Chapter 1 Kinematics of fluid motion ............................................. 1 1.1 Introduction to Continuum Motion ............................................ 1 1.2 Fluid Particles ............................................................. 1 1.3 Inertial Coordinate Frames .................................................. 2 1.4 Motion of a Continuum ....................................................... 2 1.5 The Time Derivatives ........................................................ 6 1.6 Velocity and Acceleration ................................................... 6 1.7 Steady and Nonsteady Flow .................................................. 10 1.8 Trajectories of Fluid Particles and Streamlines ............................ 11 1.9 Material Volume and Surface ................................................ 12 1.10 Relation Between Elemental Volumes ......................................... 13 1.11 Kinematics Formulas of Euler and Reynolds .................................. 13 1.12 Control Volume and Surface ................................................. 16 1.13 Kinematics of Deformation .................................................. 17 1.14 Kinematics of Vorticity and Circulation .................................... 22 Vortex Line ................................................................ 22 Vortex Tube ................................................................ 22 Circulation of Velocity .................................................... 24 Rate of Change of Circulation .............................................. 24 Reference ....................................................................... 25 Problems ........................................................................ 26 Chapter 2 The conservation laws and the kinetics of flow ........................ 33 2.1 Fluid Density and the Conservation of Mass ................................. 33 2.2 Principle of Mass Conservation ............................................. 33 Time Variation of ρP ....................................................... 34 Particular Forms of the Continuity Equation ................................ 35 2.3 Mass Conservation Using a Control Volume ................................... 35 2.4 Kinetics of Fluid Flow ..................................................... 36 Stress Principle of Cauchy ................................................. 36 2.5 Conservation of Linear and Angular Momentum ................................ 37 Conservation of Linear Momentum ............................................ 37 Conservation of Angular Momentum ........................................... 38 Nature of the Stress Vector ................................................ 38 Symmetry of T .............................................................. 41 2.6 Equations of Linear and Angular Momentum ................................... 42 2.7 Momentum Conservation Using a Control Volume ............................... 44 2.8 Conservation of Energy ..................................................... 44 2.9 Energy Conservation Using a Control Volume ................................. 47 2.10 General Conservation Principle ............................................. 47 2.11 The Closure Problem ........................................................ 48 2.12 Stokes' Law of Friction .................................................... 51 The Postulates of Stokes ................................................... 52 Stokesian Stress Tensor .................................................... 52 2.13 The Interpretation of Pressure ............................................. 57 2.14 The Dissipation Function ................................................... 58 2.15 Constitutive Equation for Non-Newtonian Fluids ............................. 59 2.16 Thermodynamic Aspects of Pressure and Viscosity ............................ 61 Ideal Gases ................................................................ 62 Concept of Viscosity in Fluids ............................................. 64 Sutherland Formula for Viscosity ........................................... 66 2.17 Equations of Motion in Lagrangian Coordinates .............................. 67 References ...................................................................... 71 Problems ........................................................................ 71 Chapter 3 The Navier-Stokes Equations ........................................... 75 3.1 Formulation of the Problem ................................................. 75 3.2 Viscous Compressible Flow Equations ........................................ 78 Conservation of Mass ....................................................... 78 Conservation of Momentum ................................................... 78 Equations of Mechanical Energy ............................................. 78 Equations of Internal Energy ............................................... 78 Equations of Entropy and Enthalpy .......................................... 79 Conservation of Total Kinetic Energy ....................................... 80 3.3 Viscous Incompressible Flow Equations ...................................... 80 Conservation of Mass ....................................................... 80 Conservation of Momentum ................................................... 80 Equation of Vorticity ...................................................... 81 Equation of Internal Energy ................................................ 82 Equation for Pressure ...................................................... 82 3.4 Equations of Inviscid Flow (Euler's Equations) ............................. 83 Conservation of Mass ....................................................... 83 Conservation of Momentum ................................................... 84 Equations of Entropy and Enthalpy .......................................... 84 Conservation of Energy ..................................................... 84 Conservation of Total Kinetic Energy ....................................... 