Vector Fields with Applications to Thermodynamics and Irreversibility

Description

Cover Half Title Series Page Title Page Copyright Page Dedication Table of Contents List of Notes, Tables, and Diagrams Series Preface Preface to Volume V Relationship with Previous Volumes of the Series Objectives and Contents of Volume V Organization and Presentation of the Subject matter About the Authors Acknowledgments List of Mathematical Symbols 1 Sets, Quantifiers and Logic 1.1 Sets 1.2 Quantifiers 1.3 Logic 1.4 Constants 2 Numbers, Ordering and Operations 2.1 Types of Numbers 2.2 Complex Numbers 2.3 Relations and Ordering 2.4 Operations between Numbers 3 Functions, Limits and Convergence 3.1 Limits and Values 3.2 Iterated Sums and Products 3.3 Convergence 3.4 Integrals 4 Vector, Matrices and Tensors 4.1 Vectors 4.2 Matrices 4.3 Tensors 4.4 Operations 5 Derivatives and Differential Operators 5.1 Differentials and Derivatives 5.2 Vector Operators 5.3 Tensor Operators 5.4 Adjointness 6 Functional Spaces 7 Geometries with N Dimensions 8 Generalized Functions 9 Auxiliary Functions List of Physical Quantities 1 Suffixes 2 Small Arabic Letters 3 Capital Arabic Letters 4 Small Greek Letters 5 Capital Greek Letters and Others Chapter 1: Classes of Equations and Similarity Solutions 1.1 Hierarchy of Partial Differential Equations 1.1.1 Single Equations and Simultaneous Systems 1.1.2 Partial Differential Equation in N Variables 1.1.3 One Dependent and Two Independent Variables 1.2 General Integral and Arbitrary Functions 1.2.1 First-Order P.D.E. and One Arbitrary Function 1.2.2 P.D.E. of Order N and N Arbitrary Functions 1.3 Unforced P.D.E. with First-Order Derivatives 1.3.1 Classification of P.D.E.S of First Order 1.3.2 Family of Hypersurfaces Tangent to a Vector Field 1.3.3 Characteristic Curve Tangent to a Vector Field 1.3.4 Characteristic Variables as Solutions of the Characteristic Equations 1.3.5 Plane Curve Tangent to a Vector Field 1.3.6 Example of the Family of Surfaces Tangent to the Position Vector 1.4 Quasi-Linear and Forced First-Order P.D.E.s 1.4.1 Quasi-Linear P.D.E. of the First-Order 1.4.2 Solution as Implicit Function of N Variables 1.4.3 Linear Forced P.D.E. with First-Order Derivatives 1.4.4 Plane Curves with Unit Projection on the Position Vector 1.5 Differentials of First-Degree in Three Variables 1.5.1 Exact, Inexact and Non-Integrable Differentials 1.5.2 Linear and Angular Velocities and Helicity 1.5.3 Exact Differential and Immediate Integrability 1.5.4 Inexact Differential with an Integrating Factor 1.5.5 Inexact Differential without Integrating Factor 1.5.6 Irrotational/Rotational and Non-Helical/Helical Vector Fields 1.5.7 Irrotational or Conservative Vector Field as the Gradient of a Scalar Potential 1.5.8 Rotational Non-Helical Vector Field and Two Scalar Potentials 1.5.9 Helical Vector Field and Three Scalar Potentials 1.5.10 Existence and Determination of the Three Scalar Potentials 1.5.11 Scalar Poisson Equation Forced by the Divergence of a Vector Field 1.5.12 Vector Poisson Equation Forced by the Curl of a Vector Field 1.5.13 Potential Vector Field and Laplace Equation 1.5.14 Scalar and Vector Potentials for a General Vector Field 1.5.15 Three Alternative Scalar Euler or Clebsch Potentials 1.5.16 Eight Cases of Three-Dimensional Vector Fields 1.5.17 Potential Vector Field with Scalar or Vector Potentials 1.5.18 Irrotational Vector Field and One Scalar Potential 1.5.19 Solenoidal Vector Field and Vector Potential 1.5.20 Non-Solenoidal, Rotational Vector Field and Scalar and Vector Potentials 1.5.21 Rotational, Non-Helical Vector Field and Two Euler Scalar Potentials 1.5.22 Helical Vector Field and Three Scalar Euler Potentials 1.5.23 Rotational, Non-Solenoidal Vector Field and Three Scalar Clebsch Potentials 1.5.24 Partial Differential Equations and Characteristic Systems 1.5.25 Unicity of Scalar and Vector Potentials 1.5.26 Characteristic Systems for Euler and Clebsch Potentials 1.6 P.D.E.s with Constant Coefficients and All Derivatives of Same Order 1.6.1 P.D.E. of Constant Order and Characteristic Polynomial 1.6.2 Similarity Solutions for a Linear Combination of Variables 1.6.3 General Integral