Continuum Thermomechanics

"The purpose of this textbook is to give a modern and unified approach to continuum mechanics and thermodynamics in a rigorous mathematical framework. The intended audience are students, researchers and engineers interested in these fields or in mathematical modelling and numerical simulation in solid and fluid mechanics, acoustics, heat transfer, chemistry, or combustion."--BOOK JACKET.

Preface

xi

General Definitions, Conservation Laws

1

(12)

Motion of a Body

1

(3)

Conservation of Mass

4

(1)

Balance of Linear and Angular Momentum

5

(2)

Balance of Energy. First Principle of Thermodynamics

7

(4)

Second Principle of Thermodynamics. The Clausius-Duhem Inequality

11

(2)

Lagrangian Coordinates

13

(4)

The Piola-Kirchhoff Stress Tensors

13

(1)

The Conservation Equations in Lagrangian Coordinates

14

(3)

Constitutive Laws

17

(10)

Thermodynamic Process. Material Body

17

(1)

Coleman-Noll Materials

18

(9)

The Principle of Material Frame-Indifference

27

(6)

Change in the Observer. The Indifference Principle

27

(1)

Consequences for Coleman-Noll Materials

28

(5)

Replacing Entropy with Temperature

33

(4)

The Conservation Equations in Terms of Temperature

33

(4)

Isotropy

37

(6)

The Extended Symmetry Group

37

(2)

Isotropic Bodies

39

(4)

Equations in Lagrangian Coordinates

43

(4)

Linearized Models

47

(10)

Linear Approximation of the Motion Equation

47

(5)

Linear Approximation of the Energy Equation

52

(2)

Isotropic Linear Thermoviscoelasticity

54

(3)

Quasi-static Thermoelasticity

57

(4)

Statement of the Equations

57

(1)

Time Discretization

57

(2)

A Particular Case

59

(2)

Fluids

61

(20)

The Concept of Fluid, First Properties

61

(2)

Motion Equation. Thermodynamic Pressure

63

(1)

Energy Equation, Enthalpy

64

(2)

Thermodynamic Coefficients and Equalities

66

(11)

Gibbs Free Energy

77

(1)

Statics of Fluids

77

(1)

The Boussinesq Approximation, Natural Convection

78

(3)

Linearized Models for Fluids, Acoustics

81

(12)

General Equations, Dissipative Acoustics

81

(4)

The Isentropic Case, Non-Dissipative Acoustics

85

(2)

Linearized Models under Gravity

87

(6)

Perfect Gases

93

(8)

Definition, General Properties

93

(1)

Entropy and Free Energy

94

(3)

The Compressible Navier-Stokes Equations

97

(1)

The Compressible Euler Equations

98

(3)

Incompressible Fluids

101

(4)

Isochoric Processes

101

(1)

Newtonian Fluids

101

(2)

Ideal Fluids

103

(2)

Turbulent Flow of Incompressible Newtonian Fluids

105

(4)

Turbulence Models

105

(2)

The k -- ε Model

107

(2)

Mixtures of Coleman-Noll Fluids

109

(10)

General Definitions

109

(4)

Mixture of Perfect Gases

113

(6)

Chemical Reactions in a Stirred Tank

119

(6)

Chemical Kinetics. The Mass Action Law

119

(3)

Conservation of Chemical Elements

122

(1)

Reacting Mixture of Perfect Gases

123

(2)

Chemical Equilibrium of a Reacting Mixture of Perfect Gases in a Stirred Tank

125

(10)

The Least Action Principle for the Gibbs Free Energy

125

(1)

Equilibrium for a Set of Reversible Reactions, Equilibrium Constants

126

(5)

The Stoichiometric Method

131

(4)

Flow of a Mixture of Reacting Perfect Gases

135

(10)

Mass Conservation Equations

135

(2)

Motion Equation

137

(1)

Energy Conservation Equation

137

(3)

Conservation of Elements

140

(1)

Equilibrium Chemistry

141

(1)

The Case of Low Mach Number

142

(3)

The Method of Mixture Fractions

145

(8)

General Equations

145

(3)

Examples

148

(1)

The Adiabatic Case

149

(1)

The Case of Equilibrium Chemistry

149

(4)

Turbulent Flow of Reacting Mixtures of Perfect Gases, The PDF Method

153

(8)

Elements of Probability

153

(2)

The Mixture Fraction/PDF Method

155

(6)

A Vector and Tensor Algebra

161

(12)

Vector Space. Basis

161

(2)

Inner Product

163

(1)

Tensors

164

(6)

The Affine Space

170

(3)

B Vector and Tensor Analysis

173

(16)

Differential Operators

173

(2)

Curves and Curvilinear Integrals

175

(2)

Gauss' and Green's Formulas. Stokes' Theorem

177

(1)

Change of Variable in Integrals

178

(1)

Transport Theorems

178

(1)

Localization Theorem

179

(1)

Differential Operators in Coordinates

179

(10)

Cartesian Coordinates

179

(3)

Cylindrical Coordinates

182

(2)

Spherical Coordinates

184

(5)

C Some Equations of Continuum Mechanics in Curvilinear Coordinates

189

(6)

Mass Conservation Equation

189

(1)

Motion Equation

190

(1)

Constitutive Law for Newtonian Viscous Fluids in Cooordinates

191

(4)

D Arbitrary Lagrangian-Eulerian (ALE) Formulations of the Conservation Equations

195

(8)

ALE Configuration

195

(2)

Conservative ALE Form of Conservation Equations

197

(3)

Mixed Conservative ALE Form of the Conservation Equations

199

(1)

Mixed Nonconservative Form of ALE Conservation Equations

200

(3)

Bibliography

203

(2)

Index

205

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Continuum Thermomechanics

Summary

Table of Contents

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