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.
Author(s): Meinhard T. Schobeiri
Publisher: Springer
Year: 2021
Language: English
Pages: 260
City: Cham
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
