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2015 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Rational Extended Thermodynamics beyond the Monatomic Gas

verfasst von: Tommaso Ruggeri, Masaru Sugiyama

Verlag: Springer International Publishing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book is dedicated to the recent developments in RET with the aim to explore polyatomic gas, dense gas and mixture of gases in non-equilibrium. In particular we present the theory of dense gases with 14 fields, which reduces to the Navier-Stokes Fourier classical theory in the parabolic limit. Molecular RET with an arbitrary number of field-variables for polyatomic gases is also discussed and the theory is proved to be perfectly compatible with the kinetic theory in which the distribution function depends on an extra variable that takes into account a molecule’s internal degrees of freedom. Recent results on mixtures of gases with multi-temperature are presented together with a natural definition of the average temperature. The qualitative analysis and in particular, the existence of the global smooth solution and the convergence to equilibrium are also studied by taking into account the fact that the differential systems are symmetric hyperbolic. Applications to shock and sound waves are analyzed together with light scattering and heat conduction and the results are compared with experimental data.

Rational extended thermodynamics (RET) is a thermodynamic theory that is applicable to non-equilibrium phenomena. It is described by differential hyperbolic systems of balance laws with local constitutive equations. As RET has been strictly related to the kinetic theory through the closure method of moment hierarchy associated to the Boltzmann equation, the

applicability range of the theory has been restricted within rarefied monatomic gases.

The book represents a valuable resource for applied mathematicians, physicists and engineers, offering powerful models for potential applications like satellites reentering the atmosphere, semiconductors and nano-scale phenomena.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
Rational extended thermodynamics (RET), which is compatible with the kinetic theory of gases and is also closely related to the mathematical theory of hyperbolic systems, has been a successful theory of rarefied monatomic gases to explain nonequilibrium phenomena such as light scattering, sound dispersion, shock wave structure, nonequilibrium radiation. These subjects were treated in detail in the book “Rational Extended Thermodynamics” by Ingo Müller and Tommaso Ruggeri. The other subjects remained to be explored in RET were those of polyatomic gases, of dense gases in general, and of mixtures of gases with multi-temperature. This is the aim of the present book to discuss such new RET. In this chapter, before going into the details, we give some introductory perspective on these subjects starting with a short history of nonequilibrium thermodynamics. The new RET theory includes the 14-field theory of dense gases that reduces to the classical Navier-Stokes Fourier theory in the parabolic limit (Maxwellian iteration), to the singular limit of a monatomic gas with 13 fields, and to the subsystem with 6 fields. The 6-field theory is the minimal dissipative system, where the dissipation is only due to the dynamic pressure, after the Euler system of perfect fluids. The concept of nonequilibrium temperature is also discussed. For rarefied polyatomic gases, we discuss a theory of molecular RET with arbitrary number of field variables by using the methods of closure based on both the maximum entropy principle and the entropy principle. It can be proved that the two methods are equivalent to each other. Several applications of the new RET theory are reviewed as well. Moreover we discuss the theory of a mixture of gases with multi-temperature, i.e., a mixture in which each constituent has its own temperature. In the new approach, the qualitative analysis of the differential system is also done by taking into account the fact that, due to the convexity of the entropy, there exists a privileged field (main field) such that the system becomes symmetric hyperbolic. Existence of global smooth solutions and convergence to equilibrium are also discussed.
Tommaso Ruggeri, Masaru Sugiyama

Mathematical Structure and Waves

Frontmatter
Chapter 2. Mathematical Structure
Abstract
In this chapter, we give a survey on the mathematical structure of the system of RET, which is strictly related to the mathematical problems of hyperbolic systems in balance form with a convex entropy density. We summarize the main results: The proof of the existence of the main field in terms of which a system becomes symmetric, and several properties derived from the qualitative analysis concerning symmetric hyperbolic systems. In particular, the Cauchy problem is well-posed locally in time, and if the so-called K-condition is satisfied, there exist global smooth solutions provided that the initial data are sufficiently small. Moreover the main field permits to identify natural subsystems and in this way we have a structure of nesting theories. The main property of these subsystems is that the characteristic velocities satisfy the so-called sub-characteristic conditions that imply, in particular, that the maximum characteristic velocity does not decrease when the number of equations increases. Another beautiful general property is the compatibility of the balance laws with the Galilean invariance that dictates the precise dependence of the field equations on the velocity.
Tommaso Ruggeri, Masaru Sugiyama
Chapter 3. Waves in Hyperbolic Systems
Abstract
Wave propagation phenomena give us an important mean to check the validation of the nonequilibrium thermodynamics theory. In this chapter, we present a short review on the modern theory of wave propagation for hyperbolic systems. Firstly, we present the theory of linear waves emphasizing the role of the dispersion relation. The high frequency limit in the dispersion relation is also studied. Secondly, nonlinear acceleration waves are discussed together with the transport equation and the critical time. Thirdly we present the main results concerning shock waves as a particular class of weak solutions and the admissibility criterion to select physical shocks (Lax condition, entropy growth condition, and Liu condition). The chapter finishes with the discussion of traveling waves, in particular, shock waves with structure. The sub-shock formation is particularly interesting. The Riemann problem and the large time asymptotic behavior are also discussed.
Tommaso Ruggeri, Masaru Sugiyama

