Thermodynamics of Fluids Under Flow (eBook)

Preface 5
Contents 10
Non-equilibrium Thermodynamics and Rheology 15
1.1 A Short Review of Rheological Concepts 16
1.1.1 Basic Rheological Quantities 17
1.1.2 Basic Rheological Models 18
1.2 Extended Irreversible Thermodynamics 24
1.2.1 Viscous Pressure 25
1.2.2 Viscous Pressure and Diffusion Flux 28
1.3 Rational Extended Thermodynamics 31
1.4 Theories with Internal Variables 37
1.5 Hamiltonian Formulations 40
1.5.1 Microscopic Level: Distribution Function as an Internal Variable 42
1.5.2 Mesoscopic Level: Configuration Tensor as an Internal Variable 43
Non-equilibrium Temperature and Entropy in Flowing Ideal Gases: Maximum-Entropy Approach 46
2.1 Review of Some Basic Concepts 47
2.2 Information Theory: General Formalism 48
2.3 Information Analysis of an Ideal Gas Under Viscous Pressure 52
2.3.1 Non-equilibrium Entropy and Chemical Potential: General Formalism 55
2.3.2 Analysis of Plane Couette Flow: Pure Shear Effects 57
2.3.3 Plane Couette Flow: Shear and Normal Effects 60
2.4 Non-equilibrium Temperatures in Flowing Gases and Mixtures 63
2.5 Partition Function for a Flowing Relativistic Ideal Gas 67
Kinetic Theory of Flowing Gases and Phonons. Phonon Hydrodynamics and Heat Transport in Nanosystems 69
3.1 Kinetic Theory: Basic Concepts 69
3.1.1 H Theorem 70
3.1.2 Non-equilibrium Distribution Function 72
3.2 Grad’s Approach 73
3.3 Comparison with Exact Results 75
3.4 Kinetic Theory of Phonons and Phonon Hydrodynamics 77
3.5 Poiseuille Phonon Flow and Heat Transport in Nanosystems 81
3.6 Boundary Conditions and Effective Thermal Conductivity in Smooth and Rough Nanowires 82
3.6.1 Heat Transfer in Thin Smooth Nanowires 83
3.6.2 Heat Transfer in Thin Rough Nanowires 84
3.7 Thermal Conductivity of Porous Silicon 86
Non-ideal Fluids and Nuclear Collisions 90
4.1 Modified Equations of State and Shift of Critical Point 91
4.1.1 Van der Waals Fluids 91
4.1.2 Regular Binary Solutions 94
4.1.3 Experimental Results 95
4.2 Kinetic Theory of Dilute Non-ideal Gases 96
4.3 Comparison with Computer Simulations 101
4.4 Nuclear Collisions 106
4.4.1 Internal Collective Flows and Information Theory 106
4.4.2 Generalised Gibbs Equation 107
4.4.3 Causal Dissipative Hydrodynamics 110
Polymeric Solutions and Blends 112
5.1 Kinetic Theory of Dilute Polymeric Solutions 113
5.1.1 Freely Jointed Chain 113
5.1.2 The Bead-and-Spring Rouse–Zimm Model 116
5.2 Derivation of the Steady-State Compliance 124
5.3 Maximum-Entropy Approach 126
5.4 Entangled Solutions. Reptation Model 127
5.5 Polymer Blends. Double Reptation Model 130
Non-equilibrium Chemical Potential and Shear-Induced Effects in Polymer Solutions and Blends 133
6.1 Survey of Experimental Results 134
6.2 Equilibrium Chemical Potential and Stability Analysis 137
6.3 Non-equilibrium Chemical Potential and Stability Analysis 139
6.3.1 The Choice of Non-equilibrium Variables: Viscous Pressure or Configuration Tensor 139
6.3.2 Stability Analysis 142
6.3.3 Two-Fluids Model 143
6.4 Phase Diagram of Polymer Solutions Under Shear Flow 144
6.4.1 Dilute and Semidilute Polymer Solutions 145
6.4.2 Entangled Solutions 148
6.5 A Practical Illustration: Flow Effects in Polymer Extraction from a Porous Matrix 151
6.6 Flow-Induced Effects in Polymer Blends: Two-Fluid Approach and Extended Approach 154
6.6.1 Two-Fluid Approach 155
6.6.2 Extended Approach 157
6.7 Non-Newtonian Effects in Phase Separation 159
6.8 Other Approaches: Flexibility and Droplet Approaches 161
6.8.1 The Flexibility Approach 161
6.8.2 The Droplet Approach 162
Comparison of Thermodynamical and Dynamical Approaches 164
7.1 Dynamical Derivation and Generalization of Thermodynamical Stability Criteria 165
7.1.1 Situations Without Coupling Between Diffusion and Shear 166
7.1.2 Situations with Coupling Between Diffusion and Shear 168