84 Inviscid Barotropic Flow ................................................... 84 3.5 Initial and Boundary Conditions ............................................ 85 3.6 Mathematical Nature of the Equations ....................................... 86 3.7 Vorticity and Circulation .................................................. 86 Vorticity and Circulation for Inviscid Fluids .............................. 87 The Bernoulli Equation ................................................. 89 3.8 Some Results Based on the Equations of Motion .............................. 90 Force Acting on a Solid Body ............................................... 90 Stress Vector and Tensor at a Surface ...................................... 91 Vorticity Vector at a Surface .............................................. 92 Rate-of-Strain Tensor at a Surface ......................................... 93 3.9 Nondimensional Parameters in Fluid Motion .................................. 94 Principle of Similarity .................................................... 97 Dynamic Similarity ......................................................... 97 Variable Nondimensional Parameters ..................................... 97 Principle of Reynolds Number Similarity ................................ 98 3.10 Coordinate Transformation ................................................. 99 Orthogonal Coordinates .................................................... 100 Navier-Stokes Equations in Orthogonal Coordinates ......................... 105 Nonorthogonal Curvilinear Coordinates ..................................... 107 Steady Eulerian Coordinates ........................................... 107 Nonsteady Eulerian Coordinates ........................................ 110 Equations in General Coordinates .......................................... 115 Equations in General Coordinates Using Contravariant Components ....... 117 Equations in General Coordinates Using Covariant Components ........... 117 Equations in General Coordinates with Vectors and Tensor Densities .... 118 Equations in Nonsteady Eulerian Coordinates ............................... 120 Equations in Curvilinear Coordinates with Cartesian Velocity Components ... 124 3.11 Streamlines and Stream Surfaces ........................................... 125 Two-Dimensional Stream Function ........................................... 125 Stream Functions in Three Dimensions ...................................... 127 3.12 Navier-Stokes Equations in Stream Function Form ........................... 129 Two-Dimensional and Axially Symmetric Flows ............................... 129 Flows in Three Dimensions ................................................. 130 Profile Drag .............................................................. 131 Free Surface Problem Formulation .......................................... 139 Kinematic Conditions .................................................. 139 Dynamic Conditions .................................................... 144 References ..................................................................... 146 Problems ....................................................................... 146 Chapter 4 Flow of Inviscid Fluids .............................................. 161 4.1 Introduction .............................................................. 161 Part I: Inviscid Incompressible Flow ...................................... 162 4.2 The Bernoulli Constant .................................................... 162 4.3 Irrotational Flows ........................................................ 163 Boundary Conditions ....................................................... 164 Irrotational Flows in Two Dimensions ...................................... 165 Examples of Analytic Functions for Inviscid Flows ......................... 167 Blasius Formulas for Force and Moment ..................................... 173 4.4 Method of Conformal Mapping in Inviscid Flows ............................. 176 Kutta-Joukowskii Transformation ........................................... 178 Pure Circulatory Motion around a Plate ................................ 180 Flow Past a Wing Profile .............................................. 181 An Iterative Method for the Numerical Generation of z = f(ζ) .............. 184 4.5 Sources, Sinks, and Doublets in Three Dimensions .......................... 185 Sources and Sinks in Three Dimensions ..................................... 187 Doublets in Three Dimensions .............................................. 188 Induced Velocities Due to Line and Sheet Vortices ......................... 189 Part II: Inviscid Compressible Flow ....................................... 191 4.6 Basic Thermodynamics ...................................................... 191 First Law of Thermodynamics ............................................... 192 Second Law of Thermodynamics .............................................. 194 Deductions from the Two Thermodynamic Laws ................................ 196 Specific Heats ............................................................ 198 Enthalpy .................................................................. 199 Maxwell Equations ......................................................... 200 Isentropic State .......................................................... 202 Speed of Sound ............................................................ 202 Thermodynamic Relations for an Ideal Gas .................................. 203 Perfect Gases ............................................................. 204 4.7 Subsonic and Supersonic Flow .............................................. 205 4.8 Critical and Stagnation Quantities ........................................ 207 4.9 Isentropic Ideal Gas Relations ............................................ 208 4.10 Unsteady Inviscid Compressible Flow in One-dimension ...................... 210 4.11 Steady Plane Flow of Inviscid Gases ....................................... 219 Stream Function Formulation ............................................... 219 Irrotational Flow of an Inviscid Gas ...................................... 