vectors and tensors i.e., in D, g and h (while the dependence on scalars p, T may be nonlinear) [9, 14, 23, 24, 27, 45]. We denote this model as a linear fluid or fluid with linear transport properties because the results describe the classical Navier-Stokes (Newtonian) and Fourier fluid with linear viscosity and heat conduction at the same time the classical thermodynamic relations (local equilibrium) are valid. 

Therefore, in this section, we study the regular linear fluid body or the fluid with linear transport properties with regular responses consisting of the linear fluid body of Sect. 3.7 to which we add the regularity properties (3.232)-(3.234). 

This is the core chapter of our book. Here we discuss rational thermodynamics of mixtures and our main interest is the classical subject— the chemically reacting fluid mixture composed from fluids with linear transport properties (linear fluid mixture). In the last section, we discuss the relation of our results to those classical. 

Chemically Reacting Mixture of Fluids with Linear Transport Properties... 

In this book, we confine ourselves only to the special case of fluids mixture (4.128) which is linear in vector and tensor variables.We denote it as the chemically reacting mixture of fluids with linear transport properties or simply the linear fluid mixture [56, 57, 64, 65]. Then (see Appendix A.2) the scalar, vector and tensor isotropic functions (4.129) linear in vectors and tensors (symmetrical or skew-symmetrical) have the forms ... 

We now deduce basic thermodynamic properties of the mixture of fluids with linear transport properties discussed in Sect. 4.5. Among others, we show that Gibbs equations and (equilibrium) thermodynamic relationships in such mixtures are valid also in any non-equilibrium process including chemical reactions (i.e. local equilibrium is proved in this model) [56, 59, 64, 65, 79, 138]. 

Therefore, the classical relations of thermochemistry were obtained. Especially, the Gibbs equations (4.201)-(4.206) are valid in arbitrary process in this chemically reacting mixture of fluids with linear transport properties, i.e. the principle of local equilibrium is valid in this mixture. But we show in the following relations that this accord with classical thermochemistry (e.g. [138]) is not quite identical indeed, if we differentiate (4.211) and use (4.22), (4.23) we obtain... 

At the end, we summarize the results of the model of a reacting mixture of fluids with linear transport properties from Sects. 4.5 and 4.6 (properties such as kinematics, stoichiometry and balances of mass, momentum and their moment, energy and entropy inequality are as in Sects. 4.2, 4.3 and 4.4). Constitutive equations, their properties and final form of entropy production are given in the end of Sect. 4.5 (from Eq. (4.156)), further thermodynamic quantities and properties are given at the... 

From the model of (chemically) reacting (non-simple) mixture of fluids with linear transport properties simpler models may follow, e.g. the non-reacting mixture (where (4.15) is valid identically and regularity 2. plays no role), the incompressible fluid mixture (which should have similar properties as incompressible fluid from the end of Sect. 3.7.) or the simple mixtures (where density gradients are not a priori present in constitutive equations, see below (4.129) in Sect. 4.5). These simplified models will be thoroughly discussed in Sect.4.8. 

Chapter 3 adds also the description of spatial distribution (gradients). Only single fluid is considered for the sake of simplicity and preparation of the basics for the subsequent treatment of mixtures. Mathematics necessary for the spatial description is introduced in Sect. 3.1. Section 3.2 in the same chapter stresses the importance of the referential frame (coordinate system) and its change in the mathematical description. Sections 3.3—3.6 shows the development of final material model (of a fluid) within our thermodynamic framework, consistent with general laws (balances) as well as with thermodynamic principles (the First and Second Laws and the principles of rational thermodynamics). The results of this development are simplified in Sect. 3.7 to the model of (single) fluid with linear transport properties. Sections 3.6 and 3.7 also show that the local equilibrium hypothesis is proved for fluid models. The linear fluid model is used in Sect. 3.8 to demonstrate how the stability of equilibrium is analysed in our approach. 

---