Survey of Rational Extended Thermodynamics of Monatomic Gas

Frontmatter
Chapter 4. RET of Rarefied Monatomic Gas
Abstract
In this chapter, we give a survey of the main results of RET concerning rarefied monatomic gases, some of which are explained in the Müller–Ruggeri book of RET (Müller and Ruggeri, Rational Extended Thermodynamics, Springer, New York, 1998). We start from the phenomenological RET theory with 13 fields and prove that the closure of RET coincides with the one obtained by Grad using kinetic arguments and with the MEP procedure. The theory with N-moments is also presented with the proof of nesting theories that emerge from the concept of principal subsystem. The problematic of bounded domain in RET is also considered, and a simple example of heat conduction is explained to show a significant difference of the results between RET and NSF. A lower bound for the maximum characteristic velocity is obtained in terms of the truncation tensor index N. This quantity increases as the number of moments grows and it is unbounded when \(N \rightarrow \infty \). The relativistic counterpart is also described briefly. In this framework, the maximum characteristic velocity is bounded for any number of moments, and converges to the light velocity from the below for \(N \rightarrow \infty \). The chapter contains also comparison between the RET theory and experiments in sound waves and light scattering.
Tommaso Ruggeri, Masaru Sugiyama

RET of Polyatomic Gas and Dense Gas with 14 Fields

Frontmatter
Chapter 5. RET 14-Field Theory of Polyatomic Gas and Dense Gas
Abstract
The objective of the present chapter is to explain in detail the new ET theory of rarefied polyatomic gases and dense gases with 14 independent fields (ET14), that is, the mass density, the velocity, the temperature, the shear stress, the dynamic pressure, and the heat flux. We adopt the system of field equations with a binary hierarchy structure. We show that the constitutive equations can be determined explicitly by the caloric and thermal equations of state as in the monatomic ET13 theory. We also analyze physically important systems, that is, a rarefied polyatomic gas, a gas with the virial equations of state, a hard-sphere system, and a van der Waals gas. Lastly we show that the ET13 theory of rarefied monatomic gases is derived from the ET14 theory as a singular limit.
Tommaso Ruggeri, Masaru Sugiyama
Chapter 6. Maximum Entropy Principle for Rarefied Polyatomic Gas
Abstract
In this chapter, we prove, in the case of polyatomic rarefied gases, that the maximum entropy principle (MEP) gives the same closure of the system as that obtained in the phenomenological ET theory with 14 fields discussed in Chap. 5 The main idea is to consider a generalized distribution function depending not only on the velocity but also on an extra variable that connects with the internal degrees of freedom of a constituent molecule. On the basis of MEP, we again obtain the same binary hierarchy introduced in the previous chapter: the one is the usual momentum-type, F-series, and the other is the energy-type, G-series. The extra variable plays a role in the G-series. Thus we prove the perfect agreement between the ET theory and the molecular ET theory at least within 14-field theories. The agreement for any number of moments will be proved in Chap. 10
Tommaso Ruggeri, Masaru Sugiyama