221 Case of Small Perturbations ............................................... 222 Subsonic Flow ............................................................. 223 Supersonic Flow ........................................................... 224 4.12 Theory of Shock Waves ..................................................... 228 Shock Relations for an Arbitrarily Moving Shock ........................... 229 First Shock Condition ................................................. 230 Second Shock Condition ................................................ 230 Third Shock Condition ................................................. 231 Fourth Shock Condition ................................................ 231 Shock Surface, Slip Surface, and Contact Discontinuity .................... 233 Energy Equation for a Shock Surface ....................................... 233 Hugonoit Equation ......................................................... 233 Summary of All Shock Relations ............................................ 234 Case I: Shock Relations Without Using an Equation of State ............ 234 Case II: Shock Relations While Using an Equation of State ............. 235 The Role of Entropy ....................................................... 236 Stationary Shocks ......................................................... 238 Stationary Normal Shock ............................................... 238 Stationary Oblique Shocks ............................................. 238 Prandtl's Relation ........................................................ 240 Shock Polar for Stationary Oblique Shocks ................................. 242 References ..................................................................... 243 Problems ....................................................................... 243 Chapter 5 Laminar Viscous Flow ................................................. 263 Part I: Exact Solutions ................................................... 263 5.1 Introduction .............................................................. 263 5.2 Exact Solutions ........................................................... 264 Flow on an Infinite Plate ................................................. 264 Flow Between Two Infinite Parallel Plates ................................. 264 Flow Between Rotating Coaxial Cylinders (Circular Couette Flow) ........... 266 Steady Flow through a Cylindrical Pipe (Hagen-Poiseuille Flow) ............ 267 Flow in the Entrance Region of a Circular Pipe ............................ 270 Nonsteady Unidirectional Flow ............................................. 271 Stokes Problems ........................................................... 272 Ekman Layer Problem ....................................................... 274 Motion Produced Due to a Vortex Filament .................................. 276 Two-Dimensional Stagnation Point Flow (Hiemenz Flow) ...................... 278 Axially Symmetric Stagnation Point Flow (Homann Flow) ..................... 279 Motion between Two Inclined Plates ........................................ 280 5.3 Exact Solutions for Slow Motion ........................................... 284 Flow Past a Rigid Sphere .................................................. 285 Flow Past a Rigid Circular Cylinder ....................................... 289 Part II: Boundary Layers .................................................. 294 5.4 Introduction .............................................................. 294 5.5 Formulation of the Boundary Layer Problem ................................. 296 Method of Inner and Outer Limits .......................................... 301 5.6 Boundary Layer on 2-D Curved Surfaces ..................................... 302 Boundary Layer Parameters ................................................. 305 5.7 Separation of the 2-D Steady Boundary Layers .............................. 307 5.8 Transformed Boundary Layer Equations ...................................... 312 Similar Boundary Layers ................................................... 314 Boundary Layer on a Semi-Infinite Plate ................................... 316 Solution of the Blasius Equation .......................................... 316 Boundary Layer on a Wedge ................................................. 320 Numerical Solution of the Falkner-Skan Equation ........................... 322 Nonsimilar Boundary Layers ................................................ 324 Gortler's Series Solution ................................................. 325 5.9 Momentum Integral Equation ................................................ 330 Solution of the Momentum Integral Equation ................................ 332 Choice of the Velocity Profile ............................................ 335 5.10 Free Boundary Layers ...................................................... 336 Flow in the Wake of a Flat Plate .......................................... 337 Two-Dimensional Jet ....................................................... 338 Axially Symmetric Jet ..................................................... 340 5.11 Numerical Solution of the Boundary Layer Equation ......................... 342 Numerical Solution of the Diffusion Equation .............................. 342 Errors: Truncation and Round Off .......................................... 343 Crank and Nicolson ........................................................ 345 Dufort and Frankel ........................................................ 345 Three-Point Scheme ........................................................ 345 Solution of the Boundary Layer Equation ................................... 345 The Box Method ............................................................ 349 5.12 Three-Dimensional Boundary Layers ......................................... 352 The Metric Coefficients ................................................... 352 The Matching Conditions ................................................... 353 Equations in Rotating Coordinates ......................................... 