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 Kinematic 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
References ..................................................... 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 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 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 (ξ) ................. 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 ū2 f (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 Flows ............................ 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-Î) 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 Systems ...... 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 ................................. 812
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
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