Rational Extended Thermodynamics

ISBN-13: 9781461274605 / Angielski / Miękka / 2011 / 396 str.

Ingo Mueller; Tommaso Ruggeri

Opis

Ordinary thermodynamics provides reliable results when the thermodynamic fields are smooth, in the sense that there are no steep gradients and no rapid changes. In fluids and gases this is the domain of the equations of Navier-Stokes and Fourier. Extended thermodynamics becomes relevant for rapidly varying and strongly inhomogeneous processes. Thus the propagation of high frequency waves, and the shape of shock waves, and the regression of small-scale fluctuation are governed by extended thermodynamics. The field equations of ordinary thermodynamics are parabolic while extended thermodynamics is governed by hyperbolic systems. The main ingredients of extended thermodynamics are field equations of balance type, constitutive quantities depending on the present local state and entropy as a concave function of the state variables. This set of assumptions leads to first order quasi-linear symmetric hyperbolic systems of field equations; it guarantees the well-posedness of initial value problems and finite speeds of propaga tion. Several tenets of irreversible thermodynamics had to be changed in subtle ways to make extended thermodynamics work. Thus, the entropy is allowed to depend on nonequilibrium vari ables, the entropy flux is a general constitutive quantity, and the equations for stress and heat flux contain inertial terms. New insight is therefore provided into the principle of material frame indifference. With these modifications an elegant formal structure can be set up in which, just as in classical thermostatics, all restrictive conditions--derived from the entropy principle-take the form of integrability conditions.

Spis treści

1 Tour d’Horizon.- 2 Early Version of Extended Thermodynamics and Kinetic Theory of Gases.- 1 Paradoxes of Heat Conduction and Shear Diffusion.- 1.1 Heuristic Derivation of the Laws of Fourier and Navier-Stokes.- 1.2 Parabolic Laws of Heat Conduction and Shear Diffusion.- 2 Paradox Removed.- 2.1 The Cattaneo Equation.- 2.2 Extended TIP.- 2.3 Finite Pulse Speeds in Extended TIP.- 2.4 Conclusion and Criticism.- 3 Kinetic Theory of Monatomic Gases.- 3.1 Boltzmann Equation and Moments.- 3.2 Equations of Balance for Moments.- 3.3 Balance of Entropy and Possible Equilibria.- 3.4 The Grad Distribution.- 3.5 Entropy and Entropy Flux in Grad’s 13-Moment Theory.- 3.6 Phenomenological Equations derived from the Kinetic Theory.- 3.7 Pulse Speeds.- 3.8 Conclusions.- 3 Formal Structure of Extended Thermodynamics.- 1 Field Equations.- 1.1 Thermodynamic Processes and Principles of the Constitutive Theory.- 1.2 Universal Principles of the Constitutive Theory.- 2 Entropy Inequality and Symmetric Hyperbolic Systems.- 2.1 Exploitation of the Entropy Inequality.- 2.2 Symmetric Hyperbolic Field Equations.- 2.3 Discussion.- 2.4 Characteristic Speeds.- 3 Main Subsystems.- 3.1 Constraints on the Main Field.- 3.2 A Main Subsystem Implies an Entropy Inequality.- 3.3 A Main Subsystem Is Symmetric Hyperbolic.- 3.4 Characteristic Speeds of the Subsystems.- 3.5 Other Subsystems.- 4 Galilean Invariance.