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Register a new userThe Macroscopic Approach to Extended Thermodynamics with 14 Moments, Up to Whatever Order
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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.
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The 14 moments model for dense gases, introduced in the last years by Ruggeri, Sugiyama and collaborators, is here considered. They have found the closure of the balance equations up to second order with respect to equilibrium; subsequently, Carrisi has found the closure up to whatever order with respect to equilibrium, but for a more constrained system where more symmetry conditions are imposed. Here the closure is obtained up to whatever order and without imposing the supplementary conditions. It comes out that the first non symmetric parts appear only at third order with respect to equilibrium, even if Ruggeri and Sugiyama found a non symmetric part proportional to an arbitrary constant also at first order with respect to equilibrium. Consequently, this constant must be zero, as Ruggeri, Sugiyama assumed in the applications and on an intuitive ground.
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.
Recently the 14 moments model of Extended Thermodynamics for dense gases and macromolecular fluids has been considered and an exact solution, of the restrictions imposed by the entropy principle and that of Galilean relativity, has been obtained through a non relativistic limit. Here we prove uniqueness of the above solution and exploit other pertinent conditions such us the convexity of the function $h$ related to the entropy density, the problem of subsystems and the fact that the flux in the conservation law of mass must be the moment of order 1 in the conservation law of momentum. Also the solution of this last condition is here obtained without using expansions around equilibrium. The results present interesting aspects which were not suspected when only approximated solutions of this problem were known.
In this paper, we survey our recent results on the variational formulation of nonequilibrium thermodynamics for the finite dimensional case of discrete systems as well as for the infinite dimensional case of continuum systems. Starting with the fundamental variational principle of classical mechanics, namely, Hamiltons principle, we show, with the help of thermodynamic systems with gradually increasing level complexity, how to systematically extend it to include irreversible processes. In the finite dimensional cases, we treat systems experiencing the irreversible processes of mechanical friction, heat and mass transfer, both in the adiabatically closed and in the open cases. On the continuum side, we illustrate our theory with the example of multicomponent Navier-Stokes-Fourier systems.
In the recent literature there has been a resurgence of interest in the fourth-order field-theoretic model of Pais-Uhlenbeck cite {Pais-Uhlenbeck 50 a}, which has not had a good reception over the last half century due to the existence of {em ghosts} in the properties of the quantum mechanical solution. Bender and Mannheim cite{Bender 08 a} were successful in persuading the corresponding quantum operator to `give up the ghost. Their success had the advantage of making the model of Pais-Uhlenbeck acceptable to the physical community and in the process added further credit to the cause of advancement of the use of ${cal PT} $ symmetry. We present a case for the acceptance of the Pais-Uhlenbeck model in the context of Diracs theory by providing an Hamiltonian which is not quantum mechanically haunted. The essential point is the manner in which a fourth-order equation is rendered into a system of second-order equations. We show by means of the method of reduction of order cite {Nucci} that it is possible to construct an Hamiltonian which gives rise to a satisfactory quantal description without having to abandon Dirac.
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