for Distinct Roots 1.6.4 Single or multiple Roots of the Characteristic Polynomial 1.6.5 Method of Variation of Parameters 1.6.6 Method of Parametric Differentiation 1.6.7 General Integral for Multiple Roots 1.6.8 Linear P.D.E. with Constant Coefficients and Second-Order Derivatives 1.7 Harmonic and Biharmonic Functions on The Plane 1.7.1 Laplace Equation in Cartesian Coordinates 1.7.2 Real/Complex Harmonic Functions and Boundary Conditions 1.7.3 Biharmonic Equation in the Cartesian Plane 1.7.4 Real/Complex Biharmonic Functions and Forcing 1.8 Forced Linear P.D.E. with of Derivatives Constant Order 1.8.1 Forcing of P.D.E. by a Similarity Function 1.8.2 Similarity Forcing in the Non-Resonant Case 1.8.3 Parametric Differentiation for Multiple Resonance 1.8.4 Multiple Resonance via L’Hôspital Rule 1.8.5 Comparison of Resonant and Non-Resonant Cases 1.8.6 Forcing by an Exponential Similarity Function 1.8.7 Forcing by a Sinusoidal Similarity Function 1.8.8 General, Particular and Complete Integrals 1.8.9 Complete Integrals with Simple/Double Resonances 1.9 Forced Harmonic and Biharmonic Equations 1.9.1 General Forcing of the Laplace Equation 1.9.2 Complete Integral of the Harmonic Equation 1.9.3 General Forcing of the Double Laplace Equation 1.9.4 Complete Integral of the Biharmonic Equation 1.9.5 Method of Similarity Variables for Arbitrary Forcing 1.9.6 Fourth-Order Equation with Two Independent and Three Similarity Variables 1.9.7 Two Methods for Exponential Forcing 1.10 Conclusion Chapter 2: Thermodynamics and Irreversibility 2.1 Work, Heat, Entropy and Temperature 2.1.1 Work of Conservative and Non-Conservative Forces 2.1.2 Internal Energy and First Principle of Thermodynamics 2.1.3 Inertia Force and Kinetic Energy 2.1.4 Mass and Gravity Field, Potential, Force and Energy 2.1.5 Electric Charge, Field, Scalar Potential and Force 2.1.6 Electric Displacement, Work and Energy 2.1.7 Dielectric Permittivity Tensor and Scalar 2.1.8 Work in Terms of the Electric Field and Displacement 2.1.9 Electric Current and Magnetic Vector Potential 2.1.10 Magnetic Field, Induction and Energy 2.1.11 Magnetic Permeability Tensor and Scalar 2.1.12 Permutation Symbol, Curl and Outer Vector Product 2.1.13 Anisotropic Inhomogeneous Poisson Equation 2.1.14 Work in Terms of the Magnetic Field and Induction 2.1.15 Work of the Pressure in a Volume Change 2.1.16 Volume Forces Associated with Surface Pressure or Stresses 2.1.17 Work of the Surface Stresses in a Displacement 2.1.18 Balance of Moment of Forces and Symmetric Stress Tensor 2.1.19 Displacement and Strain Tensors and Rotation Vector 2.1.20 Mole and Avogadro Numbers and Chemical Work, Potential and Affinity 2.1.21 Matter with Two Phases and a Combustion Reaction 2.1.22 Total Work and Augmented Internal Energy: The Total Work ( 2.96) 2.1.23 Energy Balance and Modified Internal Energy 2.1.24 Extensive and Intensive Thermodynamic Parameters 2.1.25 Thermal, Electrical, Magnetic, Mechanical and Chemical Equilibrium 2.1.26 Temperature, Entropy, Heat and Internal Energy 2.2 Functions of State and Constitutive Properties 2.2.1 Duality Transformation of First-Order Differentials (Legendre) 2.2.2 Free Energy (Helmholtz) and Enthalpy (Gibbs) 2.2.3 Free Enthalpy as a Fourth Function of State 2.2.4 First-Order Derivatives and Conjugate Thermodynamic Variables 2.2.5 Second-Order Derivatives and Constitutive Properties 2.2.6 Specific Heat, Dielectric Permittivity, Magnetic Permeability and Elastic Stiffness 2.2.7 Pyroelectric/Magnetic Vectors, Electromagnetic Coupling, Thermoelastic, and Piezoelectric/Magnetic Tensors 2.2.8 Maximum Number of Constitutive Coefficients for Anisotropic Matter 2.2.9 Constitutive Properties of Anisotropic and Isotropic Matter 2.2.10 Absence of Electromagnetic Coupling in Isotropic Matter 2.2.11 Absence of Electro/Magnetoelastic Interaction in Isotropic Matter 2.2.12 Lamé Moduli, Young Modulus and Poisson Ratio 2.2.13 Constitutive Relations for Isotropic Matter 2.2.14 Basic Thermodynamic System with Thermomechanical Coupling 2.2.15 Non-Linear, Anisotropic, Inhomogeneous and Unsteady Matter 2.2.16 Twenty-Four Cases of Constitutive Relations 2.2.17 Analogies among