- 4.1 Tensors, Galilean Tensors, and Euclidean Tensors.- 4.2 Principle of Relativity.- 4.3 Exploitation of the Principle of Relativity for the Entropy Balance.- 4.4 Exploitation of the Principle of Relativity for the Field Equations.- 4.5 Field Equations for Internal Quantities.- 4.6 Galilei Invariance for Subsystems.- 4.7 Galilean Invariance and Entropy Principle.- 4.8 Explicit Velocity Dependence of Constitutive Quantities. The Determination of Ar.- 5 Thermodynamics of an Euler Fluid.- 5.1 The Euler Fluid.- 5.2 Lagrange Multipliers.- 5.3 Internal Lagrange Multipliers.- 5.4 Absolute Temperature.- 5.5 Vector Potential.- 5.6 Convexity.- 5.7 Characteristic Speed.- 5.8 Subsystems.- 5.9 Discussion.- 4 Extended Thermodynamics of Monatomic Gases.- 1 The Equations of Extended Thermodynamics of Monatomic Gases.- 1.1 Thermodynamic Processes.- 1.2 Discussion.- 1.3 Galilean Invariance. Convective and Nonconvective Fluxes.- 1.4 Euclidean Invariance. Inertial Effects.- 2 Constitutive Theory.- 2.1 Restrictive Principles.- 2.2 Exploitation of the Principle of Material Frame-Indifference.- 2.3 Exploitation of the Entropy Principle.- 2.4 Exploitation of the Requirement of Convexity and Causality.- 3 Field Equations and the Thermodynamic Limit.- 3.1 Field Equations.- 3.2 The Thermodynamic Limit.- 3.3 The Frame Dependence of the Heat Flux.- 3.4 Material Frame Indifference in Ordinary and Extended Thermodynamics.- 4 Thermal Equations of State and Ideal Gases.- 4.1 The Classical Ideal Gas.- 4.2 Comparison with the Kinetic Theory.- 4.3 Comparison with Extended TIP.- 4.4 Degenerate Ideal Gases.- 5 Thermodynamics of Mixtures of Euler Fluids.- 1 Ordinary Thermodynamics of Mixtures (TIP).- 1.1 Constitutive Equations.- 1.2 Paradox of Diffusion.- 2 Extended Thermodynamics of Mixtures of Euler Fluids.- 2.1 Balance Equations.- 2.2 Thermodynamic Processes.- 2.3 Constitutive Theory.- 2.4 Summary of Results.- 2.5 Wave Propagation in a Nonreacting Binary Mixture.- 2.6 Landau Equations. First and Second Sound in He II.- 3 Ordinary and Extended Thermodynamics of Mixtures.- 3.1 The Laws of Fick and Fourier in Extended Thermodynamics.- 3.2 Onsager Relations.- 3.3 Inertial Contribution to the Laws of Diffusion.- 6 Relativistic Thermodynamics.- 1 Balance Equations and Constitutive Restrictions.- 1.1 Thermodynamic Processes.- 1.2 Principles of the Constitutive Theory.- 2 Constitutive Theory.- 2.1 Scope and Structure.- 2.2 Lagrange Multipliers and the Vector Potential. Step i.- 2.3 Principle of Relativity and Linear Representations. Step ii.- 2.4 Stress Deviator, Heat Flux, and Dynamic Pressure. Step iii.- 2.5 Fugacity and Absolute Temperature. Step iv.- 2.6 Linear Relations Between Lagrange Multipliers and n,UA, t(AB),?,qA,e. Step v.- 2.7 The Linear Flux Tensor. Step vi.- 2.8 The Entropy Flux Vector. Step vii.- 2.9 Residual Inequality Step viii.- 2.10 Causality and Convexity. Step ix.- 2.11 Summary of Results. Step x.- 3 Identification of Viscosities and Heat-Conductivity.- 3.1 Extended Thermodynamics and Ordinary Thermodynamics.- 3.2 Transition from Extended to Ordinary Thermodynamics.- 4 Specific Results for Relativistic and Degenerate Gases.- 4.1 Equilibrium Distribution Function.