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Adiabatic Thermodynamics of Fluids ADIABATIC THERMODYNAMICS OF FLUIDS When asked to explain entropy, a teacher of engineering said “Oh, I don’t know what entropy is; nobody knows what it is; you just use it”. Quoted by J.H. Hildebrand. “Fundamental questions were settled 50 years ago.” From a letter ex- plaining to an author why his submission was rejected. Moins ils savent, moins ils doutent, mais la doute est porte de la v´erit´ee. Parot. A monograph Copyright 2012, 2013, 2014, 2015, 2016, 2017 2018 C. Fronsdal. To be published by World Scientific. – Page 1 i TABLE OF CONTENTS Forword Page 1 Acknowledgements Page 6 I. INTRODUCTION I.1. Elements of hydrodynamics Page 8 The nature of a liquid. Density and pressure. Energy. Equation of continuity. Newtonian fluids. I.2. Hamiltonian and Lagrangian forms of hydrodynamics Page 12 Action principles. Lagrange multipliers. The Lagrangian. Velocity potential. Kinetic energy. The Hamiltonian. I.3. The temperature Page 18 The ideal gas law. I.4. The internal energy and the first law Page 20 Internal Energy. Entropy. Carnot cycle. The free energy. I.5. The kinetic potential, and a warning Page 25 The energy concept. The danger of defining a theory by a set of equations. I.6. Possibility of a Lagrangian in case of dissipation Page 27 I.7. Hydrodynamics of potential flow Page 28 I.8. The ‘Lagrangian’ formulation of hydrodynamics Page 30 The limitations of potential hydrodynamics. II. ACTION PRINCIPLE FOR THERMODYNAMICS II.1. The internal energy and the free energy Page 35 Functional relations. Legendre transformations. Normalization of the energy II.2. The ideal gas Page 36 The free energy of the ideal gas II.3. The two time scales of thermodynamics Page 38 Adiabatic process. Dissipative process II.4. Towards an Action Principle for Thermodynamics Page 41 Interpretation of the action principle The free energy and the Gibbs function. Fixing the entropy and the pressure II.5. Lowest energy and maximal entropy Page 45 Gibbs’ theorem ii III. LOCALIZATION III.1. Hamiltonian density. Adiabatic equations Page 47 Adiabatic condition. Hydrostatic condition. The specific entropy density. Thermodynamic pressure and the pressure on the walls. III.2. The kinetic energy Page 52 Poisson bracket. III.3. The Lagrangian and adiabatic dynamics Page 54 On-shell Lagrangian. Legendre transformations III.4. The second law Page 56 The increasing entropy III.5. Work and pressure Page 59 Pressure on the walls. Adiabatic systems III.6. Free expansion of the expansion of the ideal gas Page 64 Entropic rubber. Some questions III.7. Local thermodynamical potentials Page 67 The Clapeyron equation. Open and closed systems. III.8. Conservation laws and the stress tensor Page 69 Noether’s theorems. The equalisation of pressure III.9. Minimal energy and maximal entropy Page 74 The classical Lagrangian paradigm. Maximal entropy at fixed energy. The two laws of thermodynamics III.10. The third law and the normalization of entropy Page 78 III.11. Galilei invariance Page 79 III.12. Remark concerning the space of variations Page82 Taking stock. assigned entropy. IV. APPLICATIONS TO SIMPLE SYSTEMS IV.1. The propagation of sound in an ideal gas Page 87 Harmonic perturbations. Adiabatic propagation. Radial waves IV.2. The van der Waals gas Page 91 Excluded volume. Critical point. Data for Helium IV.3. Saturation Page 95 Interpretation of the velocity potential IV.4. Dew point,bubble point and Maxwell’s rule Page 98 The adiabatic relation. Calculating the dew point of Argon. Normalization of f. Common tangent IV.5. The critical point Page 106 Coexistence iii IV.6. Sound propagation near the critical point Page 108 IV.7. Refinements of the internal energy Page 110 The heat capacity. The speed of sound IV.8. The cubic equation of state Page 112 IV.9. Stability Page 113 Tables Page 115 V. A THEORY OF FLUID MIXTURES V.1. Variables and potentials Page 121 Molar fractions and ‘free energy of mixing’ V.2. Entropy and equations of motion Page 125 Adiabatic systems. Entropy path. Partial pressures V.3. Mixture of 2 ideal gases Page 130 Dalton’s law V.4. The Gibbs-Dalton hypothesis for ideal gases ... Page 133 Strong Gibbs-Dalton hypothesis V.5. .. and what lies behind it Page 136 Condensation V.6. Gibbs-Dalton hypothesis for some real gases Page 140 V.7. The simplest mixtures. Boltzmann theory Page 143 V.8. Chemical reactions. Dissociation Page 144 Gibbs’ phase rule. Solving the equations. Minimal energy V.9. Other chemical reactions Page 150 Constraints. Reaction constant. General case V.10. Taking stock Page 153 Stability Project 1. Study speed of sound over a wide range of densities and temperatures. Collect data and test models Then use the same models to study critical phenomena. Project 2. Adapt the Gibbs-Dalton hypothesis to real gases. Project 3. Use the entropy model to study chemical reactions. VI. IMMISCIBLE FLUIDS VI.1. Ideal gases with mutual interaction Page 156 Data. Conditions for equilibrium. The method of the common tangent. Symmetry. Direct search. Summary. VI.2. Mixtures