357 Choice of Surface Coordinates ............................................. 358 Internal Cartesian Coordinates ............................................ 361 Nondevelopable Surfaces ................................................... 362 Physical Consequences of Three Dimensionality ............................. 363 Intrinsic Coordinates ..................................................... 363 Domains of Dependence and Influence ....................................... 365 5.13 Momentum Integral Equations in Three Dimensions ........................... 365 5.14 Separation and Attachment in Three Dimensions ............................. 366 Limiting Streamlines and Vortex Lines .................................... 368 5.15 Boundary Layers on Bodies of Revolution and Yawed Cylinders ............... 370 Mangler's Tranformation ................................................... 371 Boundary Layer on Yawed Cylinders ......................................... 373 Cross Flow ................................................................ 374 Transformed Equations for Yawed Cylinders ................................. 376 5.16 Three-Dimensional Stagnation Point Flow ................................... 376 5.17 Boundary Layer On Rotating Blades ......................................... 377 5.18 Numerical Solution of 3-D Boundary Layer Equations ........................ 378 5.19 Unsteady Boundary Layers .................................................. 380 Purely Unsteady Boundary Layers ........................................... 380 Periodic Boundary Layers .................................................. 383 Separation of Unsteady Boundary Layers .................................... 386 Mathematical Formulation of the M-R-S Principle ........................... 387 Numerical Method of Solution of Unsteady Equations ........................ 388 5.20 Second-Order Boundary Layer Theory ........................................ 389 Method of Matched Asymptotic Expansion .................................... 391 Outer Expansion ........................................................... 392 Some Important Derivatives at the Wall .................................... 395 Inner Expansion ........................................................... 396 The First- and Second-Order Boundary Layer Problems ....................... 397 Matching of Inner and Outer Solutions ..................................... 398 A Unified Second-Order-Correct Viscous Model .............................. 401 Matching .................................................................. 402 5.21 Inverse Problems in Boundary Layers ....................................... 404 Inverse Formulation with Assigned Displacement Thickness .................. 405 5.22 Formulation of the Compressible Boundary Layer Problem .................... 407 Estimation of the Viscous Terms ........................................... 409 External-Flow Equations and the Boundary Conditions ....................... 413 Particular Cases .......................................................... 413 Numerical Solution of Compressible Boundary Layer Equations ............... 414 Part III: Navier-Stokes Formulation ....................................... 418 5.23 Incompressible Flow ....................................................... 418 Formulation of the Problem in Primitive Variables ......................... 419 Ad Hoc Modifications ...................................................... 420 Formulation of the Problem in Vorticity/Potential Form .................... 421 Vorticity-Stream Function Formulation ..................................... 421 Vorticity-Potential Function Formulation .................................. 422 Integro-Differential Formulation .......................................... 424 Application of the Boundary Conditions .................................... 426 Basic Computational Aspects ............................................... 427 5.24 Compressible Flow ......................................................... 427 Determination of Temperature .............................................. 429 Case of Mr→0 .............................................................. 430 Numerical Formulation ..................................................... 431 5.25 Hyperbolic Equations and Conservation Laws ................................ 434 System of Quasi-linear Equations from the Conservation Equations .......... 442 Hyperbolic Equations in Higher Dimensions ................................. 447 5.26 Numerical Transformation and Grid Generation .............................. 448 Equations for Grid Generation ............................................. 449 Gaussian Equations for Grid Generation .................................... 450 5.27 Numerical Algorithms for Viscous Compressible Flows ....................... 451 Nature of the Difference Schemes .......................................... 456 Formulation for Compressible Navier-Stokes Equations ...................... 461 5.28 Thin-Layer Navier-Stokes Equations (TLNS) ................................. 466 Parabolized Navier-Stokes Equations (PNS) ................................. 466 References ..................................................................... 467 Problems ....................................................................... 470 Chapter 6 Turbulent Flow ....................................................... 489 Part I: Stability Theory and the Statistical Description of Turbulence .... 489 6.1 Introduction .............................................................. 489 6.2 Stability of Laminar Flows ................................................ 489 Formulation of the Problem ................................................ 490 6.3 Formulation for Plane-Parallel Laminar Flows .............................. 492 Squire's Theorem .......................................................... 495 Temporal and Spatial Instabilities ........................................ 496 Boundary Conditions for the Orr-Sommerfeld Equation ....................... 