Tensor Analysis for Engineers and Physicists - With Application to Continuum Mechanics, Turbulence, and Einstein’s Special and General Theory of Relativity

Description

This book unies the common tensor analytical aspects in engineering and physics. Using tensor analysis enables the reader to understand complex physical phenomena from the basic principles in continuum mechanics including the turbulence, its correlations and modeling to the complex Einstein' tensor equation. The development of General Theory of Relativity and the introduction of spacetime geometry would not have been possible without the use of tensor analysis. This textbook is primarily aimed at students of mechanical, electrical, aerospace, civil and other engineering disciplines as well as of theoretical physics. It also covers the special needs of practicing professionals who perform CFD-simulation on a routine basis and would like to know more about the underlying physics of the commercial codes they use. Furthermore, it is suitable for self-study, provided that the reader has a sufficient knowledge of differential and integral calculus. Particular attention was paid to selecting the application examples. The transformation of Cartesian coordinate system into curvilinear one and the subsequent applications to conservation laws of continuum mechanics and the turbulence physics prepares the reader for fully understanding the Einstein tensor equations, which exhibits one of the most complex tensor equation in theoretical physics. Preface Contents Nomenclature Symbols Greek Symbols, Operators Subscripts, Superscripts Operators 1 Vectors and Tensors 1.1 Introduction 1.1.1 Space, Euclidean Space, Curved Space 1.1.2 Cartesian Tensors in Three-Dimensional Euclidean Space 1.1.3 Physical Quantities, Order of Tensors 1.1.4 Index Notation 1.2 Vector Operations 1.2.1 Scalar Product 1.2.2 Vector or Cross Product 1.2.3 Tensor Product 1.2.4 Symmetric, Antisymmetric Behavior of Tensors 1.3 Contraction of Tensors 1.3.1 Contraction of a Second Order Tensor 1.3.2 Trace of a Second Order Tensor 1.3.3 Product of Two Second Order Tensors 1.3.4 Contraction of Higher Order Tensors 1.4 Decomposition of Second Order Tensors 1.5 Inverse of a Tensor References 2 Transformation of Tensors 2.1 Transformation of a First Order Tensor 2.1.1 Coordinate Transformation in an Absolute Frame of Reference 2.1.2 Transformation from an Absolute Frame into a Relative Frame 2.2 Transformation of a Second and Higher Order Tensor 2.3 Eigenvalue and Eigenvector of a Second Order Tensor 3 Differential Operators in Continuum Mechanics 3.1 Substantial Derivatives 3.1.1 Differential Operator 3.1.2 Transformation of Nabla Operator 3.1.3 Transformation of Gradient of a Scalar Function 3.1.4 Laplace Operator Δ 3.2 Operator Applied to Different Functions 3.2.1 Scalar Product of and a Vector 3.2.2 Transformation of Divergence of a Vector Function 3.2.3 Vector Product timesV 3.2.4 Tensor Product of and V 3.2.5 Scalar Product of and a Second Order Tensor 3.2.6 Stress Vector 3.2.7 Mohr Circle References 4 Tensors and Kinematics 4.1 Material and Spatial Description 4.1.1 Material Description 4.1.2 Spatial Description 4.1.3 Jacobian Transformation Function 4.2 Reynolds Transport Theorem 4.3 Translation, Deformation, Rotation 4.4 Strain Tensor References 5 Differential Balances in Continuum Mechanics 5.1 Mass Flow Balance in Stationary Frame of Reference 5.1.1 Incompressibility Condition 5.2 Momentum Balance in Stationary Frame 5.2.1 Relationship Between Stress Tensor and Deformation Tensor 5.2.2 Navier–Stokes Equation of Motion in Stationary Frame 5.2.3 Special Case: Euler Equation of Motion 5.3 Mass Flow Balance in Rotating Frame 5.4 Momentum Balance in Rotating Frame 5.4.1 Navier–Stokes Equation of Motion in Rotating Frame 5.5 Some Discussions on Navier–Stokes Equations 5.6 Energy Balance in Stationary Frame of Reference 5.6.1 Mechanical Energy 5.6.2 Thermal Energy Balance 5.6.3 Total Energy 5.6.4 Entropy Balance References 6 Tensor Operations in Orthogonal Curvilinear Coordinate Systems 6.1 Change of Coordinate System 6.2 Co- and Contravariant Base Vectors, Metric Coefficients 6.2.1 Transformation of Base Vectors 6.2.2 Transformation of Components 6.2.3 Metric Coefficients, Jacobian Determinants 6.2.4 Vectors, Components and Scalar Product 6.3 