- 4.2 The Degenerate Relativistic Gas.- 4.3 Nondegenerate Relativistic Gas.- 4.4 Degenerate Nonrelativistic Gas.- 4.5 Nondegenerate Nonrelativistic Gas.- 4.6 Strongly Degenerate Relativistic Fermi Gas.- 4.7 A Remark on the Strongly Degenerate Relativistic Bose Gas.- 4.8 Equilibrium Properties of an Ultrarelativistic Gas.- 5 An Application: The Mass Limit of a White Dwarf.- 6 The Relativistic Kinetic Theory for Nondegenerate Gases.- 6.1 Boltzmann-Chernikov Equation.- 6.2 Equations of Transfer.- 6.3 Equations of Balance for Particle Number, Energy-Momentum, Fluxes, and Entropy.- 6.4 Maxwell-Jüttner Distribution, Equilibrium Properties.- 6.5 Possible Thermodynamic Fields in Equilibrium.- 7 The Nonrelativistic Limit of Relativistic Thermodynamics.- 7.1 The Problem.- 7.2 Variables and Constitutive Quantities.- 7.3 The Dynamic Pressure.- 7.4 Order of Magnitude of the Dynamic Pressure.- 7 Extended Thermodynamics of Reacting Mixtures.- 1 Motivation, Results, and Discussion.- 1.1 Motivation.- 1.2 Results.- 1.3 Discussion.- 2 Fields.- 2.1 A Conventional Choice.- 2.2 Absolute Temperature, Fugacities, and Chemical Affinity.- 2.3 Summary of Fields.- 3 Field Equation.- 3.1 Balance Laws.- 3.2 Constitutive Theory.- 3.3 Principle of Relativity.- 4 Entropy Inequality.- 4.1 Lagrange Multipliers.- 4.2 Exploitation.- 5 Nonrelativistic Limit.- 5.1 Discussion.- 5.2 Dynamic Pressure and Bulk Viscosity.- 5.3 Thermal Conductivity and Viscosity.- 8 Waves in Extended Thermodynamics.- 1 Hyperbolicity and Symmetric Hyperbolic Systems.- 1.1 Hyperbolicity in the t-direction.- 1.2 Symmetric Hyperbolic Systems.- 2 Linear Waves.- 2.1 Plane Harmonic Waves, the Dispersion Relation.- 2.2 The High-Frequency Limit.- 2.3 Higher-Order Terms.- 2.4 Linear Waves in Extended Thermodynamics.- 3 Hyperbolicity and Nonlinear Waves..- 3.1 The Characteristic Polynomial.- 3.2 Region of Hyperbolicity.- 4 Acceleration Waves.- 4.1 Amplitude of Discontinuity Waves.- 4.2 Growth and Decay.- 4.3 Evolution of Amplitude in Extended Thermodynamics.- 4.4 Acceleration Waves in Relativistic Extended Thermodynamics.- 5 Weak Solutions and Shock Waves.- 5.1 Weak Solutions.- 5.2 Rankine-Hugoniot Equations.- 5.3 Shocks in Extended Thermodynamics.- 5.4 Selection Rules for Physical Shocks. The Entropy Growth Condition..- 5.5 Selection Rules for Physical Shocks. The Lax Conditions..- 5.6 Lax Condition in Extended Thermodynamics.- 9 Extended Thermodynamics of Moments.- 1 Field Equations for Moments.- 1.1 Densities, Fluxes, and Productions as Moments of the Phase Density.- 1.2 Extended Thermodynamics of Moments.- 1.3 Specific Phase Densities.- 1.4 Field Equations for ?? and Equations for u? near Equilibrium.- 1.5 The Case N=3: An Illustration.- 1.6 Field Equations for n=13, 14, 20, 21, 26, 35.- 2 Characteristic Speeds.- 2.1 Field Equations near Equilibrium.- 2.2 Pulse Speed.- 2.3 Discussion.- 2.4 The Relativistic Case; Speeds Smaller than c.- 3 Mean Eigenfunctions.- 3.1 Eigenfunctions and Eigenvalues.- 3.2 Mean Eigenfunctions as the Main Field.- 3.3 Linear Field Equations for the Mean Eigenfunctions.- 4 Maximization of Entropy.- 4.1 Maximizing Entropy.- 4.2 Maximizing Entropy is Equivalent to Extended Thermodynamics of Moments.