496 Temporal Stability ........................................................ 500 6.4 Temporal Stability at Infinite Reynolds Number ............................ 500 Rayleigh's First Theorem .................................................. 501 Rayleigh's Second Theorem ................................................. 501 6.5 Numerical Algorithm for the Orr-Sommerfeld Equation ....................... 505 6.6 Transition to Turbulence .................................................. 507 6.7 Statistical Methods in Turbulent Continuum Mechanics ...................... 509 Average or Mean of Turbulent Quantities ................................... 510 Time and Space Averaging .................................................. 510 Time Average .......................................................... 511 Ensemble Average ...................................................... 511 Space Average ......................................................... 513 Basic Axioms of Averaging ............................................. 515 6.8 Statistical Concepts ...................................................... 515 Probability Distribution Functions ........................................ 516 Probability Density ....................................................... 517 Mathematical Expectation .................................................. 518 Correlation Functions ..................................................... 519 Stationary Processes ...................................................... 519 Characteristic Functions .................................................. 519 Gaussian Distribution ..................................................... 521 6.9 Internal Structure in Physical Space ...................................... 522 Second- and Third-Order Correlations ...................................... 522 Dynamic Equation of Correlations .......................................... 524 Homogeneous Turbulence .................................................... 527 Homogeneous Shear Turbulence .............................................. 528 Isotropic Turbulence ...................................................... 528 Analysis of Isotropic Turbulence .......................................... 530 Longitudinal and Lateral Correlations ..................................... 532 Approximate Analysis ...................................................... 535 Dynamic Equation for Isotropic Turbulence ................................. 537 6.10 Internal Structure in the Wave-Number Space ............................... 538 Some General Definitions .................................................. 538 Dynamic Equation of Homogeneous Turbulence in k-Space ..................... 540 Analysis of Isotropic Turbulence in k-Space ............................... 542 Connection Between u2f(r, t) and E(k, t) .................................. 545 Formulation of 1-D Spectrum ............................................... 547 Taylor's Formulas ......................................................... 549 6.11 Theory of Universal Equilibrium ........................................... 550 Determination of E(k, t) Based on Kolmogorov's Hypothesis ................. 551 Transfer Theories ......................................................... 552 Heisenberg's Transfer Theory .......................................... 553 Pao's Transfer Theory ................................................. 555 Comparison of Taylor's and Kolmogorov's Dissipation Lengths ............... 556 Integral Length and Timescales ............................................ 558 Part II: Development of Averaged Equations ................................ 559 6.12 Introduction .............................................................. 559 6.13 Averaged Equations for Incompressible Flow ................................ 559 Equation of Turbulence Kinetic Energy ..................................... 562 Equation of Mean-Square Vorticity Fluctuations ............................ 565 Rate Equation for Reynolds Stresses ....................................... 567 Rate Equation for the Dissipation ......................................... 569 Physical Interpretation of the Terms ...................................... 569 Analysis of the Pressure-Strain Correlation ............................... 571 6.14 Averaged Equations for Compressible Flow .................................. 573 Equation of Turbulence Energy and the Reynolds Stresses ................... 577 Dissipation Function ...................................................... 578 6.15 Turbulent Boundary Layer Equations ........................................ 580 Equations in Rectangular Cartesian Coordinates ............................ 580 Two-Dimensional Equations ................................................. 583 Three-Dimensional Equations ............................................... 583 Equations in Orthogonal Curvilinear Coordinates ........................... 585 Part III: Basic Empirical and Boundary Layer Results in Turbulence ........ 586 6.16 The Closure Problem ....................................................... 586 6.17 Prandtl's Mixing-Length Hypothesis ........................................ 587 Turbulent Flow Near a Wall ................................................ 588 Experimental Determination of uτ .......................................... 592 Application of the Logarithmic Formula in Pipe Flow ....................... 592 Power Laws for the Velocity Distribution .................................. 594 Rough Pipes ............................................................... 595 6.18 Wall-Bound Turbulent Hows ................................................. 596 6.19 Analysis of Turbulent Boundary Layer Velocity Profiles .................... 605 Law of the Wall for Compressible Flow ..................................... 612 6.20 Momentum Integral Methods in Boundary Layers .............................. 613 Method of Truckenbrodt .................................................... 617 Method of Head ............................................................ 