Applications of Vector Analysis and Complex Variables in Engineering

This textbook presents the application of mathematical methods and theorems to solve engineering problems, rather than focusing on mathematical proofs. Applications of Vector Analysis and Complex Variables in Engineering explains the mathematical principles in a manner suitable for engineering students, who generally think quite differently than students of mathematics. The objective is to emphasize mathematical methods and applications, rather than emphasizing general theorems and principles, for which the reader is referred to the literature. Vector analysis plays an important role in engineering, and is presented in terms of indicial notation, making use of the Einstein summation convention. This text differs from most texts in that symbolic vector notation is completely avoided, as suggested in the textbooks on tensor algebra and analysis written in German by Duschek and Hochreiner, in the 1960s. The defining properties of vector fields, the divergence and curl, are introduced in terms of fluid mechanics. The integral theorems of Gauss (the divergence theorem), Stokes, and Green are introduced also in the context of fluid mechanics. The final application of vector analysis consists of the introduction of non-Cartesian coordinate systems with straight axes, the formal definition of vectors and tensors. The stress and strain tensors are defined as an application. Partial differential equations of the first and second order are discussed. Two-dimensional linear partial differential equations of the second order are covered, emphasizing the three types of equation: hyperbolic, parabolic, and elliptic. The hyperbolic partial differential equations have two real characteristic directions, and writing the equations along these directions simplifies the solution process. The parabolic partial differential equations have two coinciding characteristics; this gives useful information regarding the character of the equation, but does not help in solving problems. The elliptic partial differential equations do not have real characteristics. In contrast to most texts, rather than abandoning the idea of using characteristics, here the complex characteristics are determined, and the differential equations are written along these characteristics. This leads to a generalized complex variable system, introduced by Wirtinger. The vector field is written in terms of a complex velocity, and the divergence and the curl of the vector field is written in complex form, reducing both equations to a single one. Complex variable methods are applied to elliptical problems in fluid mechanics, and linear elasticity. The techniques presented for solving parabolic problems are the Laplace transform and separation of variables, illustrated for problems of heat flow and soil mechanics. Hyperbolic problems of vibrating strings and bars, governed by the wave equation are solved by the method of characteristics as well as by Laplace transform. The method of characteristics for quasi-linear hyperbolic partial differential equations is illustrated for the case of a failing granular material, such as sand, underneath a strip footing. The Navier Stokes equations are derived and discussed in the final chapter as an illustration of a highly non-linear set of partial differential equations and the solutions are interpreted by illustrating the role of rotation (curl) in energy transfer of a fluid. Contents Preface Acknowledgemenents Chapter 1 Vectors in Three-Dimensional Space 1.1 Unit Vectors 1.2 Comparison with Symbolic Notation 1.3 The Dot Product 1.3.1 Projection of a Vector onto a Given Direction 1.4 The Cross Product 1.5 Free Indices and Dummy Indices 1.6 Base Vectors 1.7 The Kronecker Delta 1.8 The Levi-Civita Symbol and the Kronecker Delta Chapter 2 Vector Fields 2.1 Field Lines: Path Lines and Stream Lines 2.1.1 The Lagrangian Description 2.1.2 The Eulerian Description Chapter 3 Fundamental Equations for Fluid Mechanics 3.1 The Euler and Bernoulli Equations 3.2 The Mass Balance Equation 3.3 Rotational Flow 3.4 The Bernoulli Equation for Irrotational Flow 3.4.1 Conservation of Zero Rotation 3.4.2 Transient Flow of a Compressible Inviscid Fluid with Rotation 3.5 Objects Moving through the Fluid 3.5.1 Euler’s Equation for Moving Coordinates; Irrotational Flow 3.5.2 The Energy Head 3.6 Flow with Divergence and Rotation; the General Field 3.7 Irrotational and Divergence-Free Flow 3.8 Three-Dimensional Flow 3.8.1 The Point Sink 3.8.2 The Dipole 3.8.3 Higher-Order Singularities 3.8.4 Other Harmonic Functions 3.8.5 An Impermeable Sphere in a Field of Uniform Flow 3.8.6 The Line-Sink Chapter 4 Integral Theorems 4.1 Introduction 4.2 The Integral Theorem of Stokes 4.2.1 Stokes’s Integral Theorem Applied to a Vector Potential 4.3 The Divergence Theorem 4.4 Green’s Integral Theorem 4.5 Boundary Integral Equations Chapter 5 Coordinate Transformations: Definitions of Vectors and Tensors 5.1 Coordinate Transformation 5.2 Definition of Vectors and Tensors 5.2.1 Tensors of the Second Rank 5.2.2 The Gradient Tensor 5.2.3 Principal Directions 5.3 Non-Cartesian Transformations with Straight Axes 5.3.1 Invariants 5.3.2 The Gradient 5.4 The Stress Tensor 5.4.1 The Normal and Shear Stress Components 5.4.2 Principal Directions and Principal Values of the Stress Tensor in Two Dimensions. 