Relation Between, Contravariant and Covariant Base Vectors 6.3.1 Vector Product of Base Vectors in Curvilinear Coordinate System 6.3.2 Raising and Lowering Indices 6.4 Physical Components of Tensors 6.4.1 Physical Components of First Order Tensors 6.4.2 Physical Components of a Second Order Tensor 6.4.3 Derivatives of Base Vectors, Christoffel Symbols 6.4.4 Spatial Derivatives in Curvilinear Coordinate System 6.5 Application of to Tensor Functions 6.5.1 Scalar Product of and a First Order Tensor 6.5.2 Scalar Product of and a Second Order Tensor 6.5.3 Vector Product of and a First Order Tensor 6.5.4 Tensor Product of and a First Order Tensor 6.6 Covariant Derivative 6.6.1 Covariant Derivative of a First Order Tensor 6.6.2 Covariant Derivative of a Second Order Tensor 6.7 Application Example 1: Inviscid Incompressible Flow Motion 6.7.1 Equation of Motion in Curvilinear Coordinate Systems 6.7.2 Special Case: Cylindrical Coordinate System 6.7.3 Base Vectors, Metric Coefficients 6.7.4 Christoffel Symbols 6.7.5 Introduction of Physical Components 6.8 Application Example 2: Viscous Flow Motion 6.8.1 Equation of Motion for Viscous Flow 6.8.2 Special Case: Cylindrical Coordinate System References 7 Tensor Application, Navier–Stokes Equation 7.1 Steady Viscous Flow Through a Curved Channel 7.1.1 Description of the Curved Channel Geometry 7.1.2 Case I: Solution of the Navier–Stokes Equation 7.1.3 Case I: Curved Channel, Negative Pressure Gradient 7.1.4 Case I: Curved Channel, Positive Pressure Gradient 7.1.5 Case II: Radial Flow, Positive Pressure Gradient 7.2 Temperature Distribution 7.2.1 Case I: Solution of Energy Equation 7.2.2 Case I: Curved Channel, Negative Pressure Gradient 7.2.3 Case I: Curved Channel, Positive Pressure Gradient 7.2.4 Case II: Radial Flow, Positive Pressure Gradient References 8 Curves, Curvature, Surfaces, Geodesics 8.1 Representation of the Plane Curves 8.2 Curvature of Plane Curves 8.2.1 Curvature of Plane Curves, Derivation 8.2.2 Derivation of Curvature and Torsion Using Tangent Vectors 8.3 Space Curves, Torsion, Curvature 8.3.1 Calculation of Curvature of Space Curves 8.3.2 Calculation of Torsion τ 8.4 Surfaces 8.4.1 Description of a Surface 8.4.2 Dimensions of a Surface 8.4.3 Non-uniqueness of Parametric Representation 8.5 Fundamental Forms of the Surface Theory 8.5.1 Surface Metric Tensor 8.5.2 First Fundamental Form of Surface, Arc Length 8.5.3 Surface Area 8.5.4 Second Fundamental Form, Curvature Tensor 8.6 Geodesics 8.6.1 Introduction 8.6.2 Calculation of Geodesics for N-dimensional Riemann Space 8.6.3 Calculation of Geodesics for 2-Dimensional Surfaces References 9 Turbulent Flow, Modeling 9.1 Fundamentals of Turbulent Flows 9.2 Role of Tensors in Describing Turbulence 9.2.1 Correlations, Length and Time Scales 9.2.2 Spectral Representation of Turbulent Flows 9.2.3 Spectral Tensor, Energy Spectral Function 9.3 Averaging Fundamental Equations of Turbulent Flow 9.3.1 Averaging Conservation Equations 9.3.2 Equation of Turbulence Kinetic Energy 9.3.3 Equation of Dissipation of Kinetic Energy 9.4 Turbulence Modeling 9.4.1 Two-Equation Models 9.5 Grid Turbulence References 10 Special Theory of Relativity 10.1 Introduction 10.2 Frames, Coordinate Systems, Lorenz, Transformation, Events 10.2.1 Definition of an Event 10.2.2 Lorentz Transformation 10.2.3 Consequences of Lorenz Transformation 10.3 Relativistic Length: Length Contraction 10.3.1 Relativistic Time: Time Dilation 10.3.2 Relativistic Mass: Mass Increase 10.4 Einstein's Equivalence of Energy and Mass 10.5 Einstein's Four Vectors in Spacetime Coordinate 10.5.1 Distance and Position Vectors in Spacetime Coordinate 10.5.2 Four Velocity and Momentum Vector in Spacetime Coordinate 10.6 Divergence of the Energy Stress Tensor 10.6.1 Divergence of Energy Component 10.6.2 Divergence of Momentum Component References 11 Tensors in General Theory of Relativity 11.1 Operator Commutator 11.2 Parallel Transport 11.2.1 Parallel Transport, Riemann Tensor 11.2.2 Properties of the Riemann Tensor 11.3 Construction of Einstein Space-Time Geometry 11.3.1 Ricci Tensor and Curvature Scalar 11.4 Einstein Tensor, Field Equation of General Relativity 11.5 Newton' Gravitation as the Special Case of the GTR 11.6 An Appendix from Chap.6摥映數爠eflinkchap666 11.6.1 Covariant Derivative of Second Order Tensors References Index