The Macroscopic Approach to Extended Thermodynamics with 14 Moments, Up to Whatever Order

Extended Thermodynamics is the natural framework in which to study the physics of fluids, because it leads to symmetric hyperbolic systems of field laws, thus assuming important properties such as finite propagation speeds of shock waves and well posedness of the Cauchy problem. The closure of the system of balance equations is obtained by imposing the entropy principle and that of galilean relativity. If we take the components of the mean field as independent variables, these two principles are equivalent to some conditions on the entropy density and its flux. The method until now used to exploit these conditions, with the macroscopic approach, has not been used up to whatever order with respect to thermodynamical equilibrium. This is because it leads to several difficulties in calculations. Now these can be overcome by using a new method proposed recently by Pennisi and Ruggeri. Here we apply it to the 14 moments model. We will also show that the 13 moments case can be obtained from the present one by using the method of subsystems

An exact macroscopic extended model with many moments

Extended Thermodynamics is a very important theory: for example, it predicts hyperbolicity, finite speeds of propagation waves as well as continuous dependence on initial data. Therefore, it constitutes a significative improvement of ordinary thermodynamics. Here its methods are applied to the case of an arbitrary, but fixed, number of moments. The kinetic approach has already been developed in literature; then, the macroscopic approach is here considered and the constitutive functions appearing in the balance equations are determined up to whatever order with respect to thermodynamical equilibrium. The results of the kinetic approach are a particular case of the present ones

New results in extended thermodynamics

The subject of this thesis lays in the context of Extended Thermodynamics. The classical methodology to solve the system of balance equations, in E.T., leads to difficult calculations. Recently a new methodology has been proposed to overcome these difficulties and to have more elegant equations, easier to solve. During my PhD, I worked in the direction of proving that the new methodology can be applied to the many moments case, to materials different from ideal gases and also in the relativistic case and not only in the classical one; furthermore I have been able to find the exact solutions for many of these problems. After a brief introduction on Extended Thermodynamics, the new methodology will be shown, applied to the 5, the 13 and the 14 moments case for ideal gases. The last of these is a model obtained by Pennisi and myself. Afterwards I will show how we have applied the new methodology to dense gases and macromolecular fluids for the 13 and 14 moments case. After that I will present the results of my publications regarding the cases with an arbitrary but fixed number of moments for ideal gases, in the classical and in the relativistic context, and I will also show what happens when we consider their subsystems. Finally it will be considered a new kind of system of balance equations that is suggested from the non relativistic limit of the relativistic case

An Exact Solution for the Macroscopic Approach to Extended Thermodynamics of Dense Gases with Many Moments

Extended Thermodynamics of Dense Gases with an arbitrary but fixed number of moments has been recently studied in literature; the arbitrariness of the number of moments is linked to a number N and the resulting model is called an (N)−Model. As usual in Extended Thermodynamics, in the field equations some unknown functions appear; restriction on their generalities are obtained by imposing the entropy principle, the Galilean relativity principle and some symmetry conditions. The solution of these conditions is called the ”closure problem” and it has not been written explicitly because an hard notation is necessary, but it has been shown how the theory is selfgenerating in the sense that, if we know the closure of the (N) −Model, than we will be able to find that of the (N + 1) − Model. Instead of this, we find here an exact solution which holds for every number N

A macroscopic solution for a model suggested by the non relativistc limit of relativistic extended thermodynamics

Extended thermodynamics provides a good framework for studying the physics of fluids, because it leads to symmetric hyperbolic systems of field laws having important properties such as finite propagation speeds of shock waves and well posedness of the Cauchy problem. For the case with many moments the model classically used leads to errors and problems: for example it includes models that do not have a relativistic counterpart or that cannot be represented in a kinetic way. To overcome these difficulties Pennisi and I proposed a model belonging from the relativistic one through its non relativistic limit. We proved that, even for this model, it is possible to use the classical procedures to close the system and the new methodology recently proposed by Pennisi and Ruggeri to exploit the Galilei relativity principle. We found the solution for our model by using a four-dimensional notation. It depends on a family of arbitrary scalar functions arising from integration. Here the solution will be found, by using the classical notation and it will proven that, by fixing a certain order $p$ up to equilibrium and only one scalar valued arbitrary function, everything is determined in terms of that single function. The same result has been found also with a generalized kinetic approach. Up to a fixed order $p$ the two methods lead to the same solution and then we are allowed to use the generalized kinetic method whose results are expressed in an easier and handier way. This is not the case for orders greater than $p$ but because of the arbitrariness of $p$ we can reach every desired degree of approximation even with the kinetic approach

A further condition in the extended macroscopic approach to relativistic gases

An exact macroscopic extended model, with many moments, for relativistic gases has been recently proposed in literature. However, a further condition, arising from the exploitation of the entropy principle, has not been imposed, even if its presence is evident in the case of a charged gas and when the electromagnetic field acts as an external force. In the present paper we exploit it and we prove that it amounts in many identities plus some residual conditions which allow to determine the arbitrary single variable functions present in the general theory. The result is that they are polynomials of increasing degree with respect to equilibrium, which coefficients are arbitrary constants. Even in such case the macroscopic model remains more general than the kinetic one

Monatomic gas as a singular limit of relativistic theory of 15 moments with non-linear contribution of microscopic energy of molecular internal mode

Recently a new relativistic model of polyatomic gases has been proposed, by Arima-CarrisiPennisi-Ruggeri (2022), in the context of Rational Extended Thermodynamics. It is based on a hierarchy of 15 moments of the Boltzmann-Chernikov equation that appropriately takes into account the non-linear contribution of the microscopic total energy of the molecule (the sum of the rest energy and of the energy of the molecular internal modes). In this paper, in the singular limit, under initial conditions compatible with monatomic gases, we prove that the 15-moments model for polyatomic gases leads, to the well-known 14-moments model of monatomic gases

Wave speeds in the macroscopic relativistic extended model with many moments

An exact macroscopic extended model for relativistic gases, with an arbitrary number of moments, is present in the literature. It is determined except for a numberable family F of single variable functions whose physical meaning remains an open problem; a possibility is that it allows to apply the theory to a wider set of materials. Other models appearing in literature are particular cases of this macroscopic one when all these arbitrary functions are zero. Here we exploit equations determining wave speeds for that model. We find interesting results; for example, the whole system for their determination can be divided into independent subsystems which are expressed by linear combinations, through scalar coefficients, of tensors all of the same order. As expected, these wave speeds for the macroscopic model depend on F. Moreover, some wave speeds (but not all of them) are expressed by square roots of rational numbers