Identification of an average temperature and a dynamical pressure in a multitemperature mixture of fluids

Abstract

We present a classical approach to a mixture of compressible fluids when each constituent has its own temperature. The introduction of an average temperature together with the entropy principle dictates the classical Fick law for diffusion and also novel constitutive equations associated with the difference of temperatures between the components. The constitutive equations fit with results recently obtained through a Maxwellian iteration procedure in extended thermodynamics theory of multitemperature mixtures. The differences of temperatures between the constituents imply the existence of a dynamical pressure even if the fluids have a zero bulk viscosity. The nonequilibrium dynamical pressure can be measured and may be convenient in several physical situations, such as, for example, in cosmological circumstances where--as many authors assert--a dynamical pressure played a major role in the evolution of the early universe.

Dynamical pressure for fluid mixtures with several temperatures

“…This is a main property of mixtures with multi-temperatures and this fact may have some applications in plasma of gases and in the evolution of the early universe [14]. The results are in complete accordance with the ones by Gouin & Ruggeri [9] and developed in [10]. This is an important verification of the fact that the Hamilton principle can be extended to nonconservative mixture motions when components have different temperature.…”

“…As we wrote in [9], we define the average temperature T associated with T 1 and T 2 through the implicit solution of the equationε…”

“…In Ref. [9], the internal energy is the sum of the internal energies of the components ρ ε = 2 α=1 ραεα(ρα, sα) .…”

“…The variational approach to the construction of two-fluid models has been used by many authors [6,7,8]. The method is different from the method proposed is [9] and is now developed in [10]. To study thermodynamical processes by using the Hamilton principle, the entropies of components are added to the field parameters instead of temperatures.…”

Constitutive equations for heat conduction in nanosystems and nonequilibrium processes: an overview

“…In other words, since we regard phonons and electrons as a mixture of gases flowing through the crystal lattice [32,71], each of which is endowed with its own temperature, according with the theory of fluid mixtures with different temperatures [74][75][76], we assume that each constituent obeys the same balance laws as a single fluid. The average temperature of the mixture may introduced by the consideration that the internal energy of the mixture is the same as in the case of a single-temperature mixture [75].…”