- 10 Extended Thermodynamics and Light Scattering.- 1 Basic Electrodynamics.- 1.1 Distant Field Approximation.- 1.2 Incident Plane Harmonic Wave.- 2 A Modicum of Fluctuation Theory.- 2.1 Expectation Values.- 2.2 Temporal Evolution of a Fluctuation.- 2.3 Autocorrelation of ES(R, t).- 3 Measuring the Spectral Density.- 3.1 Signal and Spectral Density.- 3.2 Measured Data and Their Dependence on Pressure.- 4 Navier-Stokes-Fourier Fluid.- 4.1 Dynamic Form Factor.- 4.2 An Alternative Form of the Dynamic Form Factor. Also: An Approximate Form for Forward Scattering.- 4.3 Graphical Representation of the Dynamic Form Factor for a Monatomic Ideal Gas.- 4.4 Comparison with Experimental Data.- 4.5 Autocorrelation.- 4.6 Heat and Sound Modes.- 5 Extended Thermodynamics.- 5.1 Introducing Extended Thermodynamics. The Case of 13 Moments.- 5.2 Dynamic Form Factors for n=20, 35, 84.- 5.3 Heat and Sound Modes in Extended Thermodynamics.- 5.4 Higher Moments by Method of Eigenfunctions.- 5.5 Dynamic Form Factors for Many Moments.- 5.6 Evaluation of Moment Theories.- 5.7 Characteristic speeds.- 5.8 More Experimental and Theoretical Evidence.- 6 Extrapolation of S(q, ?) for y ? 0.- 6.1 The Problem.- 6.2 The Boltzmann Equation in the Krook Approximation.- 6.3 The Dynamic Form Factor S(q,?); General Formula.- 6.4 Fluctuations in Phase Space.- 6.5 The Dynamic Form Factor S(q,?); Specific Form.- 6.6 Discussion.- 11 Testing Extended Thermodynamics by Sound.- 1 Basic Acoustics.- 1.1 How the Acoustic Resonator Measures Phase Speeds in Principle.- 1.2 Piezoelectric Transducer and the Mechanical Impedance.- 1.3 External Mechanical Impedance and Wavelength.- 1.4 Difficulties with Many Modes and Damping.- 2 Dispersion Relations.- 2.1 Navier-Stokes-Fourier Theory.- 2.2 Extended Thermodynamics of 13 Fields.- 2.3 Extended Thermodynamics with Many Variables.- 2.4 Conclusion and Estimate.- 3 Maximum Speed.- 3.1 Modes of Least Damping.- 3.2 The Maximum Speed.- 12 Structure of Shock Waves.- 1 Experimental Evidence.- 2 Review of Previous Work.- 2.1 Rankine-Hugoniot Relations.- 2.2 Becker’s Solutions.- 2.3 Singular Perturbation Analysis.- 2.4 Numerical Solution by Gilbarg and Paolucci.- 2.5 The 13-Moment Theory by Grad.- 2.6 The 13-Moment Theory by Anile & Majorana.- 2.7 Criticism of Moment Methods for Shock Structure.- 2.8 Alternative Methods for Shock Structure Calculations.- 3 Preliminaries on Singular Points and Characteristic Speeds.- 3.1 Field Equations and Boundary Values.- 3.2 Singular Points and Stationary Points.- 3.3 The Singularities D = 0.- 3.4 Regular and Irregular Singularities.- 4 Numerical Calculation of the Shock Structure.- 4.1 Initial and Boundary Value Problems.- 4.2 Algorithm for the Initial Value Problem.- 4.3 Algorithm for the Boundary Value Problem.- 4.4 The 13-Moment Case.- 4.5 The 14-Moment Case.- 4.6 The 21-Moment Case.- 5 Conclusion.- 6 Addendum on Initial Value Problem for 13 Moments.- 7 Quantitative Results and Conclusions.- 13 Extended Thermodynamics of Radiation.- 1 Structure of Extended Thermodynamics of Photons.- 1.1 Energy and Momentum of Individual Photons.- 1.2 Radiative Transfer Equation.- 1.3 Moments and Moment Equations. The Closure Problem.