Flows and Chemical Reactions

Roger Prud'homme is an Emeritus Research Director having previously worked as research director for Laboratoire d'Aérothermique (Laboratory of Aerothermics) and then Laboratoire de Modélisation en Mécanique (Laboratory of Mechanical Modeling) for CNRS – the French national scientific research center. His fields of interest include flames (premixed flame modeling and their behavior in microgravity), two phase flows (droplet combustion with condensation of the products, sound propagation in suspensions, vortex, chock wave structure), and the modeling of fluid interfaces.

• Preface xiiiList of the Main Symbols xvChapter 1. Simple Fluids 11.1. Introduction 11.2. Key elements in deformation theory – Lagrangian coordinates and Eulerian coordinates 21.2.1. Strain rates 21.2.2. Lagrangian coordinates and Eulerian coordinates 71.2.3. Trajectories, stream lines, emission lines  81.3. Key elements in thermodynamics Reversibility, irreversible processes: viscosity, heat conduction 91.3.1. Thermodynamic variables, definition of a system, exchanges, differential manifold of equilibrium states, transformation 91.3.2. Laws of thermodynamics 111.3.3. Properties of simple fluids at equilibrium. 141.4. Balance equations in fluid mechanics. Application to incompressible and compressible perfect fluids and viscous fluids 181.4.1. Mass balance 181.4.2. Concept of a particle in a continuous medium: local state 191.4.3. Balance for the property F 201.4.4. Application to volume, to momentum and to energy 221.4.5. Entropy balance and the expression of the rate of production of entropy 231.4.6. Balance laws for discontinuity 251.4.7. Application to incompressible perfect fluids 261.4.8. Application to dissipative fluids 311.5. Examples of problems with 2D and 3D incompressible perfect fluids 321.5.1. Planar 2D irrotational flows: description in the complex plane of steady flows 321.5.2. 3D irrotational flows of incompressible perfect fluids: source, sink, doublet 361.5.3. Rotational flows of incompressible perfect fluids 411.6. Examples of problems with a compressible perfect fluid: shockwave, flow in a nozzle, and characteristics theory 441.6.1. General theorems 441.6.2. Propagation of sound in an ideal gas 441.6.3. Discontinuities 461.6.4. Unsteady characteristics 471.6.5. Steady normal shockwave: Hugoniot and Prandtl relations 481.6.6. Flow in a de Laval nozzle 491.6.7. Simple wave 531.7. Examples of problems with viscous fluids 561.7.1. General equations 561.7.2. Incompressible viscous fluid  571.7.3. Flow of a compressible dissipative fluid: structure of a shockwave 611.8. Exercises 641.8.1. Exercises in kinematics (section 1.2) 641.8.2. Exercises in thermodynamics (section 1.3). 671.8.3. Exercises for the balance equations in fluid mechanics (section 1.4) 681.8.4. Examples of problems with 2D and 3D incompressible perfect fluids (section 1.5) 701.8.5. Examples of problems with a compressible perfect fluid (section 1.6) 741.8.6. Examples of problems with viscous fluids (section 1.7) 771.9. Solutions to the exercises 791.9.1. Solutions to the exercises in kinematics. 791.9.2. Solutions to the Exercises in thermodynamics 831.9.3. Solutions to the exercises for the balance of equations in fluid mechanics 881.9.4. Solutions to the examples of problems with 2D and 3D incompressible perfect fluids 891.9.5. Solutions to the examples of problems with a compressible perfect fluid 931.9.6. Solutions to the examples of problems with viscous fluids 95Chapter 2. Reactive Mixtures 1012.1. Introduction 1012.2. Equations of state 1032.2.1. Definition of the variables of state of a mixture 1032.2.2. Thermodynamic properties of mixtures 1082.2.3. Reactive mixture 1182.2.4. Other issues relating to the thermodynamics of mixtures 1232.3. Balance equations of flows of reactive mixtures 1242.3.1. Balance of mass of the species j and overall balance of mass 1242.3.2. General balance equation of a property F. 1272.3.3. Momentum balance 1292.3.4. Energy balance 1292.3.5. Balance relations in a discrete system. 1322.3.6. Entropy balance in a continuum 1372.3.7. Balance equations at discontinuities in continuous media 1402.4. Phenomena of transfer and chemical kinetics 1422.4.1. Introduction 1422.4.2. Presentation of the transfer coefficients by linear TIP 1432.4.3. Other presentations of the transfer coefficients 1472.4.4. Elements of chemical kinetics 1522.5. Couplings 1552.5.1. Heat transfer and diffusion 1552.5.2. Shvab-Zeldovich approximation 158Chapter 3. Interfaces and Lines 1633.1. Introduction 1633.1.1. Interfaces 1633.1.2. Lines 1653.2. Interfacial phenomena 1663.2.1. General aspects  1663.2.2. General form of an interfacial balance law 1683.2.3. Constitutive laws for interfaces whose variables directly satisfy the classical equations in thermostatics and in 2D-TIP 1733.2.4. Constitutive laws for interfaces deduced from classical thermostatics and 3D-TIP. Stretched flame example 1773.2.5. Interfaces manifesting resistance to folding 1793.2.6. Numerical modeling  1793.2.7. Interfaces and the second gradient theory. 