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Continuum Thermomechanics

Alfredo Bermúdez de Castro
Publisher: 
Birkhäuser
Publication Date: 
2005
Number of Pages: 
209
Format: 
Hardcover
Series: 
Progress in Mathematical Physics 43
Price: 
79.95
ISBN: 
3-7643-7265-6
Category: 
Monograph
We do not plan to review this book.

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

Dummy View - NOT TO BE DELETED