Introduction .................................................... 1
1 Navier-Stokes-Fourier Exact Model ............................ 11
1.1 The Transport Theorem .................................... 11
1.2 The Equation of Continuity ............................... 12
1.3 The Cauchy Equation of Motion ............................ 12
1.4 The Constitutive Equations of a Viscous Fluid ............ 13
1.4.1 Stokes's Four Postulates: Stokesian Fluid .......... 14
1.4.2 Classical Linear Viscosity Theory: Newtonian
Fluid .............................................. 15
1.5 The Energy Equation and Fourier's Law .................... 17
1.5.1 The Total Energy Equation .......................... 17
1.5.2 Heat Conduction and Fourier's Law .................. 18
1.6 The Navier-Stokes-Fourier Equations ...................... 19
1.6.1 The NSF Equation for an Ideal Gas when Cv and Cp
are Constants ...................................... 20
1.6.2 Dimensionless NSF Equations ........................ 21
1.6.3 Reduced Dimensionless Parameters ................... 22
1.7 Conditions for Unsteady-State NSF Equations .............. 25
1.7.1 The Problem of Initial Conditions .................. 26
1.7.2 Boundary Conditions ................................ 28
2 Some Features and Various Forms of NSF Equations ............. 35
2.1 Isentropicity, Polytropic Gas, Barotropic Motion, and
Incompressibility ........................................ 35
2.1.1 NS Equations ....................................... 35
2.1.2 Navier System ...................................... 36
2.1.3 Navier System with Time-Dependent Density .......... 37
2.1.4 Fourier Equation ................................... 38
2.2 Some Interesting Issues in Navier Incompressible Fluid
Flow ..................................................... 39
2.2.1 The Pressure Poisson Equation ...................... 41
2.2.2 ψN — ωN and uN — ωN Formulations .................... 42
2.2.3 The Omnipotence of the Incompressibility
Constraint ......................................... 43
2.2.4 A First Statement of a Well-Posed Initial,
Boundary-Value Problem (IBVP) for Navier
Equations .......................................... 46
2.2.5 Cauchy Formula for Vorticity ....................... 47
2.2.6 The Navier Equations as an Evolutionary Equation
for Perturbations .................................. 48
2.3 From NSF to Hyposonic and Oberbeck-Boussinesq (OB)
Equations ................................................ 50
2.3.1 Model Equations for Hyposonic Fluid Flows .......... 50
2.3.2 The Oberbeck-Boussinesq Model Equations ............ 52
3 Some Simple Examples of Navier, NS and NSF Viscous Fluid
Flows ........................................................ 57
3.1 Plane Poiseuille Flow and the Orr-Sommerfeld Equation .... 57
3.1.1 The Orr-Sommerfeld Equation ........................ 58
3.1.2 A Double-Scale Technique for Resolving
the Orr-Sommerfeld Equation ........................ 60
3.2 Steady Flow Through an Arbitrary Cylinder under
Pressure ................................................. 61
3.2.1 The Case of a Circular Cylinder .................... 62
3.2.2 The Case of an Annular Region Between Concentric
Cylinders .......................................... 63
3.2.3 The Case of a Cylinder of Arbitrary Section ........ 63
3.3 Steady-State Couette Flow Between Cylinders in Relative
Motion ................................................... 64
3.3.1 The Classic Taylor Problem ......................... 65
3.3.2 The Taylor Number .................................. 66
3.4 The Bénard Linear Problem and Thermal Instability ........ 68
3.5 The Bénard Linear Problem with a Free Surface and
the Marangoni Effect ..................................... 71
3.5.1 The Case when the Neutral State is Stationary ...... 73
3.5.2 Free-Surface Deformation ........................... 75
3.6 Flow due to a Rotating Disc .............................. 75
3.6.1 Small Values of ζ .................................. 77
3.6.2 Large Values of ζ .................................. 77
3.6.3 Joining (Matching) ................................. 78
3.7 One-Dimensional Unsteady-State NSF Equations and
the Rayleigh Problem ..................................... 78
3.7.1 Small M2 Solution - Close to the Flat Plate but
far from the Initial Time .......................... 81
3.7.2 Small M2 Solution - Far from a Flat Plate .......... 83
3.7.3 Small M2 Solution - Close to the Initial Time ...... 86
3.8 Complementary Remarks .................................... 87
4 The Limit of Very Large Reynolds Numbers ..................... 89
4.1 Introduction ............................................. 89
4.2 Classical Hierarchical Boundary-Layer Concept and
Regular Coupling ......................................... 93
4.2.1 A 2-D Steady-State Navier Equation for the Stream
Function ........................................... 93
4.2.2 A Local Form of the 2-D Steady-State Navier
Equation for the Stream Function ................... 94