Applications of the ET14 Theory

Frontmatter
Chapter 7. Linear Wave in a Polyatomic Gas
Abstract
In this chapter, we study a linear sound wave in a rarefied polyatomic gas in equilibrium with the aim of clarifying the validity and the features of the ET14 theory established in Chap. 5 We derive the dispersion relations on the basis of the ET14 theory and of the classical Navier-Stokes Fourier (NSF) theory. Comparison of these relations with experimental data reveals clearly the superiority of the ET14 theory to the NSF theory. We confine our analysis within sound waves in some rarefied diatomic gases (hydrogen, deuterium, and hydrogen deuteride gases) because suitable experimental data are scarce and are mainly restricted to rarefied gases. We also evaluate the relaxation times, and the shear and bulk viscosities and the heat conductivity of the gases.
Tommaso Ruggeri, Masaru Sugiyama
Chapter 8. Shock Wave in a Polyatomic Gas
Abstract
In this chapter we study the shock wave structure in a rarefied polyatomic gas by using the ET14 theory. We show how the ET14 theory can overcome the difficulties encountered in the previous approaches: Bethe-Teller approach and Gilbarg-Paolucci approach. Firstly, the predictions derived from the ET14 theory are shown and compared with the results from the NSF theory. Secondly, the Bethe-Teller theory is reexamined in the light of the ET14 theory. Lastly, comparison between the theoretical predictions derived from the ET14 theory and the experimental data is made, where we show a very good agreement. We are able to explain in a unified manner the three different shock wave profiles Types A, B and C for increasing Mach number.
Tommaso Ruggeri, Masaru Sugiyama
Chapter 9. Light Scattering, Heat Conduction, and Fluctuation
Abstract
In this chapter, we discuss briefly some other interesting applications of the ET14 theory: light scattering, stationary heat conduction, and fluctuating hydrodynamics. However, since these studies have just begun recently and is still quite primitive, this chapter should be read as an outline of these promising research fields.
Tommaso Ruggeri, Masaru Sugiyama

Maximum Entropy Principle and Nesting Theories of Many Moments

Frontmatter
Chapter 10. Molecular ET Theory of Rarefied Polyatomic Gas
Abstract
Molecular extended thermodynamics of rarefied polyatomic gases is characterized by the system composed of two hierarchies of balance equations for the moments of a distribution function. The internal degrees of freedom of a molecule are properly taken into account in the distribution function. By the reasoning of physical relevance, the truncation orders of the two hierarchies are proved to be not independent of each other. And the two closure procedures based on the maximum entropy principle (MEP) and on the entropy principle are also proved to be equivalent to each other. Characteristic velocities of a hyperbolic system of the balance equations for a polyatomic gas are compared to those obtained for a monatomic gas. The lower bound estimate for the maximum equilibrium characteristic velocity established for a monatomic gas is proved to be valid also for a rarefied polyatomic gas, that is, the estimate is independent of the degrees of freedom of a molecule. As a consequence, also for polyatomic gases, when the number of moments increases the maximum characteristic velocity becomes unbounded.
Tommaso Ruggeri, Masaru Sugiyama

ET6: A Theory of Far-from-Equilibrium Thermodynamics

Frontmatter
Chapter 11. Non-linear ET6 and the Role of the Dynamic Pressure: Phenomenological Approach
Abstract
In this chapter, we present ET of real gases with six independent fields, i.e., the mass density, the velocity, the temperature and the dynamic pressure, without adopting the near-equilibrium approximation. We prove its compatibility with the universal principles (the entropy principle, the Galilean invariance and the stability), and obtain the symmetric hyperbolic system with respect to the main field. The correspondence between the ET 6-field (ET6) theory and the Meixner theory of relaxation processes is discussed. The internal variable and the nonequilibrium temperature in the Meixner theory are expressed in terms of the quantities of the ET6 theory, in particular, the dynamic pressure. As an example, we present the case of rarefied polyatomic gases and study the monatomic-gas limit where the system converges to the Euler system of a perfect fluid.
Tommaso Ruggeri, Masaru Sugiyama
Chapter 12. Molecular Non-linear ET6 for Rarefied Polyatomic Gas
Abstract
We establish extended thermodynamics of rarefied polyatomic gases with six independent fields via the maximum entropy principle. The distribution function is not necessarily near equilibrium. The result is in perfect agreement with the phenomenological ET theory explained in the previous Chap. 11 This is the first example of molecular extended thermodynamics with a non-linear closure. The integrability condition of the moments requires that the dynamical pressure should be bounded from below and from above.
Tommaso Ruggeri, Masaru Sugiyama
Chapter 13. Application of ET6: Shock Wave and Sub-shock Formation
Abstract
In this chapter, we show the usefulness of the ET6 theory for the analysis of the shock wave structure in a rarefied polyatomic gas. We compare the theoretical prediction derived from the ET6 theory with that from the ET14 theory. We see, in particular, that the thin layer in Type C with finite thickness described by the ET14 theory is replaced by a discontinuous jump, the sub-shock. The strength and the stability of a sub-shock is also discussed. Lastly the temperature overshoot at a sub-shock in terms of Meixner’s temperature is shown.
Tommaso Ruggeri, Masaru Sugiyama
Chapter 14. Acceleration Wave, K-Condition, and Global Existence in ET6
Abstract
We verify the K-condition for the non-linear ET6 model and show for any gas the existence of global smooth solutions provided that initial data are sufficiently small. As an example, in the case of polyatomic gases, we study acceleration waves. We evaluate the Bernoulli equation for the amplitude of the wave. If the initial amplitude of an acceleration wave is sufficiently small compared with the critical amplitude, the acceleration wave exists for all time and decays to zero as the time t becomes large. Vice versa, for large initial amplitude, there exists a critical time at which we have the blow up of the solution and the formation of a shock wave. We show the peculiarity of this model, that is, the velocity of a disturbance and the critical time are universal: these are independent of the degrees of freedom of the constituent molecule D.
Tommaso Ruggeri, Masaru Sugiyama
Chapter 15. Nonequilibrium Temperature and Chemical Potential
Abstract
In this chapter, we propose a natural definition of nonequilibrium temperature and chemical potential. The main field, with which the generalized Gibbs equation is expressed in a differential form, is the key quantity in the definition. In the ET6 theory, in particular, the nonequilibrium quantities coincide exactly with those in the Meixner theory explained in Chap. 11
Tommaso Ruggeri, Masaru Sugiyama