7.2 Structure Factor 170
7.2.1 Generalized Ginzburg–Landau Potential in Flowing Systems 171
7.2.2 Dynamical Contributions to the Chemical Potential 173
7.3 Derivation of the Structure Factor 174
7.3.1 Evolution Equations 174
7.3.2 Equations of State 175
7.3.3 Flow Contribution to the Structure Factor 176
8.1 Shear-Induced Migration of Polymers 180
Shear-Induced Migration and Flow Chromatography 180
8.1.1 The Simplest Model for Shear-Induced Migration 181
8.1.2 Non-equilibrium Chemical Potential and Effective Diffusion Coefficient 183
8.2 Shear-Induced Concentration Banding and Macromolecular Separation in Cone-and-Plate Flows 189
8.3 Shear-Induced Migration and Molecular Separation in Tubes 193
Taylor Dispersion and Anomalous Diffusion 196
9.1 Brownian Motion in Shear Flow 197
9.2 Taylor Dispersion and Microfluidics 199
9.3 Taylor Dispersion for Short and Intermediate Times 201
9.3.1 Evolution Equation for the Flow of Matter 202
9.3.2 Evolution of Effective Diffusion Coefficient: From Reversible to Irreversible Behaviours 202
9.3.3 Entropy and Entropy Flux: From Exhaustive Information to the Relevant Information 203
9.4 Anomalous Diffusion and Non-equilibrium Thermodynamics 206
9.4.1 Classical Irreversible Thermodynamics and Diffusion 206
9.4.2 Non-conventional Statistical Mechanics 207
9.4.3 Generalized Thermodynamics and Anomalous Diffusion 209
9.5 Anomalous Diffusion in Flowing Systems 212
9.6 Taylor Dispersion and Anomalous Diffusion 214
9.7 Diffusion on Fractals 215
10.1 Thermodynamic Formulation 219
Chemical Reactions and Polymer Degradation Under Flow 219
10.2 Shear-Induced Polymer Degradation: Kinetic Analysis 222
10.3 Shear-Induced Polymer Degradation: Thermodynamic Analysis 226
10.4 Kinetic Theory of Chemical Reactions 230
10.5 Recurrence Method for Probability Weight Distribution Under Viscous Pressure 233
Non-equilibrium Thermodynamics of Laminar and Turbulent Superfluids 236
11.1 Essential Concepts and Phenomena of Superfluids 237
11.2 The Two-Fluid Model and Second Sound 238
11.2.1 Evolution Equations and Wave Propagation 239
11.2.2 Thermodynamics of Superfluid Helium 242
11.3 The Extended One-Fluid Model of Liquid Helium II 243
11.4 Quantized Vortices in Rotation and Counterflow 246
11.4.1 Macroscopic Description of Vortex Friction 248
11.4.2 Rotating Frame 249
11.4.3 Counterflow Turbulence 249
11.5 Second Sound Propagation in the Presence of Quantized Vortices 251
11.5.1 Rotating Cylinders 251
11.5.2 Second Sound and Counterflow Turbulence 252
11.6 Evolution Equation for the Vortex Line Density 253
11.6.1 Transition from the Laminar to the Turbulent Regime 254
11.6.2 Simultaneous Rotation and Counterflow 256
11.6.3 Non-equilibrium Thermodynamics of Vortex Tangles 257
11.7 Hydrodynamics of Turbulent Superfluids 260
A.1 Polystyrene in Dioctyl-Phthalate (PS/DOP) 263
Experimental Data on Polymer Solutions 263
A.1 Polystyrene in Dioctyl-Phthalate (PS/DOP) 263
A.2 Polystyrene in Transdecalin (PS/TD) 264
A.3 Polystyrene Dissolved in Oligomeric Polystyrene 266
Liquid Crystals 268
B.1 Equilibrium Thermodynamics and the Isotropic-Nematic Phase Transition 268
B.1.1 Phase Transition Induced by Temperature Changes 269
B.1.2 Phase Transition Induced by Density Changes 270
B.2 Dynamic Equations in the Presence of a Flow 271
B.3 Thermodynamic Formulation 274
B.4 Maximum-Entropy Approach 275
Summary of Vector and Tensor Notation 278
C.1 Symmetric and Antisymmetric Tensors 278
C.2 Decomposition of a Tensor 278
C.3 Scalar (or Dot) and Tensorial (Inner) Products 279
C.4 (Inner) Tensorial Product (Also Named Dyadic Product) 279
C.5 Cross Multiplication Between Two Vectors and Between a Tensor and a Vector 280
C.6 Differentiation 280
C.7 Tensor Invariants 281
Useful Integrals in the Kinetic Theory of Gases 282
Some Physical Constants 283
References 284
Index 300