Mechanics, Electricity, Magnetism and Elasticity 2.2.18 Inequalities for the Isotropic Constitutive Coefficients 2.2.19 Vector and Tensor Quadratic Forms for Energies 2.2.20 Definite, Semi-Definite and Indefinite Quadratic Forms 2.2.21 Eigenvalues, Eigenvectors, Diagonalization and Sum of Squares (Sylvester) 2.2.22 Classification of Quadratic Forms by the Eigenvalues of the Matrix 2.2.23 Conditions for Positive Electric, Magnetic and Elastic Energies 2.2.24 Biaxial, Uniaxial and Isotropic Materials 2.2.25 Principal Submatrices and Subdeterminants of a Square Matrix 2.2.26 Subdeterminants and Positive/Negative Definiteness 2.2.27 Necessary and Sufficient Conditions for Positive/Negative Definiteness 2.2.28 Inequalities for Constitutive Tensors Ensuring Positive Energies 2.2.29 Indefinite Second-Order Differential of the Free Energy 2.3 Three Principles and Four Processes of Thermodynamics 2.3.1 Complete and Basic Thermodynamic System 2.3.2 Five Functions of State 2.3.3 Cases When Work and Heat Are Functions of State 2.3.4 Adiabatic Process and Pressure Equilibrium 2.3.5 Isochoric Process and Thermal Equilibrium 2.3.6 Work and Heat in an Isobaric Process 2.3.7 Heat and Work in an Isothermal Process 2.3.8 Four Thermodynamic Variables and Four Relations between Derivatives (Maxwell 1867) 2.3.9 Skew-Symmetry, Inversion and Product Properties of Jacobians 2.3.10 Specific Heats at Constant Volume and Pressure 2.3.11 Coefficients of Thermal Expansion and Isothermal Compression 2.3.12 Adiabatic and Isothermal Sound Speeds 2.3.13 Non-Adiabatic Pressure and Volume Coefficients 2.3.14 Twelve Non-Inverse Thermodynamic Derivatives 2.3.15 Three Independent Thermodynamic Derivatives 2.3.16 The Second Principle of Thermodynamics and Entropy Growth 2.3.17 Paths on the Convex Thermodynamic Surface 2.3.18 Stable Equilibrium and Minimum Internal Energy 2.3.19 Inequalities for Thermodynamic Derivatives 2.3.20 First and Second-Order Derivatives of Entropy 2.3.21 Thermodynamic Derivatives at Constant Internal Energy 2.3.22 Thermodynamic Stability and Maximum Entropy 2.3.23 Inexistence of Extremals for Enthalpy and Free Energy and Maximum for Free Enthalpy 2.3.24 Fluid Transfer Between Full and Empty Reservoirs 2.3.25 Transfer at Constant Pressure through a Porous Wall (Joule–Thomson 1882) 2.3.26 Body Reducing Environmental Disturbances (Le Chatelier 1898) 2.3.27 Body in Pressure/Thermal Equilibrium with Environment 2.3.28 Thermodynamic Properties at Zero Absolute Temperature 2.3.29 Third Principle of Thermodynamics (Nerst 1907) 2.3.30 System with a Variable Number of Particles 2.3.31 Differentials of Chemical Potentials (Gibbs 1876–1878, Duhem 1886) 2.4 Entropy Production and Diffusive Properties 2.4.1 Heat Conduction and Flux in a Domain 2.4.2 Thermal Conductivity (Fourier 1818) Scalar and Tensor 2.4.3 Electromagnetic Energy Density, Flux and Dissipation 2.4.4 Joule (1847) Effect, Ohm (1827) Law and Electrical Conductivity/Resistivity 2.4.5 Coupling of Heat Flux and Electric Current 2.4.6 Gradients, Fluxes and Reciprocity (Onsager 1931) 2.4.7 Isotropic Thermoelectric Diffusion Scalars 2.4.8 Anisotropic Thermoelectric Diffusion Tensors 2.4.9 Stresses and Rates-of-Strain for a Viscous Fluid 2.4.10 Shear and Bulk Viscosities for a Newtonian Fluid 2.4.11 Positive Viscosities in the Navier (1822) – Stokes (1845) Equation 2.4.12 Decoupling of Viscosity from Thermal and Electrical Conduction 2.4.13 Decoupling of Thermoelectric Conduction from the Pressure Gradient 2.4.14 Electric Current in Linear and Non-Linear Media 2.4.15 Ohm (1827) and Hall (1879) Effects and Ionic Propulsion 2.4.16 Thomson (1851) Thermoelectric Effect Including Convection 2.4.17 Volta (1821) – Seeback (1822) – Peltier (1834) Effect and Thermoelectromotive Force 2.4.18 Mass Diffusion (Fick 1855) in a Two-Phase Medium 2.4.19 Entropy Production by Mass Diffusion 2.4.20 Coupled Thermoelectric Diffusion Equations 2.4.21 Anisothermal Dissipative Piezoelectromagnetism (Campos, Silva & Moleiro 2020) 2.4.22 Isotropic Media and Steady Fields 2.4.23 Thermoelastic Electromagnetism in a Slab References Bibliography 1. Generic Encyclopedias 2. Partial Differential Equations Index