Hydromechanics: Theory and Fundamentals

Hydromechanics......Page 1
Dedication......Page 7
Contents......Page 9
Preface......Page 15
List of Symbols......Page 19
1.1 Goals and Methods of Continuum Mechanics......Page 31
1.2 The Main Hypotheses of Continuum Mechanics......Page 33
2.1 Dynamics of the Continuum in the Lagrangian Perspective......Page 35
2.3 Scalar and Vector Fields and Their Characteristics......Page 38
2.4 Theory of Strains......Page 43
2.5 The Tensor of Strain Velocities......Page 54
2.6 The Distribution of Velocities in an Infinitesimal Continuum Particle......Page 55
2.7 Properties of Vector Fields. Theorems of Stokes and Gauss......Page 60
3.1 Equation of Continuity......Page 69
3.2 Equations of Motion......Page 73
3.3 Equation of Motion for the Angular Momentum......Page 81
4.1 Ideal Fluid and Gas......Page 85
4.2 Linear Elastic Body and Linear Viscous Fluid......Page 88
4.3 Equations in Curvilinear Coordinates......Page 93
4.3.1 Equation of Continuity......Page 94
4.3.2 Equation of Motion......Page 95
4.3.4 Laplace Operator......Page 96
4.3.5 Complete System of Equations of Motion for a Viscous, Incompressible Medium in the Absence of Heating......Page 97
5.1 Theorem of the Living Forces......Page 99
5.2 Law of Conservation of Energy and First Law of Thermodynamics......Page 102
5.3 Thermodynamic Equilibrium, Reversible and Irreversible Processes......Page 106
5.4 Two Parameter Media and Ideal Gas......Page 107
5.5 The Second Law of Thermodynamics and the Concept of Entropy......Page 110
5.6 Thermodynamic Potentials of Two-Parameter Media......Page 113
5.7 Examples of Ideal and Viscous Media, and Their Thermodynamic Properties, Heat Conduction......Page 116
5.7.1 The Model of the Ideal, Incompressible Fluid......Page 117
5.7.2 The Model of the Ideal, Compressible Gas......Page 118
5.7.3 The Model of Viscous Fluid......Page 120
5.8 First and Second Law of Thermodynamics for a Finite Continuum Volume......Page 123
5.9 Generalized Thermodynamic Forces and Currents, Onsager's Reciprocity Relations......Page 124
6.1 Initial Conditions and Boundary Conditions......Page 127
6.2 Typical Simplifications for Some Problems......Page 131
6.3 Conditions on the Discontinuity Surfaces......Page 135
6.4 Discontinuity Surfaces in Ideal Compressible Media......Page 141
6.5 Dimensions of Physical Quantities......Page 148
6.6 Parameters that Determine the Class of the Phenomenon......Page 150
6.7 Similarity and Modeling of Phenomena......Page 157
7.1 Equilibrium Equations......Page 161
7.2 Equilibrium in the Gravitational Field......Page 162
7.3 Force and Moment that Act on a Body from the Surrounding Fluid......Page 163
7.4 Equilibrium of a Fluid Relative to a Moving System of Coordinates......Page 165
8.1 Bernoulli's Integral......Page 167
8.2 Examples of the Application of Bernoulli's Integral......Page 169
8.3 Dynamic and Hydrostatic Pressure......Page 171
8.4 Flow of an Incompressible Fluid in a Tube of Varying Cross Section......Page 172
8.5 The Phenomenon of Cavitation......Page 173
8.6 Bernoulli's Integral for Adiabatic Flows of an Ideal Gas......Page 174
8.7 Bernoulli's Integral for the Flow of a Compressible Gas......Page 177
9.1 Integral Relations......Page 181
9.2 Interaction of Fluids and Gases with Bodies Immersed in the Flow......Page 183
10 Potential Flows for Incompressible Fluids......Page 189
10.1 The Cauchy–Lagrange Integral......Page 190
10.2 Some Applications for the General Theory of Potential Flows......Page 191
10.3 Potential Movements for an Incompressible Fluid......Page 193
10.4 Movement of a Sphere in the Unlimited Volume of an Ideal, Incompressible Fluid......Page 201
10.5 Kinematic Problem of the Movement of a Solid Body in the Unlimited Volume of an Incompressible Fluid......Page 206
10.6 Energy, Movement Parameters and Moments of Movement Parameters for a Fluid during the Movement of a Solid Body in the Fluid......Page 207
11.1 Method of Complex Variables......Page 211
11.2 Examples of Potential Flows in the Plane......Page 213
11.3 Application of the Method of Conformal Mapping to the Solution of Potential Flows around a Body......Page 222
11.4 Examples of the Application of the Method of Conformal Mapping......Page 225
11.5 Main Moment and Main Vector of the Pressure Force Exerted on a Hydrofoil Profile......Page 229
12.1 Movement of an Ideal Gas Under Small Perturbations......Page 233
12.2 Propagation of Waves with Finite Amplitude......Page 237
12.3 Plane Vortex-Free Flow of an Ideal Compressible Gas......Page 241
12.4 Subsonic Flow around a Thin Profile......Page 245
