Preface ........................................................ xi
1 Definitions of Continuum Mechanics ........................... 1
1.1 Vectors and Tensors ..................................... 1
1.1.1 Covariant Differentiation ........................ 5
1.1.2 The Levi-Civita Tensor ........................... 7
1.1.3 Differential Operations .......................... 9
1.1.4 Physical Components of Vectors and Tensors ....... 9
1.1.5 Eigenvalues and Eigenvectors of a Symmetric
Tensor .......................................... 10
1.1.6 The Ostrogradsky-Gauss Theorem .................. 12
1.1.7 The Stokes Theorem .............................. 14
1.1.8 The Weyl Formula ................................ 15
1.2 Eulerian and Lagrangian Description of a Continuum:
Strain Tensor .......................................... 24
1.2.1 Lagrangian and Eulerian Description of
a Continuum ..................................... 24
1.2.2 Strain Tensor ................................... 28
1.2.3 A Condition for Compatibility of Deformations ... 35
1.2.4 Rate-of-Strain Tensor: Cauchy-Helmholtz
Theorem ......................................... 37
1.3 Stress Tensor .......................................... 55
1.3.1 The Cauchy Stress Tensor in the Accompanying
Coordinate System ............................... 55
1.3.2 Piola-Kirchhoff Stress Tensors in
the Reference Frame and in the Eulerian
Coordinates ..................................... 59
1.3.3 Principal Values and Invariants of the Stress
Tensor .......................................... 61
1.3.4 Differentiation of the Stress Tensor with
Respect to Time ................................. 63
References .................................................. 73
2 Fundamental Principles and Laws of Continuum Mechanics ...... 75
2.1 Equations of Continuity, Motion, and Energy for a
Continuum .............................................. 75
2.1.1 Continuity Equation ............................. 76
2.1.2 Equations of Motion and of Momentum Moment ...... 78
2.1.3 The Energy Conservation Law: The First and
Second Laws of Thermodynamics ................... 84
2.1.4 Equation of State (General Relations) ........... 92
2.1.5 Equations of an Ideal and Viscous, Heat-
Conducting Gas .................................. 95
2.2 The Hamilton-Ostrogradsky's Variational Principle in
Continuum Mechanics ................................... 115
2.2.1 Euler-Lagrange Equations in Lagrangian
Coordinates .................................... 115
2.2.2 Hamilton's Equations in Lagrangian
Coordinates .................................... 121
2.2.3 Euler-Lagrange Equations in Eulerian
Coordinates and Murnaghan's Formula ............ 125
2.3 Conservation Laws for Energy and Momentum in
Continuum Mechanics ................................... 135
2.3.1 Conservation Laws in Cartesian Coordinates ..... 135
2.3.2 Conservation Laws in an Arbitrary Coordinate
System ......................................... 144
References ................................................. 152
3 The Features of the Solutions of Continuum Mechanics
Problems ................................................... 155
3.1 Similarity and Dimension Theory in Continuum
Mechanics ............................................. 155
3.2 The Characteristics of Partial Differential
Equations ............................................. 163
3.3 Discontinuity Surfaces in Continuum Mechanics ......... 171
References ............................................ 185
4 Ideal Fluid ................................................ 187
4.1 Integrals of Motion Equations of Ideal Fluid and
Gas ................................................... 187
4.1.1 Motion Equations in the Gromeka-Lamb Form ...... 188
4.1.2 The Bernoulli Integral ......................... 188
4.1.3 The Lagrange Integral .......................... 189
4.2 Planar Irrotational Steady Motions of an Ideal
Incompressible Fluid .................................. 193
4.2.1 The Governing Equations of Planar Flows ........ 193
4.2.2 The Potential Flow past the Cylinder ........... 202
4.2.3 The Method of Conformal Mappings ............... 208
4.2.4 The Problem of the Flow around a Slender
Profile ........................................ 219
4.3 Axisymmetric and Three-Dimensional Potential Ideal
Incompressible Fluid Flows ............................ 223
4.3.1 Axially Symmetric Flows ........................ 223
4.3.2 The Method of Sources and Sinks ................ 231
4.3.3 The Program prog4-5.nb ......................... 233
4.3.4 The Transverse Flow around the Body of
Revolution: The Program prog4-6.nb ............. 235
4.4 Nonstationary Motion of a Solid in the Fluid .......... 242
4.4.1 Formulation of a Problem on Nonstationary
Body Motion in Ideal Fluid ..................... 242
4.4.2 The Hydrodynamic Reactions at the Body
Motion ......................................... 244
4.4.3 Equations of Solid Motion in a Fluid under
the Action of Given Forces ..................... 247
4.5 Vortical Motions of Ideal Fluid ....................... 250
4.5.1 The Theorems of Thomson, Lagrange, and
Helmholtz ...................................... 250
4.5.2 Motion Equations in Friedmann's Form ........... 257
4.5.3 The Biot-Savart Formulas and the Straight
Vortex Filament ................................ 258
References ................................................. 265
5 Viscous Fluid .............................................. 267
5.1 General Equations of Viscous Incompressible Fluid ..... 268