622 6.21 Differential Equation Methods in 2-D Boundary Layers ...................... 624 Zero-Equation Modeling in Boundary Layers ................................. 626 One-Equation Model of Glushko ............................................. 628 Part IV: Turbulence Modeling .............................................. 630 6.22 Generalization of Boussinesq's Hypothesis ................................. 630 Specification of the Length Scale ......................................... 632 6.23 Zero-Equation Modeling in Shear Layers .................................... 633 Thin Shear Layers ......................................................... 634 6.24 One-Equation Modeling ..................................................... 635 Choice of the Constants b1, b3, and b5 ..................................... 636 Modifications Due to the Explicit Effects of Viscosity .................... 638 6.25 Two-Equation (K - Ε) Modeling ............................................. 641 Modeling of the Dissipation Rate Equation ................................. 641 Modeling for Separated Flows .............................................. 643 6.26 Reynolds' Stress Equation Modeling ........................................ 643 Determination of the Constants c1 and c2 ................................... 646 Another Modeling of the Energy Equation ................................... 648 The Wall Boundary Conditions .............................................. 649 6.27 Application to 2-D Thin Shear Layers ...................................... 650 6.28 Algebraic Reynolds' Stress Closure ........................................ 652 6.29 Development of A Nonlinear Constitutive Equation .......................... 655 Extension to Compressible Flow ............................................ 657 Turbulence Energy Equation ............................................ 659 Assumptions To Be Justified ........................................... 661 Implicit Algebraic Stress Model ........................................... 661 Explicit Algebraic Stress Model ........................................... 662 The Dissipation Equation .............................................. 663 The Total Energy Equation ............................................. 664 Modeling of the Correlations in the Total Energy Equation ............. 664 6.30 Current Approaches to Nonlinear Modeling ................................. 665 6.31 Heuristic Modeling ........................................................ 669 6.32 Modeling for Compressible Flow ............................................ 671 Stokes' Law of Friction ................................................... 671 Complete Stress Tensor .................................................... 672 Heat Flux ................................................................. 672 Production of Turbulence Energy ........................................... 673 Model Equations ........................................................... 674 Justification of the Modeling Constants for Compressible Flow ............. 675 6.33 Three-Dimensional Boundary Layers ......................................... 676 Eddy Viscosity Approach to 3-D Boundary Layers ............................. 680 6.34 Illustrative Analysis of Instability ...................................... 682 Reynolds-Orr Equation ..................................................... 682 Choas and Lorenz Model .................................................... 684 6.35 Basic Formulation of Large Eddy Simulation ................................ 689 Filters ................................................................... 689 Filtered Navier-Stokes Equations .......................................... 693 Linear Model .............................................................. 697 Scale-Similarity Model .................................................... 698 Dynamic Modeling .......................................................... 699 Algebraic Model ........................................................... 701 Nonlinear Constitutive Equation ........................................... 702 References ..................................................................... 703 Problems ....................................................................... 706 Mathematical Exposition 1 Base Vectors and Various Representations ............. 721 1.1 Introduction .............................................................. 721 1.2 Representations in Rectangular Cartesian System ........................... 723 1.3 Scalars, Vectors, and Tensors ............................................. 723 1.4 Differential Operations On Tensors ........................................ 725 Gradient .................................................................. 725 Divergence ................................................................ 726 Curl ...................................................................... 727 1.5 Multiplication of A Tensor and A Vector ................................... 727 1.6 Scalar Multiplication of Two Tensors ...................................... 728 1.7 A Collection of Usable Formulas ........................................... 729 1.8 Taylor Expansion in Vector Form ........................................... 731 1.9 Principal Axes of a Tensor ................................................ 732 1.10 Transformation of T to the Principal Axes ................................. 734 1.11 Quadratic Form and the Eigenvalue Problem ................................. 735 1.12 Representation in Curvilinear Coordinates ................................. 736 Fundamental Metric Components ............................................. 739 Elemental Displacement Vector ............................................. 741 Differentiation of Base Vectors ........................................... 742 Gradient of a Vector ..................................................... 744 Divergence and Curl of a Vector ........................................... 745 Divergence of Second-Order Tensors ........................................ 747 1.13 Christoffel Symbols in Three Dimensions ................................... 