12.5 Supersonic Flow around a Thin Profile......Page 246
13.1 Rheological Laws of the Viscous Incompressible Fluid......Page 249
13.2 Equations of the Newtonian Viscous Fluid and Similarity Numbers......Page 251
13.3 Integral Formulation for the Effect of Viscous Fluids on a Moving Body......Page 253
13.4 Stationary Flow of a Viscous Incompressible Fluid in a Tube......Page 256
13.5 Oscillating Laminar Flow of a Viscous Fluid through a Tube......Page 261
13.6 Simplification of the Navier–Stokes Equations......Page 263
14.1 General Properties of Stokes Flows......Page 267
14.2 Flow of a Viscous Fluid around a Sphere......Page 270
14.3 Creeping Spatial Flow of a Viscous Incompressible Fluid......Page 277
15.1 Equation of Motion for the Fluid in the Boundary Layer......Page 281
15.2 Asymptotic Boundary Layer on a Plate......Page 285
15.3 Problem of the Injected Beam......Page 287
16.1 General Information on Laminar and Turbulent Flows......Page 293
16.2 Momentum Equation of a Viscous Incompressible Fluid......Page 294
16.3 Equations of Heat Inflow, Heat Conduction and Diffusion......Page 297
16.4 The Condition for the Beginning of Turbulence......Page 299
16.5 Hydrodynamic Instability......Page 300
16.6 The Reynolds Equations......Page 302
16.7 The Equation of Turbulent Energy Balance......Page 307
16.8 Isotropic Turbulence......Page 311
16.9 The Local Structure of Fully Developed Turbulence......Page 321
16.10 Models of Turbulent Flow......Page 331
16.10.1 Semi-empirical Theories of Turbulence......Page 332
16.10.2 The Use of Transport Equations......Page 338
References......Page 342
Appendix A Foundations of Vectorial and Tensorial Analysis......Page 345
A.1 Vectors......Page 346
A.2 Tensors......Page 355
A.3 Curvilinear Systems of Coordinates and Physical Components......Page 368
A.4 Calculation of Lengths, Surface Areas and Volumes......Page 371
A.5 Differential Operators and Integral Theorems......Page 374
B.1 Curves on a Plane......Page 379
B.2 Vectorial Definition of Curves......Page 380
B.3 Curvature of a Curve in the Plane......Page 383
B.4 Curves in Space......Page 385
B.5 Curvature of Spatial Curves......Page 388
B.6 Surfaces in Space......Page 390
B.7 Fundamental Forms of the Surface......Page 393
B.8 Curvature of a Curve on the Surface......Page 397
B.9 Internal Geometry of a Surface......Page 401
B.10 Surface Vectors......Page 406
B.11 Geodetic Lines on a Surface......Page 409
B.12 Vector Fields on the Surface......Page 414
B.13 Hybrid Tensors......Page 416
C.1 Events and Set of Events......Page 419
C.2 Probability......Page 420
C.3 Common and Conditional Probability, Independent Events......Page 421
C.4 Random Variables......Page 422
C.5 Distribution of Probability Density and Mean Values......Page 423
C.6 Generalized Functions......Page 424
C.7 Methods of Averaging......Page 426
C.8 Characteristic Function......Page 428
C.9 Moments and Cumulants of Random Quantities......Page 430
C.10 Correlation Functions......Page 432
C.11 Poisson, Bernoulli and Gaussian Distributions......Page 434
C.12 Stationary Random Functions and Homogeneous Random Fields......Page 438
C.13 Isotropic Random Fields......Page 440
C.14 Stochastic Processes, Markovian Processes and Chapman--Kolmogorov Integral Equation......Page 442
C.15 Differential Equations of Chapman--Kolmogorov et al.......Page 445
C.16 Stochastic Differential Equations and the Langevin Equation......Page 457
D.1.1 Operations with Complex Numbers......Page 463
D.1.2 Geometrical Interpretation of Complex Numbers......Page 464
D.2.1 Geometrical Notions......Page 466
D.2.2 Functions of a Complex Variable......Page 467
D.2.3 Differentiation and Analyticity of Complex Functions......Page 468
D.3.1 Functions......Page 469
D.3.2 Joukowski Function......Page 472
D.4.1 Integral of Complex Variable Functions......Page 473
D.4.2 Some Theorems of Integral Calculus in Simply Connected Regions......Page 474
D.4.3 Extension of Integral Calculus to Multiply Connected Regions......Page 476
D.4.4 Cauchy Formula......Page 478
D.5.2 Laurent Series......Page 480
D.6 Singular Points......Page 482
D.6.1 Theorem about Residues......Page 483
D.6.2 Infinitely Remote Point......Page 486
D.7.1 Notion of Conformal Transformation......Page 488
D.7.2 Main Problem......Page 491
D.7.4 Linear Fractional Function......Page 492
D.7.5 Particular Cases......Page 494
D.8.1 Harmonic Functions......Page 497
D.8.2 Dirichlet Problem......Page 498
D.9.1 Plane Field and Complex Potential......Page 500
D.9.2 Examples of Plane Fields......Page 504
References to Appendix......Page 511
Index......Page 513