of two van der Waals gases Page 165 The spinodal. Mixtures. Calculation of the line of coexistence iv VI.3. Sample applications - first example Page 169 Binary van der Waals mixtures. Points that are critical to the experimenter. VI.4. Second mixture of van der Waals fluids Page 175 The square root interaction density. Nitrogen and Argon VI.5. Helium and Xenon Page 183 Interaction. Highest temperature. The adiabatic condition. Summary. Entropy. Coexistence. Van der Waals mixture. Project 4. Collect all experimental data for one or more of the mixtures that are dealt with above. Make a detailed study ′ of the success or lack of it of the interaction αρ1ρ2 + α √ρ1ρ2. VII. SPEED OF SOUND IN MIXTURES VII.1. Propagation of sound in a Gibbs-Dalton mixture Page 185 Equations of motion. Propagation matrix. Two types of mixtures. Summary. Type 1 mixture, similar gases. Type 2, disparate gases. VII.2. A simple model of interactions Page 193 The square root. Variants VII.3. Two van der Waals gases. Sound speed Page 196 VIII. SUPERFLUID HELIUM PART A VIII.1. Experiments and interpretation Page 203 Fountain experiment. Entropy. Equilibrium configuration. Second sound. Thermodynamic pressure. Critical behavior. VIII.2. First analytic model. Page 214 Thermodynamic variables. Structure of the Lagrangian. The pressure. Entropy. Sound speeds. Equilibrium. Heat capacity. Equilibrium. Heat capacity. VIII.3. Dynamical equations of the model. Page 221 The adiabatic index. Fountain. The adiabatic index. Tables. Page 224 Project 5. Try to understand flux quantization. IX. ATMOSPHERES IX.1. Introducing gravitation Page 225 The van der Waals atmosphere IX.2. Mixed, isentropic atmospheres Page 227 v IX.3. Nitrogen and Oxygen Page 228 Gibbs-Dalton and gravity IX.4. Air and Argon Page 231 The significance of a uniform lapse rate. van der Waals atmospheres IX.5. Air and H2O. Introduction Page 234 van der Waals parameters for water. Standard method. IX.6. Phase separation in mixture of Air and Water Page 237 Calculation. Interaction Appendix to chapter IX. Polytropic atmospheres Page 240 Emden’s theorem. The effect of entropic forces. Isentropic atmospheres. What physics is involved? IX.8. The isentropic atmosphere Page 206 What physics is involved? Reply to Maxwell. The U Project 6. Build a model of our atmosphere with interaction, then use it to calculate the speed of propagation in air and compare with measurements at various elevations and atmospheric condition. Compare the value of the interaction strength with that found in other applications. Try to find a single, simple model that fits all the data. X. CONSERVATIVE HYDRODYNAMICS X.1. The variables of hydrodynamics Page 248 Importance of velocity potential. Gravitation. Gauge theories. Why the scalar velocity potential is essential. X.2. Constraint and Euler-Lagrange equations Page 251 X.3. Cylindrical Couette flow Page 252 Laminar Couette flow. A classical mistake and what it tells us. X.4. The action principle provides a kinetic potential Page 259 Viscosity X.5. Stability of Couette flow by action principle Page 260 Boundary conditions. X.6. A proposal Page 263 Main result. Tension. Entropic forces. Future experiments Project 7. Apply the results about angular momentum to a new analysis of experiment with rotating, superfluid Helium. X.7. Conclusions Page 270 Action Methodology. Other applications. A theory of incompressible fluids? vi X.8. The energy - momentum tensor Page 315 Irrotational case. Angular momentum. The general case. Appendix to Chapter 10. Plane Couette flow Page 276 Interpretation XI. ROTATING PLANETS XI.1. Introduction Page 281 The equatorial bulge. Conservative Hydrodynamics XI.2. Compressible fluid rotating in gravitational field Page 285 Equatorial bulge. Regularity at the poles. Overview of results XI.3. Earth and Mars Page 291 Quality of fits, Density profiles. Rings or not. Evolution XI.4. Neptune and Uranus. Venus and Pluto Page 295 XI.5. Saturn and Jupiter. The Sun. Haumea. Galaxies Page 295 Saturn’s hexagon XI.6. Summary and conclusions Page 298 XII. SPECIAL RELATIVITY XII.1. The relativistic, antisymmetric tensor field Page 301 Gauge theory. Field equations and constraints. Photon mass. The covariant gauge. Conservative hydrodynamics XII.2. Non Relativistic limit and Galilei transformations Page 307 Constraints XII.3. Dark Matter– Page310 XIII. GENERAL RELATIVITY XIII.1. Astrophysics and hydrodynamics Page 311 XIII.2. Introduction to General Relativity, the metric Page 312 XIII.3. Introduction, the dynamical metric Page 313 Connections XIII.4. The Bianchi identity Page 315 Implications of the Bianchi identity. Particles as sources. Gravity waves XIII.5. Sources for G.R. Tolman’s formula Page 318 XIII.6. Conservative sources for Relativity Page 319 XIV. Superfluid Helium Part B Page 321 vii REFERENCES Page 234 INDEX Page ACKNOWLEDGEMENTS. It is a pleasure to thank my friends Joseph Rudnick, Gary Williams, Tore Haug-Warberg, Alexander Zheltukhin, Frank Hansen, Hidenori Sonoda and Joel Lebovitz for illuminating conversations.
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