Foreword ....................................................... xi
Preface ...................................................... xiii
Introduction .................................................... 1
1 Idealized continuous media: the basic concepts .............. 10
1.1 The idealized model of a continuous medium ............. 10
1.2 Properties of a continuum and its motion. Density,
flux and velocity. Law of mass balance ................. 18
1.3 Law of momentum balance. Stress tensor ................. 24
2 Dimensional analysis and physical similitude ................ 29
2.1 Examples ............................................... 29
2.2 Dimensional analysis ................................... 37
2.3 Physical similitude .................................... 40
2.4 Examples. Classical parameters of similitude ........... 43
3 The ideal incompressible fluid approximation: general
concepts and relations ...................................... 48
3.1 The fundamental idealization (model). Euler equations .. 48
3.2 Decomposition of the velocity field in the vicinity
of an arbitrary point. The vorticity. The strain-rate
tensor ................................................. 51
3.3 Irrotational motions. Lagrange's theorem. Potential
flows .................................................. 53
3.4 Lagrange-Cauchy integral. Bernoulli integral ........... 56
3.5 Plane potential motions of an ideal incompressible
fluid .................................................. 58
4 The ideal incompressible fluid approximation: analysis
and applications ............................................ 63
4.1 Physical meaning of the velocity potential. The
Lavrentiev problem of a directed explosion ............. 63
4.2 Lift force on a wing ................................... 66
5 The linear elastic solid approximation. Basic equations
and boundary value problems in the linear theory of
elasticity .................................................. 79
5.1 The fundamental idealization ........................... 79
5.2 Basic equations and boundary conditions of the linear
theory of elasticity ................................... 86
5.3 Plane problem in the theory of elasticity .............. 89
5.4 Analytical solutions of some special problems in
plane elasticity ....................................... 95
6 The linear elastic solid approximation. Applications:
brittle and quasi-brittle fracture; strength of
structures ................................................. 101
6.1 The problem of structural integrity ................... 101
6.2 Defects and cracks .................................... 102
6.3 Cohesion crack model .................................. 109
6.4 What is fracture from the mathematical viewpoint? ..... 113
6.5 Time effects; lifetime of a structure; fatigue ........ 119
7 The Newtonian viscous fluid approximation. General
comments and basic relations ............................... 124
7.1 The fundamental idealization. The Navier-Stokes
equations ............................................. 124
7.2 Angular momentum conservation law ..................... 128
7.3 Boundary value and initial value problems for the
Newtonian viscous incompressible fluid approximation.
Smoothness of the solutions ........................... 129
7.4 The viscous dissipation of mechanical energy into
heat .................................................. 135
8 The Newtonian viscous fluid approximation. Applications:
the boundary layer ......................................... 137
8.1 The drag on a moving wing. Friedrichs'example ......... 137
8.2 Model of the boundary layer at a thin weakly
inclined wing of infinite span ........................ 140
8.3 The boundary layer on a flat plate .................... 143
9 Advanced similarity methods: complete and incomplete
similarity ................................................. 150
9.1 Examples .............................................. 150
9.2 Complete and incomplete similarity .................... 153
9.3 Self-similar solutions of the first and second kind ... 157
9.4 Incomplete similarity in fatigue experiments (Paris'
law) .................................................. 158
9.5 A note concerning scaling laws in nanomechanics ....... 161
10 The ideal gas approximation. Sound waves; shoclc waves ..... 164
10.1 Sound waves ........................................... 164
10.2 Energy equation. The basic equations of the ideal
gas model ............................................. 167
10.3 Simple waves. The formation of shock waves ............ 168
10.4 An intense explosion at a plane interface: the
external intermediate asymptotics ..................... 171
10.5 An intense explosion at a plane interface: the
internal intermediate asymptotics ..................... 173
11 Tbrbulence: generalities; scaling laws for shear flows ..... 182
11.1 Kolmogorov's example .................................. 185
11.2 The Reynolds equation. Reynolds stress ................ 187
11.3 Turbulent shear flow .................................. 189
11.4 Scaling laws for turbulent flows at very large
Reynolds numbers. Flow in pipes ....................... 190
11.5 Turbulent flow in pipes at very large Reynolds
numbers: advanced similarity analysis ................. 195
11.6 Reynolds-number dependence of the drag in pipes
following from the power law .......................... 201
11.7 Further comparison of the Reynolds-number-dependent
scaling law and the universal logarithmic law ......... 204
11.8 Modification of the Izakson-Millikan-von Mises
analysis of the flow in the intermediate region ....... 208
11.9 Further comparison of scaling laws with experimental
data .................................................. 211
11.10 Scaling laws for turbulent boundary layers ........... 219
12 Turbulence: mathematical models of turbulent shear flows
and of the local structure of turbulent flows at very
large Reynolds numbers ..................................... 225
12.1 Basic equations for wall-bounded turbulent shear
flows. Wall region .................................... 225
12.2 Kolmogorov-Prandtl semi-empirical model for the wall
region of a shear flow ................................ 227
12.3 A model for drag reduction by polymeric additives ..... 230
12.4 The local structure of turbulent flows at very large
Reynolds numbers ...................................... 234
Bibliography and References ................................... 243
Index ......................................................... 253
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