Tommaso Ruggeri

Relativistic Thermodynamics of Gases

Abstract Relativistic thermodynamics of degenerate gases is presented here as a field theory of the 14 fields of particle density—particle flux, and stress—energy—momentum. The field equations are based on the conservation laws of particle numbers, and energy-momentum and on a balance of fluxes. The necessary constitutive equations are strongly restricted by the principle of relativity, entropy principle, requirement of hyperbolicity. It turns out that the resulting field equations contain only viscosity, bulk viscosity and heat conductivity as unknown functions. All other constitutive coefficients may be calculated from the equilibrium equations of state that are known from statistical arguments. The paper offers a more systematic version of relativistic thermodynamics of gases than the earlier papers by Muller and Israel. At the same time the present version contains less unknown functions than those earlier papers. All speeds of propagation are finite. The relation between the present theory and the classical one formulated by Eckart is described.

Galilean invariance and entropy principle for systems of balance laws

The Galilean invariance of a generic system of balance laws dictates a specific dependence of the densities and fluxes on velocity. Thus these quantities decompose in a unique manner into convective and non-convetive parts. Such a decomposition permits the elimination of velocity dependencies in the entropy principle, which becomes a constraint on the constitutive functions only. These results clarify the mathematical structure of extended thermodynamics. They also provide a connection between the equations of continuum thermodynamics and the Boltzmann equation.

On the evolution law of weak discontinuities for hyperbolic quasi-linear systems

Abstract We show that the law of propagation of weak discontinuities obtained by A. Jeffrey is in agreement with the Bernoullis law found by other authors.

Rational Extended Thermodynamics beyond the Monatomic Gas

1 Introduction.- 2 Mathematical Structure.- 3 Waves in Hyperbolic Systems.- 4 Survey of ET of Rarefied Monatomic Gases.- 5 ET Theory of Polyatomic Rarefied Gas and Dense Gas.- 6 Maximum entropy principle for rarefied polyatomic gases.- 7 Stationary heat conduction in a polyatomic gas at rest.- 8 Linear Wave in Polyatomic Gas.- 9 Shock Wave.- 10 Fluctuating hydrodynamics.- 11 Light scattering in Polyatomic Gas.- 12 ET6 Theory of Polyatomic Rarefied Gas and Dense Gas.- 13 Non linear ET6.- 14 Molecular ET Theory of Polyatomic Rarefied Gas.- 15 ET of dense gas with 6 moments.- 16 Mixture of Gases.- 17 Hyperbolic Parabolic Limit, Max wellian iteration and Objectivity.

Maximum entropy principle for rarefied polyatomic gases

The aim of this paper is to show that procedure of maximum entropy principle for closure of moments equations for rarefied monatomic gases can be extended also to polyatomic gases. The main difference with respect to the usual procedure is existence of two hierarchies of macroscopic equations for moments of suitable distribution function, in which the internal energy of a molecule is taken into account. The field equations for 14 moments of distribution function, which include dynamic pressure, are derived. The entropy and the entropy flux are shown to be a generalization of the ones for classical Grad’s distribution. The results are in perfect agreement with the recent macroscopic approach of extended thermodynamics for real gases.

Hyperbolicity of the 3 + 1 System of Einstein Equations

By a suitable choice of the lapse, which in a natural way is connected to the space metric, we obtain a hyperbolic system from the 3+1 system of Einstein equations with zero shift; this is accomplished by combining the evolution equations with the constraints.

Reflection and transmission of discontinuity waves through a shock wave. General theory including also the case of characteristic shocks

An incident wave creates a discontinuity in the acceleration of the shock front. The amplitudes of the reflected and transmitted waves are also determined. Special attention is given to the case of the weak shocks and the characteristic shocks.

Effect of the dynamic pressure on the shock wave structure in a rarefied polyatomic gas

We study the shock wave structure in a rarefied polyatomic gas based on a simplified model of extended thermodynamics in which the dissipation is due only to the dynamic pressure. In this case the differential system is very simple because it is a variant of Euler system with a new scalar equation for the dynamic pressure [T. Arima, S. Taniguchi, T. Ruggeri, and M. Sugiyama, Phys. Lett. A 376, 2799–2803 (2012)]. It is shown that this theory is able to describe the three types of the shock wave structure observed in experiments: the nearly symmetric shock wave structure (Type A, small Mach number), the asymmetric structure (Type B, moderate Mach number), and the structure composed of thin and thick layers (Type C, large Mach number).

Second sound and internal energy in solids

Abstract A non-linear thermodynamic model of heat-conducting anisotropic solid is elaborated which turns out to be in a conservative form. Then, through the associated main field variables, the symmetry and the hyperbolicity properties are investigated. As outstanding applications, the analysis of the grow of the wave discontinuities and the evaluation of the critical time are performed. Finally, the Rankine-Hugoniot conditions for the system of equations are given in detail.

Continuum approach to phonon gas and shape changes of second sound via shock waves theory

SummaryA continuum approach, based on the principles of modern extended thermodynamics, describing the model of a phonon gas is performed. The main difference with the ideal phonon gas theory consists in the presence of athermal inertia. We apply the shock wave theory and discuss the selection rules for physical shocks (theLax conditions and theentropy growth). In this way the existence of two new kinds of shocks (hot andcold shocks) in rigid heat conductors at low temperature is pointed out. In particular a critical temperature, characteristic of each material, changing the structure of the previous types of shocks is analytically deduced. This characteristic temperature permits also to explain the modification of the received second sound wave form with respect to the initial wave profile. Finally, the results are applied to the case of high-purity crystals (NaF, Bi,3He and4He) and compared with experimental results.