748 Christoffel Symbols of the First Kind ..................................... 748 Christoffel Symbols of the Second Kind .................................... 749 1.14 Some Derivative Relations ................................................. 754 Normal Derivative of Functions ............................................ 755 Physical Components in Curvilinear Coordinates ............................ 756 1.15 Scalar and Double Dot Products of Two Tensors ............................. 756 Mathematical Exposition 2 Theorems of Gauss, Green, and Stokes ................. 759 2.1 Gauss' Theorem ............................................................ 759 2.2 Green's Theorem ........................................................... 760 2.3 Stokes' Theorem ........................................................... 760 Mathematical Exposition 3 Geometry of Space and Plane Curves ................... 763 3.1 Basic Theory of Curves .................................................... 763 Tangent Vector ............................................................ 763 Principal Normal .......................................................... 764 Binormal Vector ........................................................... 765 Serret-Frenet Equations ................................................... 765 Plane Curves .............................................................. 766 Mathematical Exposition 4 Formulas for Coordinate Transformation ............... 769 4.1 Introduction .............................................................. 769 4.2 Transformation Law for Scalars ............................................ 769 4.3 Transformation Laws for Vectors ........................................... 770 4.4 Transformation Laws for Tensors ........................................... 772 4.5 Transformation Laws for the Christoffel Symbols ........................... 775 4.6 Some Formulas in Cartesian and Curvilinear Coordinates .................... 775 Laplacian of an Absolute Scalar ........................................... 776 Mathematical Exposition 5 Potential Theory ..................................... 779 5.1 Introduction .............................................................. 779 5.2 Formulas of Green ......................................................... 779 Green's Formulas for Laplace Operator ..................................... 780 5.3 Potential Theory .......................................................... 781 Integral Representation ................................................... 781 The Delta Function ........................................................ 782 Integral Representation of the Delta Function ......................... 784 The Delta Function in Higher Dimensions ............................... 785 Delta Function and the Fundamental Solution of the Laplace Equation ... 785 The Dirichlet Problem for the Poisson Equation ............................ 786 Particular Solution of Poisson's Equation ............................. 787 5.4 General Representation of a Vector ........................................ 787 5.5 An Application of Green's First Formula ................................... 788 Mathematical Exposition 6 Singularities of the First-Order ODEs ................ 791 6.1 Introduction .............................................................. 791 6.2 Singularities and Their Classification .................................... 791 Mathematical Exposition 7 Geometry of Surfaces ................................. 795 7.1 Basic Definitions ......................................................... 795 7.2 Formulas of Gauss ......................................................... 795 Christoffel Symbols Based on Surface Coefficients ......................... 796 7.3 Formulas of Weingarten .................................................... 798 7.4 Equations of Gauss ........................................................ 799 7.5 Normal and Geodesic Curvatures ............................................ 799 Longitudinal and Transverse Curvatures .................................... 802 7.6 Grid Generation in Surfaces ............................................... 803 Mathematical Exposition 8 Finite Difference Approximation Applied to PDEs ...... 805 8.1 Introduction .............................................................. 805 8.2 Calculus of Finite Differences ............................................ 805 Methods of Interpolation .................................................. 808 Cubic Spline Functions .................................................... 809 8.3 Iterative Root Finding .................................................... 810 8.4 Numerical Integration ..................................................... 813 8.5 Finite Difference Approximations of Partial Derivatives ................... 813 First Derivatives ......................................................... 813 Second Derivatives ........................................................ 814 8.6 Finite Difference Approximation of Parabolic PDEs ......................... 814 Stable Schemes for Parabolic Equations .................................... 818 8.7 Finite Difference Approximation of Elliptic Equations ..................... 819 Mathematical Exposition 9 Frame Invariancy ..................................... 825 9.1 Introduction .............................................................. 825 9.2 Orthogonal Tensor ......................................................... 825 Time Differentiation ...................................................... 826 Change of Basis ........................................................... 827 9.3 Arbitrary Rectangular Frames of Reference ................................. 828 9.4 Check for Frame Invariancy ................................................ 829 9.5 Use of Q .................................................................. 830 References for the Mathematical Expositions .................................... 831 Index .......................................................................... 833