1823.2.8. Boundary conditions of the interfaces 1853.2.9. Conclusion 1853.3. Solid and fluid curvilinear media: pipes, fluid lines and filaments 1863.3.1. General aspects 1863.3.2. Establishing the balance equations in a curvilinear medium. 1883.3.3. Simplified theories 2093.3.4. Triple line and second gradient theory 2163.3.5. Conclusion 2203.4. Exercises 2223.4.1. Exercises regarding solid curvilinear media 2223.4.2. Exercises regarding fluid curvilinear media 2223.5. Solutions to the exercises 2233.5.1. Solutions to exercises regarding solid curvilinear media. 2233.5.2. Solutions to the exercises regarding fluid curvilinear media 225APPENDICES 229Appendix 1. Tensors, Curvilinear Coordinates, Geometry and Kinematics of Interfaces and Lines 231A1.1. Tensor notations 231A1.1.1. Tensors and operations on tensors 231A1.2. Orthogonal curvilinear coordinates. 234A1.2.1. General aspects 234A1.2.2. Curl of a vector field  236A1.2.3. Divergence of a vector field 237A1.2.4. Gradient of a scalar 238A1.2.5. Laplacian of a scalar  238A1.2.6. Differentiation in a curvilinear basis 238A1.2.7. Divergence of a second order tensor 239A1.2.8. Gradient of a vector 239A1.2.9. Cylindrical coordinates and spherical coordinates 240A1.3. Interfacial layers  242A1.3.1. Prevailing directions of an interfacial medium 242A1.3.2. Operators of projection for interfaces 244A1.3.3. Surface gradients of a scalar field 245A1.3.4. Curvature vector of a curve 245A1.3.5. Normal and tangential divergences of a vector field 246A1.3.6. Extension of surface per unit length 246A1.3.7. Average normal curvature of a surface 247A1.3.8. Breakdown of the divergence of a vector field 248A1.3.9. Breakdown of the Laplacian of a scalar field 249A1.3.10. Breakdown of the divergence of a second order tensor 249A1.3.11. Projection operators with the intrinsic definition of a surface 252A1.3.12. Comparison between the two descriptions 253A1.4. Curvilinear zones 254A1.4.1. Presentation 254A1.4.2. Geometry of the orthogonal curvilinear coordinates 256A1.4.3. Projection operators and their consequences 257A1.5. Kinematics in orthogonal curvilinear coordinates 260A1.5.1. Kinematics of interfacial layers 260A1.5.2. Kinematics of curvilinear zones 266A1.5.3. Description of the center line 269Appendix 2. Additional Aspects of Thermostatics 277A2.1. Laws of state for real fluids with a single constituent 277A2.1.1. Diagram of state for a pure fluid 277A2.1.2. Approximate method to determine the thermodynamic functions 278A2.1.3. Van der Waals fluid  279A2.1.4. Other laws for dense gases and liquids 279A2.2. Mixtures of real fluids  280A2.2.1. Mixture laws for a real mixture 280A2.2.2. Expression of the free energy of a real mixture 281Appendix 3. Tables for Calculating Flows of Ideal Gas ƒ× ƒ­1.4 283A3.1. Calculating the parameters in continuous steady flow (section 1.6.6.2)  286A3.2. Formulae for steady normal shockwaves 288Appendix 4. Extended Irreversible Thermodynamics. 289A4.1. Heat balance equations in a non-deformable medium in EIT 290A4.2. Application to a 1D case of heat transfer 293A4.3. Application to heat transfer with the evaporation of a droplet 296A4.3.1. Reminders about evaporating droplets 296A4.3.2. Evaporating droplet with EIT. 300A4.4. Application to thermal shock 302A4.4.1. Presentation of the problem and solution using CIT 302A4.4.2. Thermal shock and EIT 303A4.4.3. Application of the second order approximation into two examples of thermal shock 305A4.5. Outline of EIT  307A4.6. Applications and perspectives of EIT 310Appendix 5. Rational Thermodynamics 313A5.1. Introduction 313A5.2. Fundamental hypotheses and axioms 314A5.2.1. Basic hypotheses 314A5.2.2. Basic axioms  316A5.3. Constitutive laws  318A5.4. Case of the reactive mixture 320A5.4.1. Principle of material frame indifference 320A5.4.2. Constitutive laws for a reactive mixture 321A5.5. Critical remarks  324Appendix 6. Torsors and Distributors in Solid Mechanics 325A6.1. Introduction 325A6.1.1. Torsor 325A6.1.2. Distributor 325A6.1.3. Power  326A6.2. Derivatives of torsors and distributors which depend on a single position parameter 326A6.2.1. Derivative of the velocity distributor 327A6.2.2. Derivative of the tensor of forces 328A6.3. Derivatives of torsors and distributors dependent on two positional parameters  328A6.3.1. Expression of the velocity distributor 329A6.3.2. Derivative of the velocity distributor 329Appendix 7. Virtual Powers in a Medium with a Single Constituent 331A7.1. Introduction 331A7.2. Virtual powers of a system of n material points 332A7.3. Virtual power law  333A7.4. The rigid body and systems of rigid bodies 333A7.4.1. The rigid body 333A7.4.2. System of rigid bodies, concept of a link 334A7.5. 3D deformable continuous medium 335A7.5.1. First gradient theory  335A7.5.2. A 3D case of perfect internal linkage: the incompressible perfect fluid  337A7.5.3. Second gradient theory 337A7.6. 1D continuous deformable medium 338A7.6.1. First gradient theory  338A7.6.2. A 1D case of perfect internal linkage: perfectly flexible and inextensible wires 340A7.7. 2D deformable continuous medium 340Bibliography 343Index 355