4.2.3 A Large Reynolds Number and "Principal" and
"Local" Approximations ............................. 94
4.2.4 Matching ........................................... 96
4.2.5 The Prandtl-Blasius and Blasius BL Problems ........ 97
4.3 Asymptotic Structure of Unsteady-State NSF Equations
at Re >> 1 .............................................. 103
4.3.1 Four Significant Degeneracies of NSF Equations .... 105
4.3.2 Formulation of a Simplified Initial
Boundary-Value Problem for the NSF Full
Unsteady-State Equations .......................... 108
4.3.3 Various Facets of Large Reynolds Number
Unsteady-State Flow ............................... 109
4.3.4 The Two Adjustment Problems ....................... 114
4.4 The Triple-Deck Concept and Singular Interactive
Coupling ................................................ 118
4.4.1 The Triple-Deck Theory in 2-D Steady-State
Navier Flow ....................................... 120
4.5 Complementary Remarks ................................... 126
4.5.1 Three-Dimensional Boundary-Layer Equations ........ 130
4.5.2 Unsteady-State Incompressible Boundary-Layer
Formulation ....................................... 137
4.5.3 The Inviscid Limit: Some Mathematical Results ..... 140
4.5.4 Rigorous Results for the Boundary-Layer Theory .... 144
5 The Limit of Very Low Reynolds Numbers ...................... 145
5.1 Large Viscosity Limits and Stokes and Oseen Equations ... 145
5.1.1 Steady-State Stokes Equation ...................... 145
5.1.2 Unsteady-State Oseen Equation ..................... 146
5.1.3 Unsteady-State Stokes and Steady-State Oseen
Equations ......................................... 147
5.1.4 Unsteady-State Matched Stokes-Oseen Solution
at Re << 1 for the Flow Past a Sphere ............. 147
5.2 Low Reynolds Number Flow due to an Impulsively Started
Circular Cylinder ....................................... 149
5.2.1 Formulation of the Steady-State Problem ........... 150
5.2.2 The Unsteady-State Problem ........................ 152
5.3 Compressible Flow ....................................... 153
5.3.1 The Stokes Limiting Case and Steady-State
Compressible Stokes Equations ..................... 154
5.3.2 The Oseen Limiting Case and Steady-State
Compressible Oseen Equations ...................... 155
5.4 Film Flow on a Rotating Disc: Asymptotic Analysis for
Small Re .......................................... 158
5.4.1 Solution for Small Re << 1: Long-Time Scale
Analysis .......................................... 159
5.4.2 Solution for Small Re << 1: Short-Time Scale
Analysis .......................................... 160
5.5 Some Rigorous Mathematical Results ...................... 164
6 Incompressible Limit: Low Mach Number Asymptotics ........... 165
6.1 Introduction ............................................ 165
6.2 Navier-Fourier Asymptotic Model ......................... 168
6.2.1 The Initialization Problem and Equations of
Acoustics ......................................... 171
6.2.2 The Fourier Model ................................. 175
6.2.3 Influence of Weak Compressibility:
Second-Order Equations for u and π ................ 178
6.2.4 Concluding Remarks ................................ 179
6.3 Compressible Low Mach Number Models ..................... 181
6.3.1 Hyposonic Model for Flow in a Bounded Cavity ...... 181
6.3.2 Large Channel Aspect Ratio, Low Mach Number,
Compressible Flow ................................. 183
6.4 Viscous Nonadiabatic Boussinesq Equations ............... 184
6.4.1 The Basic State ................................... 184
6.4.2 Asymptotic Derivation of Viscous, Nonadiabatic
Boussinesq Equations .............................. 186
6.5 Some Comments ........................................... 187
7 Some Viscous Fluid Motions and Problems ..................... 191
7.1 Oscillatory Viscous Incompressible Flow ................. 191
7.1.1 Acoustic Streaming Effect ......................... 191
7.1.2 Study of the Steady-State Streaming Phenomenon .... 196
7.1.3 The Role of Parameters αRe = Res and Re/α = β2 .... 198
7.1.4 Other Examples of Viscous Oscillatory Flow ........ 202
7.2 Unsteady-State Viscous, Incompressible Flow past
a Rotating and Translating Cylinder ..................... 203
7.2.1 Formulation of the Governing Problem .............. 203
7.2.2 Method of Solution ................................ 204
7.2.3 Determination of the Initial Flow ................. 205
7.2.4 Results of Calculations and Comparison with the
Visualization of Coutanceau and Ménard (1985) ..... 206
7.2.5 A Short Comment ................................... 207
7.3 Ekman and Stewartson Layers ............................. 208
7.3.1 General Equations and Boundary Conditions ......... 210
7.3.2 The Ekman Layer ................................... 211
7.3.3 The Stewartson Layer .............................. 211
7.3.4 The Inner, Outer, and Upper Regions ............... 213
