Maxwellian Iteration Research Articles

On two possible ways to recover ordinary thermodynamics from extended thermodynamics of polyatomic gases

We consider two possible ways, i.e. the Maxwellian iteration (MI) and the Chapman–Enskog method (CEM), to recover relativistic ordinary thermodynamics from relativistic extended thermodynamics of Polyatomic gases with N moments. Both of these methods give the Eckart equations which are the relativistic version of the Navier–Stokes and Fourier laws as a first iteration. However, these methods do not lead to the same expressions of the heat conductivity χ, the shear viscosity µ, and the bulk viscosity ν which appear as coefficients in the Eckart equations. In particular, we prove that the expressions of χ, µ, and ν obtained via the CEM do not depend on N, while those obtained through the MI depend on N. Moreover, we also prove that these two methods lead to the same results in the nonrelativistic limit.

Eckart equations, Maxwellian iteration and relativistic causal theories of divergence type

We consider a general causal relativistic theory of divergence type in the framework of rational extended thermodynamics (RET) for a compressible, possibly dense, gas. We require that the system converges in the Maxwellian iteration’s first step to the parabolic Eckart equations. This requirement implies a constraint between the two coefficients present in the triple tensor evaluated at equilibrium. Moreover, the production tensor is determined for prescript thermal and caloric state equations and given heat conductivity, shear and bulk viscosities. In the second part, we prove that if the original hyperbolic system satisfies the universal principles of RET, as can be put in the symmetric form using the main field , it always satisfies the previous compatibility condition. Therefore, any causal system of divergence type that satisfies the entropy principle with a convex entropy converges to the Eckart system in the Maxwellian iteration also when we have no information at the mesoscopic scale from the kinetic theory. The obtained results are tested on the RET theories of rarefied monatomic and polyatomic gases. This article is part of the theme issue ‘Foundational issues, analysis and geometry in continuum mechanics’.

Navier–Stokes–Fourier equations as a parabolic limit of a general hyperbolic system of rational extended thermodynamics

This paper aims to prove that a general system of 14 balance laws for a compressible, possibly dense, gas that satisfies the universal principles of Rational Extended Thermodynamics (RET) converges to the Navier–Stokes–Fourier equations in the first step of the Maxwellian iteration. Moreover, in a theory not far from equilibrium, we show that the production terms of the hyperbolic system are uniquely determined as soon as the heat conductivity, the shear viscosity, and the bulk viscosity are assigned. The obtained results are tested on the RET theories for rarefied monatomic and polyatomic gases.

Relativistic Rational Extended Thermodynamics of Polyatomic Gases with a New Hierarchy of Moments.

A relativistic version of the rational extended thermodynamics of polyatomic gases based on a new hierarchy of moments that takes into account the total energy composed by the rest energy and the energy of the molecular internal mode is proposed. The moment equations associated with the Boltzmann–Chernikov equation are derived, and the system for the first 15 equations is closed by the procedure of the maximum entropy principle and by using an appropriate BGK model for the collisional term. The entropy principle with a convex entropy density is proved in a neighborhood of equilibrium state, and, as a consequence, the system is symmetric hyperbolic and the Cauchy problem is well-posed. The ultra-relativistic and classical limits are also studied. The theories with 14 and 6 moments are deduced as principal subsystems. Particularly interesting is the subsystem with 6 fields in which the dissipation is only due to the dynamical pressure. This simplified model can be very useful when bulk viscosity is dominant and might be important in cosmological problems. Using the Maxwellian iteration, we obtain the parabolic limit, and the heat conductivity, shear viscosity, and bulk viscosity are deduced and plotted.