Mixture of Gases with Multi-Temperature

Frontmatter
Chapter 16. Multi-Temperature Mixture of Fluids
Abstract
We present a survey on recent results concerning some different models of a mixture of compressible fluids. In particular, we discuss the most realistic case of a mixture where each constituent has its own temperature (MT). We first compare the solutions of this model with the one with 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 system. Global behavior of smooth solutions for large time for both systems will also be discussed through the application of the Shizuta-Kawashima K-condition. Then we introduce the concept of the average temperature of a mixture based on the consideration that the internal energy of the mixture is the same as that 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 the Maxwellian iteration, a new constitutive equation for nonequilibrium temperatures of constituents is obtained in a classical limit, together with the Fick law for the diffusion flux. Finally, in order to justify the Maxwellian iteration, we present, for dissipative fluids, a possible approach to a classical theory of mixtures with the multi-temperature. We prove that the differences of temperatures between the constituents imply the existence of a new dynamic pressure even if fluids have zero bulk viscosities.
Tommaso Ruggeri, Masaru Sugiyama
Chapter 17. Shock Structure and Temperature Overshoot in Macroscopic Model of Mixtures
Abstract
In this chapter, we study the shock structure in a mixture on the basis of the model of multi-temperature mixtures explained in the previous Chap. 16 For simplicity, the study is restricted to weak and moderately strong shocks in a binary mixture of ideal gases without viscosity and heat conductivity. The model predicts the existence of the temperature overshoot of the heavier constituent, which was also predicted by other sophisticated approaches. This phenomenon is a consequence of weak energy exchange between the constituents, either due to large mass difference, or large rarefaction of the mixture. In the range of small Mach number, it is also shown that the shock thickness (or equivalently, the inverse of Knudsen number) decreases with the increase of the Mach number: a behavior similar to a single fluid.
Tommaso Ruggeri, Masaru Sugiyama

Maxwellian Iteration and Objectivity

Frontmatter
Chapter 18. Hyperbolic Parabolic Limit, Maxwellian Iteration and Objectivity
Abstract
In this chapter, we discuss the parabolic limit of extended thermodynamics via the Maxwellian iteration, and we observe that the usual constitutive equations, which are nonlocal in space, are approximations of some balance laws of ET when some relaxation times are negligible. An important consequence is that these equations need not satisfy the objectivity principle. To avoid misunderstanding, we should mention that the principle still continues to be valid for constitutive equations. We also discuss the point that, under suitable assumptions, the conditions dictated by the entropy principle in the hyperbolic case guarantee the validity of the entropy principle also in the parabolic limit. Lastly we express our opinion concerning the limitation of the parabolic regularized version of ET theories.
Tommaso Ruggeri, Masaru Sugiyama
Chapter 19. Open Problems
Abstract
We list up some open problems and try to discuss the perspective on the ET theory.
Tommaso Ruggeri, Masaru Sugiyama
Backmatter
Metadaten
Titel
Rational Extended Thermodynamics beyond the Monatomic Gas
verfasst von
Tommaso Ruggeri
Masaru Sugiyama
Copyright-Jahr
2015
Electronic ISBN
978-3-319-13341-6
Print ISBN
978-3-319-13340-9
DOI
https://doi.org/10.1007/978-3-319-13341-6

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