- 1.4 Entropy and Maximization of Entropy.- 1.5 Closure.- 2 Equilibrium.- 2.1 The First Few Moments.- 2.2 Equilibrium of Radiation with Matter.- 3 Near Equilibrium.- 3.1 Phase Density in Near-Equilibrium.- 3.2 Approximate Lagrange Multipliers.- 4 Field Equations.- 4.1 Closure for Moments.- 4.2 Closure for Productions.- 4.3 The Hierarchies of Field Equations.- 4.4 Absorption and Emission of Bremsstrahlung. Thomson Scattering..- 5 Local Radiative Equilibrium.- 5.1 The Rosseland Mean Value of the Absorption Coefficient.- 5.2 Maxwell Iteration.- 5.3 Conclusion.- 6 Compression of Radiation.- 6.1 A Thought Experiment.- 6.2 Solution of the Radiative Transfer Equation.- 6.3 Solution of Moment Equations.- 6.4 Conclusion.- 7 Penetration of a Beam of Radiation into Matter.- 7.1 Field Equations.- 7.2 Characteristic Speeds and Amplitudes of the Propagating Beam.- 7.3 Plane Harmonic Waves and Dispersion Relation (General).- 7.4 Intense Absorption. The Damped Wave Limit.- 7.5 Intense Scattering. The Diffusion Limit.- 7.6 General Case and a Simple Example.- 8 Radiative Entropy in Gray Bodies.- 8.1 Photon Gas and an Eulerian Fluid.- 8.2 Equilibrium of Radiation with Matter at Rest.- 8.3 Entropy Production due to Matter-Photon Interaction.- 8.4 Thermodynamic Fields of Radiation in the Neighborhood of a Spherical Source.- 8.5 Absorption of Radiation from a Spherical Source in an Eulerian Fluid at Temperature T.- 8.6 Entropic Production for Incident Rays.- 8.7 Pseudo-Temperature.- 8.8 Entropy Flux and Entropy.- 14 Extended Thermodynamics of Phonons.- 1 Phonon Transfer Equation.- 1.1 Energy and Momentum of Phonons.- 1.2 Phonon Transfer Equation, Energy and Momentum.- 1.3 The Phase Density of Production.- 2 Moments and Moment Equations.- 2.1 Moments and their Equilibrium Values.- 2.2 Moment Equations and Conservation Laws.- 2.3 Closure Problem.- 3 The Heat Pulse Experiment.- 3.1 Experimental Results and One-Dimensional Equations.- 3.2 Ballistic Phonons.- 3.3 Second Sound in Its Purest Form.- 3.4 Damped Second Sound and Pure Diffusion.- 3.5 The 9-Field Theory of Extended Thermodynamics.- 3.6 Heat Pulses. Numerical Solutions.- 15 Thermodynamics of Metal Electrons.- 1 Equations of Balance.- 1.1 Kinetic Theory of Metal Electrons.- 1.2 Equations of Balance of Mass, Momentum, Energy, and Energy Flux.- 1.3 Entropy Principle and Phase Density Close to Equilibrium.- 2 Extended Thermodynamics and Kinetic Theory.- 2.1 Toward Extended Thermodynamics of Electrons in Metals.- 2.2 A Convenient Shortcut via the Kinetic Theory of Electrons.- 2.3 Characteristic Speeds.- 2.4 The Laws of Ohm and Fourier.- 2.5 Hall and Coriolis Effects.- 2.6 Discussion.- 16 Viscoelastic Fluids.- 1 Viscoelastic Fluids of Second Grade.- 1.1 The Stress of a Second Grade Fluid.- 1.2 Ordinary Thermodynamics of Second Grade Fluids.- 1.3 Discussion.- 2 Rate-Type versus Differential-Type Constitutive Equations.- 2.1 Cattaneo and Stability.- 2.2 Viscoelasticity and Stability.- 2.3 Conclusion.- 3 Toward Extended Thermodynamics of Viscoelasticity.- 3.1 Fields and Field Equations.- 3.2 Incompressible Adiabatic Fluid.- 3.3 Entropy Inequality.- 3.4 Partial Exploitation of the Entropy Inequality.- 3.5 Evaluation.- 3.6 Criticism and Outlook.