Maxwellian iteration of a causal relativistic model of polyatomic gases and evaluation of bulk, shear viscosity and heat conductivity

In the present paper, we give the parabolic limit of the field equations of a recent hyperbolic model of relativistic polyatomic gas in the framework of Rational Extended Thermodynamics (RET) theory. We obtain in this way the corresponding constitutive equations of the Thermodynamics of Irreversible Processes (TIP) obtained first in the context of relativity by Eckart. The limit is reached via the Maxwellian iteration procedure to make a connection between the phenomenological coefficients (shear, bulk viscosity, and heat conductivity) and the relaxation times. The classical and ultrarelativistic limit of these coefficients are also obtained finding that, in the case of classical limit, they coincide with the ones known in the literature. As a particular case, we study the monatomic gas and we can plot the dimensionless coefficients associated with bulk, shear, and heat conductivity. In contrast to a monatomic gas in which the bulk viscosity is very small and tends to the classical regime of the order of magnitude of O(1∕c4), the bulk viscosity and the dynamical pressure for polyatomic gases, due to the rotation and vibration of internal modes, are of the order of unit (except in the ultrarelativistic limit), and this might open new perspectives in cosmology.

Six-field extended thermodynamics models representing molecular energy exchange in a dense polyatomic gas

In the framework of extended thermodynamics, we present three models of a dense polyatomic gases with 6 independent fields. The three models, respectively, represent the relaxation process due to the different molecular energy exchange among the translational, rotational and vibrational energies of a molecule. The characteristic velocities, the local exceptionality and the Maxwellian iteration are also studied. As an example, we study the six-field models of a van der Waals fluid.

Extended Thermodynamics of Rarefied Polyatomic Gases: 15-Field Theory Incorporating Relaxation Processes of Molecular Rotation and Vibration.

After summarizing the present status of Rational Extended Thermodynamics (RET) of gases, which is an endeavor to generalize the Navier–Stokes and Fourier (NSF) theory of viscous heat-conducting fluids, we develop the molecular RET theory of rarefied polyatomic gases with 15 independent fields. The theory is justified, at mesoscopic level, by a generalized Boltzmann equation in which the distribution function depends on two internal variables that take into account the energy exchange among the different molecular modes of a gas, that is, translational, rotational, and vibrational modes. By adopting the generalized Bhatnagar, Gross and Krook (BGK)-type collision term, we derive explicitly the closed system of field equations with the use of the Maximum Entropy Principle (MEP). The NSF theory is derived from the RET theory as a limiting case of small relaxation times via the Maxwellian iteration. The relaxation times introduced in the theory are shown to be related to the shear and bulk viscosities and heat conductivity.

Non-equilibrium diffusion temperatures in mixture of gases via Maxwellian iteration

This paper studies the difference of non-equilibrium temperatures between constituents, so-called diffusion temperature in the model of mixtures of gases in which each constituent has assigned its own velocity and temperature field. In a previous study (Ruggeri and Simic, Phys. Rev. E 80:026317, 2009), the constitutive equation for the diffusion temperatures, akin to Fick law for the diffusion flux, was derived by means of Maxwellian iteration. In the first order approximation the diffusion temperatures vanish if the constituents have the same ratio of specific heats, i.e. the molecules have the same number of degrees of freedom. This study proceeds to second iteration in the particular case of binary mixture, and provides a second order correction for the diffusion temperature. In this way the classical limit of non-equilibrium temperatures, which covers all the cases, is obtained at the lowest order. A comparison with known results is provided, revealing the corrections and generalizations brought by our results. A special case of constant pressure Fickian diffusion is analyzed and showed that diffusion temperature does not vanish as long as there exist mechanical diffusion between the constituents.

The NRxx method for polyatomic gases

In this paper, we propose a numerical regularized moment method to solve the Boltzmann equation with ES-BGK collision term to simulate polyatomic gas flows. This method is an extension to the polyatomic case of the method proposed in [12], which is abbreviated as the NRxx method in [11]. Based on the form of the Maxwellian, the Laguerre polynomials of the internal energy parameter are used in the series expansion of the distribution function. We develop for polyatomic gases all the essential techniques needed in the NRxx method, including the efficient projection algorithm used in the numerical flux calculation, the regularization based on the Maxwellian iteration and the order of magnitude method, and the linearization of the regularization term for convenient numerical implementation. Meanwhile, the particular integrator in time for the ES-BGK collision term is put forward. The shock tube simulations with Knudsen numbers from 0.05 up to 5 are presented to demonstrate the validity of our method. Moreover, the nitrogen shock structure problem is included in our numerical experiments for Mach numbers from 1.53 to 6.1.

