The book relates fluid flows with chemical reactions. This volume is dedicated to the establishment of consistent systems of equations with their boundary conditions and interfaces, allowing modeling and dealing with complex situations. It contains three main chapters entitled: 1. Simple fluids, 2. Reactive mixtures, and 3. Interfaces and lines. Several appendices on mathematical notation, thermodynamics and mechanics methods are grouped together in Chapter 4.

Preface xiii

List of the Main Symbols xv

Chapter 1. Simple Fluids 1

1.1. Introduction 1

1.2. Key elements in deformation theory – Lagrangian coordinates

and Eulerian coordinates 2

1.2.1. Strain rates 2

1.2.2. Lagrangian coordinates and Eulerian coordinates 7

1.2.3. Trajectories, stream lines, emission lines 8

1.3. Key elements in thermodynamics Reversibility, irreversible processes: viscosity, heat conduction 9

1.3.1. Thermodynamic variables, definition of a system, exchanges, differential manifold of equilibrium states, transformation 9

1.3.2. Laws of thermodynamics 11

1.3.3. Properties of simple fluids at equilibrium. 14

1.4. Balance equations in fluid mechanics. Application to incompressible and compressible perfect fluids and viscous fluids 18

1.4.1. Mass balance 18

1.4.2. Concept of a particle in a continuous medium: local state 19

1.4.3. Balance for the property F 20

1.4.4. Application to volume, to momentum and to energy 22

1.4.5. Entropy balance and the expression of the rate of production of entropy 23

1.4.6. Balance laws for discontinuity 25

vi Flows and Chemical Reactions

1.4.7. Application to incompressible perfect fluids 26

1.4.8. Application to dissipative fluids 31

1.5. Examples of problems with 2D and 3D incompressible perfect fluids 32

1.5.1. Planar 2D irrotational flows: description in the complex plane of steady flows 32

1.5.2. 3D irrotational flows of incompressible perfect fluids: source, sink, doublet 36

1.5.3. Rotational flows of incompressible perfect fluids 41

1.6. Examples of problems with a compressible perfect fluid: shockwave, flow in a nozzle, and characteristics theory 44

1.6.1. General theorems 44

1.6.2. Propagation of sound in an ideal gas 44

1.6.3. Discontinuities 46

1.6.4. Unsteady characteristics 47

1.6.5. Steady normal shockwave: Hugoniot and Prandtl relations 48

1.6.6. Flow in a de Laval nozzle 49

1.6.7. Simple wave 53

1.7. Examples of problems with viscous fluids 56

1.7.1. General equations 56

1.7.2. Incompressible viscous fluid 57

1.7.3. Flow of a compressible dissipative fluid: structure of a shockwave 61

1.8. Exercises 64

1.8.1. Exercises in kinematics (section 1.2) 64

1.8.2. Exercises in thermodynamics (section 1.3). 67

1.8.3. Exercises for the balance equations in fluid mechanics (section 1.4) 68

1.8.4. Examples of problems with 2D and 3D incompressible perfect fluids (section 1.5) 70

1.8.5. Examples of problems with a compressible perfect fluid

(section 1.6) 74

1.8.6. Examples of problems with viscous fluids (section 1.7) 77

1.9. Solutions to the exercises 79

1.9.1. Solutions to the exercises in kinematics. 79

1.9.2. Solutions to the Exercises in thermodynamics 83

1.9.3. Solutions to the exercises for the balance of equations in

fluid mechanics 88

1.9.4. Solutions to the examples of problems with 2D and 3D

incompressible perfect fluids 89

Table of Contents vii

1.9.5. Solutions to the examples of problems with a compressible

perfect fluid 93

1.9.6. Solutions to the examples of problems with viscous fluids 95

Chapter 2. Reactive Mixtures 101

2.1. Introduction 101

2.2. Equations of state 103

2.2.1. Definition of the variables of state of a mixture 103

2.2.2. Thermodynamic properties of mixtures 108

2.2.3. Reactive mixture 118

2.2.4. Other issues relating to the thermodynamics of mixtures 123

2.3. Balance equations of flows of reactive mixtures 124

2.3.1. Balance of mass of the species j and overall balance of mass 124

2.3.2. General balance equation of a property F. 127

2.3.3. Momentum balance 129

2.3.4. Energy balance 129

2.3.5. Balance relations in a discrete system. 132

2.3.6. Entropy balance in a continuum 137

2.3.7. Balance equations at discontinuities in continuous media 140

2.4. Phenomena of transfer and chemical kinetics 142

2.4.1. Introduction 142

2.4.2. Presentation of the transfer coefficients by linear TIP 143

2.4.3. Other presentations of the transfer coefficients 147

2.4.4. Elements of chemical kinetics 152

2.5. Couplings 155

2.5.1. Heat transfer and diffusion 155

2.5.2. Shvab-Zeldovich approximation 158

Chapter 3. Interfaces and Lines 163

3.1. Introduction 163

3.1.1. Interfaces 163

3.1.2. Lines 165

3.2. Interfacial phenomena 166

3.2.1. General aspects 166

3.2.2. General form of an interfacial balance law 168

3.2.3. Constitutive laws for interfaces whose variables directly

satisfy the classical equations in thermostatics and in 2D-TIP 173

3.2.4. Constitutive laws for interfaces deduced from classical

thermostatics and 3D-TIP. Stretched flame example 177

3.2.5. Interfaces manifesting resistance to folding 179

viii Flows and Chemical Reactions

3.2.6. Numerical modeling 179

3.2.7. Interfaces and the second gradient theory. 182

3.2.8. Boundary conditions of the interfaces 185

3.2.9. Conclusion 185

3.3. Solid and fluid curvilinear media: pipes, fluid lines and

filaments 186

3.3.1. General aspects 186

3.3.2. Establishing the balance equations in a curvilinear medium. 188

3.3.3. Simplified theories 209

3.3.4. Triple line and second gradient theory 216

3.3.5. Conclusion 220

3.4. Exercises 222

3.4.1. Exercises regarding solid curvilinear media 222

3.4.2. Exercises regarding fluid curvilinear media 222

3.5. Solutions to the exercises 223

3.5.1. Solutions to exercises regarding solid curvilinear media. 223

3.5.2. Solutions to the exercises regarding fluid curvilinear media 225

APPENDICES 229

Appendix 1. Tensors, Curvilinear Coordinates, Geometry and Kinematics of Interfaces and Lines 231