5.4.3 Stress Deviator 5.4.4 Invariants 5.4.5 The Equilibrium Conditions 5.5 The Displacement Gradient and Strain Tensors 5.5.1 The Displacement Gradient Tensor 5.5.2 The Strain Tensor and the Rotation Tensor 5.6 Integrability; The Compatibility Conditions 5.6.1 The Gradient of a Scalar 5.6.2 The Gradient of a Vector 5.7 The Velocity Gradient Tensor Chapter 6 Partial Differential Equations of the First Order 6.1 The General Differential Equation of the First Order 6.1.1 Advective Contaminant Transport Chapter 7 Partial Differential Equations of the Second Order 7.1 Types of Partial Differential Equations 7.2 Two Real Characteristics 7.3 Two Coinciding Characteristics 7.4 Complex Characteristic Directions Chapter 8 The Elliptic Case: Two Complex Characteristics 8.1 Boundary-Value Problems 8.2 The Helmholtz Decomposition Theorem 8.3 Vectors and Tensors in Complex Space 8.3.1 The Bases in Complex Space 8.3.2 Transformation of Vectors 8.3.3 Transformation of Tensors 8.3.4 A Combined Kronecker Delta and Alternating Tensor: The Combination Matrix 8.3.5 The Potential 8.3.6 The Stream Function 8.3.7 The General Vector Field 8.3.8 Helmholtz’s Decomposition Theorem 8.3.9 Areal Integration 8.3.10 Integral Theorems 8.4 Irrotational and Divergence-Free Vector Fields 8.4.1 Summary 8.5 Functions of a Single Complex Variable 8.6 Basic Complex Functions 8.6.1 The Taylor Series 8.6.2 The Asymptotic Expansion 8.6.3 The Laurent Series 8.7 Cauchy Integrals Chapter 9 Applications of Complex Variables 9.1 Flow of an in-viscid fluid 9.1.1 Uniform Flow 9.1.2 The Point Sink 9.1.3 The Vortex 9.1.4 The Dipole 9.2 Superposition of Elementary Solutions 9.2.1 Flow Around an Impermeable Cylinder 9.2.2 A Moving Cylinder 9.2.3 A Rotating Stationary Cylinder in a Moving Fluid 9.2.4 Flow with Many Cylindrical Impermeable Objects 9.2.5 Two-Dimensional Horizontal Flow toward a Sink 9.2.6 Rotational Flow with a Forced Vortex 9.3 Conformal Mapping 9.3.1 Transforming an Ellipse into a Circle 9.3.2 An Elliptical Impermeable Object in Uniform Flow 9.4 Groundwater flow 9.4.1 Horizontal Confined Flow 9.5 Linear Elasticity 9.5.1 Basic Equations 9.5.2 General Solution 9.5.3 Stresses and Strains 9.5.4 Tractions and Displacements Acting on Arbitrary Planes 9.5.5 Stresses Acting On a Half Plane 9.5.6 A Half-Space Loaded Along a Section 9.5.7 Flamant’s Problem 9.5.8 An Analytic Element for Gravity 9.5.9 A Pressurized Crack 9.5.10 An Analytic Element for Gravity in a Half-Space Chapter 10 The Parabolic Case: Two Coinciding Characteristics 10.1 The Diffusion Equation 10.1.1 Characteristics 10.2 Conduction of Heat in Solids 10.3 Transient Groundwater Flow 10.4 Consolidation of Soils 10.5 The Laplace Transform 10.5.1 The Unit Step Function 10.5.2 The Dirac Delta Function 10.6 Applications of the Laplace transform 10.6.1 One-Dimensional Consolidation 10.6.2 Transient Groundwater Flow 10.7 Separation of Variables 10.7.1 Response to a Sinusoidal Tidal Fluctuation 10.7.2 Initial Condition Chapter 11 The Hyperbolic Case: Two Real Characteristics 11.1 Longitudinal Vibration in a Bar 11.2 Transverse Vibration in a String 11.3 The Differential Equation Along the Characteristics 11.4 Initial Value ProblemWith Discontinuous Shape 11.5 Initial Value ProblemWith Smooth Shape 11.5.1 Solution Using Characteristics 11.5.2 Solution Using the Laplace Transform Chapter 12 Hyperbolic Quasi Linear Partial Differential Equations 12.1 Quasi Linear Partial Differential Equations 12.2 Granular Soils at Impending Failure 12.2.1 Conditions for Limit Equilibrium 12.3 Differential equations for impending failure of granular media 12.3.1 Examination of the Type of the Differential Equations 12.3.2 The Characteristics 12.3.3 Characteristic Directions 12.3.4 Equations Along the Characteristics 12.3.5 Application: Prandtl’s Wedge 12.3.6 Logarithmic Spirals 12.3.7 Width of the Failure Zones Next to the Strip Load Chapter 13 The Navier-Stokes Equations 13.1 Energy Transfer in a Fluid 13.1.1 Turbulence 13.2 The Bernouilli Equation with Energy Losses; A Simplified Form of the Energy Equation Appendices Appendix A Numerical Integration of the Cauchy Integral Appendix B List of Problems with Page Numbers Bibliography Index