Average temperature and Maxwellian iteration in multitemperature mixtures of fluids

This paper treats the nonequilibrium processes in mixtures of fluids under the assumption that each constituent is characterized by its own velocity and temperature field. First we discuss the concept of the average temperature of mixture based upon considerations that the internal energy of the mixture is the same as in the case of a single-temperature mixture. As a consequence, it is shown that the entropy of the mixture reaches a local maximum in equilibrium. An illustrative example of homogeneous mixtures is given to support the theoretical considerations. Through the procedure of Maxwellian iteration a new constitutive equation for nonequilibrium temperatures of constituents is obtained in a classical limit, together with the Fick's law for the diffusion flux. These results obtained for n-species are in perfect agreement with a recent classical approach of thermodynamics of irreversible processes in multitemperature case due to Gouin and Ruggeri and generalize our previous papers concerning the case of a binary mixture.

Multi-temperature mixture of fluids

We present a survey on some recent results concerning the different models of a mixture of compressible fluids. In particular we discuss the most realistic case of a mixture when each constituent has its own temperature (MT) and we first compare the solutions of this model with the one with a unique common temperature (ST ). In the case of Eulerian fluids it will be shown that the corresponding (ST ) differential system is a principal subsystem of the (MT) one. Global behavior of smooth solutions for large time for both systems will also be discussed through the application of the Shizuta-Kawashima condition. Then we introduce the concept of the average temperature of mixture based upon the consideration that the internal energy of the mixture is the same as in the case of a single-temperature mixture. As a consequence, it is shown that the entropy of the mixture reaches a local maximum in equilibrium. Through the procedure of Maxwellian iteration a new constitutive equation for non-equilibrium temperatures of constituents is obtained in a classical limit, together with the Fick's law for the diffusion flux. Finally, to justify the Maxwellian iteration, we present for dissipative fluids a possible approach of a classical theory of mixture with multi-temperature and we prove that the differences of temperatures between the constituents imply the existence of a new dynamical pressure even if the fluids have a zero bulk viscosity.

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

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.

Hyperbolicity in Extended Thermodynamics of Fermi and Bose gases

We consider the balance system of Extended Thermodynamics with 13 Moments in the case of Fermi and Bose gases, for processes not far from equilibrium. In this case, the hyperbolicity of the differential system holds only in a neighborhood of the equilibrium state. The main aim of the paper is to evaluate the hyperbolicity region of the differential system. The knowledge of this region in the state variables is mandatory to check the admissibility of the solutions and the corresponding boundary and Cauchy data in the limit of the approximation considered. The results are obtained through numerical evaluations of the Fermi and Bose integral functions \(I^{\pm}_{n}(\alpha)\) that appear in the characteristic polynomial. Particular attention is devoted to the completely degenerate case when Fermi gas reaches the 0 K and when the Bose gas is in proximity of the transition temperature Tc. In these limiting cases, the hyperbolicity requirement is lost according to previous results. In the last section we make use of the Maxwellian iteration in order to evaluate the heat conductivity and the viscosity for the degenerate Fermi and Bose gas.

Extended thermodynamics of non-ideal gases

A phenomenological extended thermodynamics theory for non-ideal gases is formulated. In this theory 13 basic fields of density, velocity, pressure tensor and heat flux are considered. The coefficients in the representations of the constitutive quantities are obtained in terms of the virial coefficients of the thermal equation of state and of the free energy density. The laws of Fourier and Navier-Stokes follow from an iteration method akin to the Maxwellian iteration in the kinetic theory of gases.

Extended thermodynamics of mixtures of ideal gases

A phenomenological extended thermodynamic theory for a mixture of v fields is formulated. In this theory 13 v fields of partial mass densities, partial velocities, partial pressure tensors and partial heat fluxes are considered. The laws of Fourier, Navier-Stokes and Fick are obtained through an interation method akin to the Maxwellian iteration method in the kinetic theory of gases. It is shown also how the Onsager reciprocity relations in the presence of axial fields came out of this phenomenological theory without the use of statistical arguments.