On linear dynamical equations of state for isotropic media I

This paper represents a contribution to the modelling of non-isothermal viscous fluids in non-equilibrium. The dissipative fluxes, i.e. the heat flux and the viscous stress tensor, are introduced as independent variables in the constitutive equations; their rate of change is governed by first order differential equations. Restrictions on the constitutive equations are, as usually, placed by the general principles of continuum mechanics and the second law of thermodynamics. The form of the second law is not Clausius-Duhem inequality but that proposed by Müller, wherein the entropy flux is given by a constitutive equation. Some extensions are developed. In particular the model is generalized to describe some classes of non-simple fluids. This is accomplished by including the gradients of the density and the dissipative fluxes into the set of independent variables. The entropy flux is then shown to contain extra-classical terms. IntroductionThe classical theory of non-equilibrium thermodynamics was essentially developed by Onsager [1], Eckart [2], Prigögine [3], Meixner and Reik [4], De Groot and Mazur [5]. It rests on the local equilibrium hypothesis. It is postulated that at a given time t, it is possible to associate a local state to each macroscopic part of the continuum. This state is defined by a complete number of variables which are the variables introduced in the thermostatic description. The hypothesis has been justified on the bases of the kinetic theory of low density gases [6]. The classical non-equilibrium thermodynamics is appropriate for describing systems not too far from equilibrium (linear range). Clearly, it is inadequate for treating more complicated situations, like nonlinear problems or systems involving complicated structures, e.g. polymers or micropolar materials. Moreover, the classical theory leads to the paradox of infinite velocity of propagation for the thermal and viscous signals. These are the reasons why many efforts have been devoted to the search of more general thermodynamic theories. The latter can be classified as follows: the extended, the rational and the entropy free thermodynamics. In the extended theory, it is assumed that the determination of the local state does not only require the introduction of the thermostatic variables, but also extra variables. Müller '[7] and later on, Lebon [8], Lebon et aL[9], Jou et al. [10], Gyarmati [11] take as additional variables the dissipative fluxes, like the heat flux and the viscous stress. Other people, Bataille and Kestin [12, 13], Kluitenberg [14], Mandel [15], introduce so-called internal or hidden variables in order to complete the description. The second class of formalisms is known under the name of "rational thermodynamics". Its main protagonists are Noll [16], [17], Coleman [18], Gurtin [19], Truesdell [20]. The notion of state is abandoned, as well as the notion of internal variables and, instead, the concept of history is introduced: the behaviour of a given material is assumed to be determined not only by the present valu...

Section: A Generalized Ideal Gas Theorysupporting

confidence: 50%

Section: Introductionmentioning

confidence: 99%

A Generalized Theory of Thermoviscous Fluids

Lebon¹,

Rubı́²

1980

“…In virtue of representation o theorems, the most general forms of a l and a 2 are 3i = a 10 q + a n P v · q + a 12 P v2 · q , (25) and «2 = «20 U + « 21 P v + <* 22 …”

Section: The Generalized Gibbs Equationmentioning

confidence: 99%

“…It is supposed that the specific non-equilibrium entropy depends, in addition to the usual variables, on quantities that vanish at equilibrium. The introduction of extra variables is not new and was widely used to describe relaxation phenomena [21][22][23][24][25][26]. Our approach differs from the latter in that the extra variables are specified from the start and identified with the usual dissipative fluxes, as the heat flux and the viscous pressure tensor.…”

Section: Introductionmentioning

confidence: 99%

A Dynamical Interpretation of Second-Order Constitutlve Equations of Hydrodynamics

Jou¹,

Casas-Vázques²

1979

The classical linear irreversible thermodynamics is extended to the non-linear range. Heat conducting viscous isotropic fluid systems are considered. It is assumed that the non-equilibrium entropy depends, besides on its classical variables, on the heat flux and the viscous pressure tensor. Representation theorems of isotropic functions are used to determine the form of the state equations, the entropy flux and the generalized constitutive equations. The latter are viewed as evolution equations for the heat flux and the viscous pressure tensor. The theory is compared with the thirteen-moments method of the kinetic theory of gases, and it is seen to generalize previous formulations of so-called extended irreversible thermodynamics.

“…During the last thirty years, some authors have proposed to include dissipative fluxes as natural variables of the fundamental thermodynamic equations of macroscopic systems, when dealing with nonequilibrium thermodynamics [1][2][3][4][5][6]. In the last six years, a wide and growing literature has developed on the grounds of such a hypothesis [7][8][9][10][11][12][13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Equilibrium Third Moments and Non-Equilibrium Second Moments of Fluctuations of Hydrodynamic Dissipative Fluxes

Jou¹,

Casas-Vázquez²

1983

We generalize a previous version of nonequilibrium entropy for viscous heat conducting fluids by including a new term of third order in the dissipative fluxes. This new term is seen to be related to equilibrium third moments and nonequilibrium second moments of fluctuations of the heat flux and the viscous pressure tensor, and to the relaxation matrix of nonequilibrium fluctuations. We also discuss the most convenient formulation of the second law in the presence of third-order terms. The positiveness of the total entropy production leads to results beyond the scope of the formulation based on the positiveness of just the second-order entropy production.