Warsi Z.U.A. Fluid dynamics: theoretical and computational approaches
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Warsi Z.U.A. Fluid dynamics: theoretical and computational approaches / Warsi Z.U.A. - 3nd ed. - Boca Raton: CRC Press/Taylor & Francis, 2006. - 845 p. - ISBN 0-8493-3397-0.
 
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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


Вверх Warsi Z.U.A. Fluid dynamics: theoretical and computational approaches / Warsi Z.U.A. - 3nd ed. - Boca Raton: CRC Press/Taylor & Francis, 2006. - 845 p. - ISBN 0-8493-3397-0.

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