A1.1. Tensor notations 231

A1.1.1. Tensors and operations on tensors 231

A1.2. Orthogonal curvilinear coordinates. 234

A1.2.1. General aspects 234

A1.2.2. Curl of a vector field 236

A1.2.3. Divergence of a vector field 237

A1.2.4. Gradient of a scalar 238

A1.2.5. Laplacian of a scalar 238

A1.2.6. Differentiation in a curvilinear basis 238

A1.2.7. Divergence of a second order tensor 239

A1.2.8. Gradient of a vector 239

A1.2.9. Cylindrical coordinates and spherical coordinates 240

A1.3. Interfacial layers 242

A1.3.1. Prevailing directions of an interfacial medium 242

A1.3.2. Operators of projection for interfaces 244

A1.3.3. Surface gradients of a scalar field 245

A1.3.4. Curvature vector of a curve 245

A1.3.5. Normal and tangential divergences of a vector field 246

A1.3.6. Extension of surface per unit length 246

A1.3.7. Average normal curvature of a surface 247

A1.3.8. Breakdown of the divergence of a vector field 248

A1.3.9. Breakdown of the Laplacian of a scalar field 249

A1.3.10. Breakdown of the divergence of a second order tensor 249

A1.3.11. Projection operators with the intrinsic definition of a surface 252

A1.3.12. Comparison between the two descriptions 253

A1.4. Curvilinear zones 254

A1.4.1. Presentation 254

A1.4.2. Geometry of the orthogonal curvilinear coordinates 256

A1.4.3. Projection operators and their consequences 257

A1.5. Kinematics in orthogonal curvilinear coordinates 260

A1.5.1. Kinematics of interfacial layers 260

A1.5.2. Kinematics of curvilinear zones 266

A1.5.3. Description of the center line 269

Appendix 2. Additional Aspects of Thermostatics 277

A2.1. Laws of state for real fluids with a single constituent 277

A2.1.1. Diagram of state for a pure fluid 277

A2.1.2. Approximate method to determine the thermodynamic functions 278

A2.1.3. Van der Waals fluid 279

A2.1.4. Other laws for dense gases and liquids 279

A2.2. Mixtures of real fluids 280

A2.2.1. Mixture laws for a real mixture 280

A2.2.2. Expression of the free energy of a real mixture 281

Appendix 3. Tables for Calculating Flows of Ideal Gas ?? ??1.4 283

A3.1. Calculating the parameters in continuous steady flow

(section 1.6.6.2) 286

A3.2. Formulae for steady normal shockwaves 288

Appendix 4. Extended Irreversible Thermodynamics. 289

A4.1. Heat balance equations in a non-deformable medium in EIT 290

A4.2. Application to a 1D case of heat transfer 293

A4.3. Application to heat transfer with the evaporation of a droplet 296

A4.3.1. Reminders about evaporating droplets 296

A4.3.2. Evaporating droplet with EIT. 300

A4.4. Application to thermal shock 302

A4.4.1. Presentation of the problem and solution using CIT 302

x Flows and Chemical Reactions

A4.4.2. Thermal shock and EIT 303

A4.4.3. Application of the second order approximation into two examples of thermal shock 305

A4.5. Outline of EIT 307

A4.6. Applications and perspectives of EIT 310

Appendix 5. Rational Thermodynamics 313

A5.1. Introduction 313

A5.2. Fundamental hypotheses and axioms 314

A5.2.1. Basic hypotheses 314

A5.2.2. Basic axioms 316

A5.3. Constitutive laws 318

A5.4. Case of the reactive mixture 320

A5.4.1. Principle of material frame indifference 320

A5.4.2. Constitutive laws for a reactive mixture 321

A5.5. Critical remarks 324

Appendix 6. Torsors and Distributors in Solid Mechanics 325

A6.1. Introduction 325

A6.1.1. Torsor 325

A6.1.2. Distributor 325

A6.1.3. Power 326

A6.2. Derivatives of torsors and distributors which depend on a single position parameter 326

A6.2.1. Derivative of the velocity distributor 327

A6.2.2. Derivative of the tensor of forces 328

A6.3. Derivatives of torsors and distributors dependent on two positional parameters 328

A6.3.1. Expression of the velocity distributor 329

A6.3.2. Derivative of the velocity distributor 329

Appendix 7. Virtual Powers in a Medium with a Single Constituent 331

A7.1. Introduction 331

A7.2. Virtual powers of a system of n material points 332

A7.3. Virtual power law 333

A7.4. The rigid body and systems of rigid bodies 333

A7.4.1. The rigid body 333

A7.4.2. System of rigid bodies, concept of a link 334

A7.5. 3D deformable continuous medium 335

A7.5.1. First gradient theory 335

A7.5.2. A 3D case of perfect internal linkage: the incompressible perfect fluid 337

A7.5.3. Second gradient theory 337

A7.6. 1D continuous deformable medium 338

A7.6.1. First gradient theory 338

A7.6.2. A 1D case of perfect internal linkage: perfectly flexible

and inextensible wires 340

A7.7. 2D deformable continuous medium 340

Bibliography 343

Index 355