5.1.1 The Navier-Stokes Equations .................... 268
5.1.2 Formulation of Problems for the System of the
Navier-Stokes Equations ........................ 275
5.2 Viscous Fluid Flows at Small Reynolds Numbers ......... 276
5.2.1 Exact Solutions of the System of Equations
for a Viscous Fluid ............................ 277
5.2.2 Viscous Fluid Motion between Two Rotating
Coaxial Cylinders .............................. 280
5.2.3 The Viscous Incompressible Fluid Flow around
a Sphere at Small Reynolds Numbers ............. 282
5.3 Viscous Fluid Flows at Large Reynolds Numbers ......... 287
5.3.1 Prandtl's Theory of Boundary Layers ............ 288
5.3.2 Boundary Layer of a Flat Plate ................. 293
5.4 Turbulent Fluid Flows ................................. 298
5.4.1 Basic Properties of Turbulent Flows ............ 298
5.4.2 Laminar Flow Stability and Transition to
Turbulence ..................................... 300
5.4.3 Turbulent Fluid Flow ........................... 302
References ................................................. 309
6 Gas Dynamics ............................................... 311
6.1 One-Dimensional Stationary Gas Flows .................. 311
6.1.1 Governing Equations for Quasi-One-Dimensional
Gas Flow ....................................... 311
6.1.2 Gas Motion in a Variable Section Duct:
Elementary Theory of the Laval Nozzle .......... 313
6.1.3 Planar Shock Wave in Ideal Gas ................. 321
6.1.4 Shock Wave Structure in Gas .................... 329
6.2 Nonstationary One-Dimensional Flows of Ideal Gas ...... 334
6.2.1 Planar Isentropic Waves ........................ 334
6.2.2 Gradient Catastrophe and Shock Wave
Formation ...................................... 342
6.3 Planar Irrotational Ideal Gas Motion (Linear
Approximation) ........................................ 346
6.3.1 Governing Equations and Their Linearization .... 346
6.3.2 The Problem of the Flow around a Slender
Profile ........................................ 348
6.4 Planar Irrotational Stationary Ideal Gas Flow
(General Case) ........................................ 354
6.4.1 Characteristics of Stationary Irrotational
Flows of Ideal Gas, Simple Wave: The Prandtl-
Meyer Flow ..................................... 355
6.4.2 Chaplygin's Equations and Method ............... 366
6.4.3 Oblique Shock Waves ............................ 377
6.4.4 Interference of Stationary Shock Waves ......... 382
6.5 The Fundamentals of the Gasdynamic Design
Technology ............................................ 386
6.5.1 The Basic Algorithm ............................ 387
6.5.2 The Superposition Procedure .................... 391
6.5.3 The Complement Procedure ....................... 395
References ................................................. 399
7 Multiphase Media ........................................... 401
7.1 Mathematical Models of Multiphase Media ............... 403
7.1.1 General Equations of the Mechanics of
Multiphase Media ............................... 403
7.1.2 Equations of a Two-Phase Medium of the Type
of Gas-Solid Particles ......................... 407
7.1.3 Equations of a Bicomponent Medium of Gas
Mixture Type ................................... 415
7.2 Correctness of the Cauchy Problem: Relations at
Discontinuities in Multiphase Media ................... 417
7.2.1 The Characteristics of a System of Equations
for Gas-Particle Mixtures and Correctness of
the Cauchy Problem ............................. 417
7.2.2 Jump Relations ................................. 431
7.3 Quasi-One-Dimensional Flows of a Gas-Particle
Mixture in Laval Nozzles .............................. 442
7.3.1 The Equations of the Quasi-One-Dimensional
Flow of a Gas-Particle Mixture ................. 442
7.3.2 The Flow of a Gas-Particle Mixture in the
Laval Nozzle with Small Velocity and
Temperature Lags of Particles .................. 447
7.4 The Continual-Discrete Model and Caustics in the
Pseudogas of Particles ................................ 456
7.4.1 The Equations of the Continual-Discrete Model
of a Gas-Particle Mixture at a Small Volume
Concentration of Particles ..................... 456
7.4.2 Investigation of Caustics in the Pseudogas
of Particles ................................... 460
7.5 Nonstationary Processes in Gas-Particle Mixtures ...... 471
7.5.1 Interaction of a Shock Wave with a Cloud
of Particles ................................... 471
7.5.2 Acoustic Approximation in the Problem of
Shock Wave Interaction with a Particle's
Cloud at a Small Volume Concentration .......... 478
7.6 The Flows of Heterogeneous Media without Regard for
Inertial Effects ...................................... 486
7.6.1 The Brownian Motion of Particles in a Fluid .... 486
7.6.2 Fluid Filtration in a Porous Medium ............ 493
7.7 Wave Processes in Bubbly Liquids ...................... 500
7.7.1 Equations of the Motion of a Bubbly Liquid ..... 500
7.7.2 Equations for Weak Nonlinear Disturbances in
Bubbly Liquids ................................. 507
7.7.3 Progressive, Weak Nonlinear Waves in Bubbly
Liquids ........................................ 512
References ................................................. 522
Appendix A: Mathematica Functions ............................. 526
Appendix B: Glossary of Programs .............................. 550
Index ......................................................... 565
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