Preface xi
1 General Definitions, Conservation Laws 1
1.1 Motion of a Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Balance of Linear and AngularMomentum . . . . . . . . . . . . . 5
1.4 Balance of Energy. First Principle of Thermodynamics . . . . . . . 7
1.5 Second Principle of Thermodynamics. The Clausius-Duhem
Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Lagrangian Coordinates 13
2.1 The Piola-Kirchhoff Stress Tensors . . . . . . . . . . . . . . . . . . 13
2.2 The Conservation Equations in Lagrangian Coordinates . . . . . . 14
3 Constitutive Laws 17
3.1 Thermodynamic Process. Material Body . . . . . . . . . . . . . . . 17
3.2 Coleman-Noll Materials . . . . . . . . . . . . . . . . . . . . . . . . 18
4 The Principle of Material Frame-Indifference 27
4.1 Change in the Observer. The Indifference Principle . . . . . . . . . 27
4.2 Consequences for Coleman-Noll Materials . . . . . . . . . . . . . . 28
5 Replacing Entropy with Temperature 33
5.1 The Conservation Equations in Terms of Temperature . . . . . . . 33
6 Isotropy 37
6.1 The Extended Symmetry Group . . . . . . . . . . . . . . . . . . . 37
6.2 Isotropic Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7 Equations in Lagrangian Coordinates 43
viii Contents
8 Linearized Models 47
8.1 Linear Approximation of the Motion Equation . . . . . . . . . . . 47
8.2 Linear Approximation of the Energy Equation . . . . . . . . . . . 52
8.3 Isotropic Linear Thermoviscoelasticity . . . . . . . . . . . . . . . . 54
9 Quasi-static Thermoelasticity 57
9.1 Statement of the Equations . . . . . . . . . . . . . . . . . . . . . . 57
9.2 Time Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 57
9.3 A Particular Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
10 Fluids 61
10.1 The Concept of Fluid, First Properties . . . . . . . . . . . . . . . . 61
10.2 Motion Equation. Thermodynamic Pressure . . . . . . . . . . . . . 63
10.3 Energy Equation, Enthalpy . . . . . . . . . . . . . . . . . . . . . . 64
10.4 Thermodynamic Coefficients and Equalities . . . . . . . . . . . . . 66
10.5 Gibbs Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
10.6 Statics of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
10.7 The Boussinesq Approximation, Natural Convection . . . . . . . . 78
11 Linearized Models for Fluids, Acoustics 81
11.1 General Equations, Dissipative Acoustics . . . . . . . . . . . . . . 81
11.2 The Isentropic Case, Non-Dissipative Acoustics . . . . . . . . . . . 85
11.3 Linearized Models under Gravity . . . . . . . . . . . . . . . . . . . 87
12 Perfect Gases 93
12.1 Definition, General Properties . . . . . . . . . . . . . . . . . . . . . 93
12.2 Entropy and Free Energy . . . . . . . . . . . . . . . . . . . . . . . 94
12.3 The Compressible Navier-Stokes Equations . . . . . . . . . . . . . 97
12.4 The Compressible Euler Equations . . . . . . . . . . . . . . . . . . 98
13 Incompressible Fluids 101
13.1 Isochoric Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
13.2 Newtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
13.3 Ideal Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
14 Turbulent Flow of Incompressible Newtonian Fluids 105
14.1 Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
14.2 The k − Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
15 Mixtures of Coleman-Noll Fluids 109
15.1 General Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
15.2 Mixture of Perfect Gases . . . . . . . . . . . . . . . . . . . . . . . . 113
Contents ix
16 Chemical Reactions in a Stirred Tank 119
16.1 Chemical Kinetics. The Mass Action Law . . . . . . . . . . . . . . 119
16.2 Conservation of Chemical Elements . . . . . . . . . . . . . . . . . . 122
16.3 Reacting Mixture of Perfect Gases . . . . . . . . . . . . . . . . . . 123
17 Chemical Equilibrium of a Reacting Mixture of Perfect Gases
in a Stirred Tank 125
17.1 The Least Action Principle for the Gibbs Free Energy . . . . . . . 125
17.2 Equilibrium for a Set of Reversible Reactions,
Equilibrium Constants . . . . . . . . . . . . . . . . . . . . . . . . . 126
17.3 The Stoichiometric Method . . . . . . . . . . . . . . . . . . . . . . 131
18 Flow of a Mixture of Reacting Perfect Gases 135
18.1 Mass Conservation Equations . . . . . . . . . . . . . . . . . . . . . 135
18.2 Motion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
18.3 Energy Conservation Equation . . . . . . . . . . . . . . . . . . . . 137
18.4 Conservation of Elements . . . . . . . . . . . . . . . . . . . . . . . 140
18.5 Equilibrium Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . 141
18.6 The Case of Low Mach Number . . . . . . . . . . . . . . . . . . . . 142
19 The Method of Mixture Fractions 145
19.1 General Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
19.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
19.3 The Adiabatic Case . . . . . . . . . . . . . . . . . . . . . . . . . . 149
19.4 The Case of Equilibrium Chemistry . . . . . . . . . . . . . . . . . 149
20 Turbulent Flow of Reacting Mixtures of Perfect Gases,
The PDF Method 153
20.1 Elements of Probability . . . . . . . . . . . . . . . . . . . . . . . . 153
20.2 The Mixture Fraction/PDF Method . . . . . . . . . . . . . . . . . 155
A Vector and Tensor Algebra 161
A.1 Vector Space. Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
A.2 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
A.3 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
A.4 The Affine Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
B Vector and Tensor Analysis 173
B.1 Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . 173
B.2 Curves and Curvilinear Integrals . . . . . . . . . . . . . . . . . . . 175
B.3 Gauss’ and Green’s Formulas. Stokes’ Theorem . . . . . . . . . . . 177
B.4 Change of Variable in Integrals . . . . . . . . . . . . . . . . . . . . 178
B.5 Transport Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 178
B.6 Localization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 179
x Contents
B.7 Differential Operators in Coordinates . . . . . . . . . . . . . . . . . 179
B.7.1 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . 179
B.7.2 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . 182
B.7.3 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . 184
C Some Equations in Curvilinear Coordinates 189
C.1 Mass Conservation Equation . . . . . . . . . . . . . . . . . . . . . 189
C.2 Motion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
C.3 Constitutive Law for Newtonian Viscous Fluids in Cooordinates . 191
D ALE Formulations of the Conservation Equations 195
D.1 ALE Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
D.2 Conservative ALE Form of Conservation Equations . . . . . . . . . 197
D.2.1 Mixed Conservative ALE Form of the Conservation
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
D.3 Mixed Nonconservative Form of ALE Conservation Equations . . . 200
Bibliography 203
Index 205