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......Page 6 Preface......Page 11 Acknowledgemenents......Page 12 Chapter 1 Vectors in Three-Dimensional Space......Page 13 1.2 Comparison with Symbolic Notation......Page 14 1.3.1 Projection of a Vector onto a Given Direction......Page 15 1.4 The Cross Product......Page 16 1.7 The Kronecker Delta......Page 19 1.8 The Levi-Civita Symbol and the Kronecker Delta......Page 20 2.1.1 The Lagrangian Description......Page 22 2.1.2 The Eulerian Description......Page 24 3.1 The Euler and Bernoulli Equations......Page 30 3.2 The Mass Balance Equation......Page 33 3.3 Rotational Flow......Page 34 3.4 The Bernoulli Equation for Irrotational Flow......Page 35 3.4.2 Transient Flow of a Compressible Inviscid Fluid with Rotation......Page 36 3.5 Objects Moving through the Fluid......Page 38 3.5.1 Euler’s Equation for Moving Coordinates; Irrotational Flow......Page 40 3.5.2 The Energy Head......Page 41 3.6 Flow with Divergence and Rotation; the General Field......Page 43 3.8.1 The Point Sink......Page 45 3.8.2 The Dipole......Page 47 3.8.5 An Impermeable Sphere in a Field of Uniform Flow......Page 49 3.8.6 The Line-Sink......Page 52 4.2 The Integral Theorem of Stokes......Page 57 4.4 Green’s Integral Theorem......Page 60 4.5 Boundary Integral Equations......Page 61 5.1 Coordinate Transformation......Page 63 5.2.1 Tensors of the Second Rank......Page 65 5.2.3 Principal Directions......Page 66 5.3 Non-Cartesian Transformations with Straight Axes......Page 67 5.3.2 The Gradient......Page 71 5.4 The Stress Tensor......Page 72 5.4.2 Principal Directions and Principal Values of the Stress Tensor in Two Dimensions.......Page 73 5.4.4 Invariants......Page 75 5.4.5 The Equilibrium Conditions......Page 76 5.5 The Displacement Gradient and Strain Tensors......Page 77 5.5.2 The Strain Tensor and the Rotation Tensor......Page 78 5.6 Integrability; The Compatibility Conditions......Page 80 5.6.2 The Gradient of a Vector......Page 81 5.7 The Velocity Gradient Tensor......Page 83 6.1 The General Differential Equation of the First Order......Page 84 6.1.1 Advective Contaminant Transport......Page 85 7.1 Types of Partial Differential Equations......Page 89 7.2 Two Real Characteristics......Page 92 7.3 Two Coinciding Characteristics......Page 95 7.4 Complex Characteristic Directions......Page 96 Chapter 8 The Elliptic Case: Two Complex Characteristics......Page 97 8.2 The Helmholtz Decomposition Theorem......Page 101 8.3.1 The Bases in Complex Space......Page 102 8.3.2 Transformation of Vectors......Page 103 8.3.3 Transformation of Tensors......Page 104 8.3.4 A Combined Kronecker Delta and Alternating Tensor: The Combination Matrix......Page 105 8.3.6 The Stream Function......Page 109 8.3.8 Helmholtz’s Decomposition Theorem......Page 110 8.3.10 Integral Theorems......Page 111 8.4 Irrotational and Divergence-Free Vector Fields......Page 114 8.5 Functions of a Single Complex Variable......Page 116 8.6.1 The Taylor Series......Page 117 8.7 Cauchy Integrals......Page 118 9.1.1 Uniform Flow......Page 122 9.1.2 The Point Sink......Page 123 9.1.3 The Vortex......Page 124 9.1.4 The Dipole......Page 125 9.2.1 Flow Around an Impermeable Cylinder......Page 126 9.2.2 A Moving Cylinder......Page 127 9.2.3 A Rotating Stationary Cylinder in a Moving Fluid......Page 130 9.2.4 Flow with Many Cylindrical Impermeable Objects......Page 131 9.2.5 Two-Dimensional Horizontal Flow toward a Sink......Page 134 9.2.6 Rotational Flow with a Forced Vortex......Page 136 9.3.1 Transforming an Ellipse into a Circle......Page 140 9.3.2 An Elliptical Impermeable Object in Uniform Flow......Page 143 9.4 Groundwater flow......Page 144 9.4.1 Horizontal Confined Flow......Page 145 9.5.1 Basic Equations......Page 147 9.5.2 General Solution......Page 151 9.5.3 Stresses and Strains......Page 152 9.5.4 Tractions and Displacements Acting on Arbitrary Planes......Page 153 9.5.5 Stresses Acting On a Half Plane......Page 154 9.5.6 A Half-Space Loaded Along a Section......Page 158 9.5.7 Flamant’s Problem......Page 160 9.5.9 A Pressurized Crack......Page 164 9.5.10 An Analytic Element for Gravity in a Half-Space......Page 169 10.1 The Diffusion Equation......Page 172 10.2 Conduction of Heat in Solids......Page 173 10.4 Consolidation of Soils......Page 174 10.5 The Laplace Transform......Page 176 10.5.2 The Dirac Delta Function......Page 177 10.6.1 One-Dimensional Consolidation......Page 178 10.6.2 Transient Groundwater Flow......Page 184 10.7.1 Response to a Sinusoidal Tidal Fluctuation......Page 187 10.7.2 Initial Condition......Page 190 11.1 Longitudinal Vibration in a Bar......Page 191 11.3 The Differential Equation Along the Characteristics......Page 192 11.4 Initial Value ProblemWith Discontinuous Shape......Page 194 11.5.1 Solution Using Characteristics......Page 198 11.5.2 Solution Using the Laplace Transform......Page 200 12.2.1 Conditions for Limit Equilibrium......Page 203 12.3 Differential equations for impending failure of granular media......Page 204 12.3.1 Examination of the Type of the Differential Equations......Page 205 12.3.2 The Characteristics......Page 206 12.3.3 Characteristic Directions......Page 207 12.3.4 Equations Along the Characteristics......Page 208 12.3.5 Application: Prandtl’s Wedge......Page 210 12.3.7 Width of the Failure Zones Next to the Strip Load......Page 212 Chapter 13 The Navier-Stokes Equations......Page 215 13.1.1 Turbulence......Page 217 13.2 The Bernouilli Equation with Energy Losses; A Simplified Form of the Energy Equation......Page 218 Appendices......Page 219 Appendix A Numerical Integration of the Cauchy Integral......Page 220 Appendix B List of Problems with Page Numbers......Page 222 Bibliography......Page 224 Index......Page 226