7.3.5 Comments .......................................... 214
7.4 Low Reynolds Number Flows: Further Investigations ....... 215
7.4.1 Unsteady-State Adjustment to the Stokes Model in
a Bounded Deformable Cavity Ω(t) .................. 215
7.4.2 On the Wake in Low Reynolds Number Flow ........... 218
7.4.3 Oscillatory Disturbances as Admissible Solutions
and their Possible Relationship to
the Von Karman Sheet Phenomenon ................... 220
7.4.4 Some References ................................... 223
7.5 The Bénard-Marangoni Problem: An Alternative ............ 224
7.5.1 Dimensionless Dominant Equations .................. 226
7.5.2 Dimensionless Dominant Boundary Conditions ........ 227
7.5.3 The Rayleigh-Bénard (RB) Thermal Shallow
Convection Problem ................................ 229
7.5.4 The Bénard-Marangoni (BM) Problem ................. 231
7.6 Some Aspects of Nonadiabatic Viscous Atmospheric Flow ... 233
7.6.1 The L-SSHV Equations .............................. 233
7.6.2 The Tangent HV (THV) Equations .................... 238
7.6.3 The Quasi-Geostrophic Model ....................... 240
7.7 Miscellaneous Topics .................................... 246
7.7.1 The Entrainment of a Viscous Fluid in
a Two-Dimensional Cavity .......................... 246
7.7.2 Unsteady-State Boundary Layers .................... 253
7.7.3 Various Topics Related to Boundary-Layer
Equations ......................................... 258
7.7.4 More on the Triple-Deck Theory .................... 260
7.7.5 Some Problems Related to Navier Equations for
an Incompressible Viscous Fluid ................... 266
7.7.6 Low and Large Prandtl Number Flow ................. 272
7.7.7 A final comment ................................... 275
8 Some Aspects of a Mathematically Rigorous Theory ............ 277
8.1 Classical, Weak, and Strong Solutions of the Navier
Equations ............................................... 278
8.2 Galerkin Approximations and Weak Solutions of
the Navier Equations .................................... 283
8.2.1 Some Comments and Bibliographical Notes ........... 287
8.3 Rigorous Mathematical Results for Navier
Incompressible and Viscous Fluid Flows .................. 289
8.3.1 Navier Equations in an Unbounded Domain ........... 295
8.3.2 Some Recent Rigorous Results ...................... 298
8.4 Rigorous Mathematical Results for Compressible and
Viscous Fluid Flows ..................................... 300
8.4.1 The Incompressible Limit .......................... 305
8.5 Some Concluding Remarks ................................. 307
9 Linear and Nonlinear Stability of Fluid Motion .............. 311
9.1 Some Aspects of the Theory of the Stability of Fluid
Motion .................................................. 311
9.1.1 Linear, Weakly Nonlinear, Nonlinear, and
Hydrodynamic Stability ............................ 312
9.1.2 Reynolds-Orr, Energy, Sufficient Stability
Criterion ......................................... 316
9.1.3 An Evolution Equation for Studying
the Stability of a Basic Solution of Fluid
Flow .............................................. 317
9.2 Fundamental Ideas on the Theory of the Stability of
Fluid Motion ............................................ 319
9.2.1 Linear Case ....................................... 320
9.2.2 Nonlinear Case .................................... 322
9.3 The Guiraud-Zeytounian Asymptotic Approach to
Nonlinear Hydrodynamic Stability ........................ 324
9.3.1 Linear Theory ..................................... 326
9.3.2 Nonlinear Theory - Confined Perturbations.
Landau and Stuart Equations ....................... 328
9.3.3 Nonlinear Theory - Unconfined Perturbations.
General Setting ................................... 331
9.3.4 Nonlinear Theory - Unconfined Perturbations.
Tollmien-Schlichting Waves ........................ 332
9.3.5 Nonlinear Theory - Unconfined Perturbations.
Rayleigh-Bénard Convection ........................ 335
9.4 Some Facets of the RB and BM Problem .................... 337
9.4.1 Rayleigh-Bénard Convective Instability ............ 337
9.4.2 Bénard-Marangoni (BM) Thermocapillary
Instability Problem for a Thin Layer (Film)
with a Deformable Free Surface .................... 356
9.5 Couette-Taylor Viscous Flow Between Two Rotating
Cylinders ............................................... 370
9.5.1 A Short Survey .................................... 370
9.5.2 Bifurcations ...................................... 376
9.6 Concluding Comments and Remarks ......................... 380
10 A Finite-Dimensional Dynamical System Approach to
Turbulence ................................................. 387
10.1 A Phenomenological Approach to Turbulence ............. 387
10.2 Bifurcations in Dissipative Dynamical Systems ......... 392
10.2.1 Normal Form of the Pitchfork Bifurcation ...... 395
10.2.2 Normal Form of the Hopf Bifurcation ........... 396
10.2.3 Bifurcation from a Periodic Orbit to
an Invariant Torus ............................ 398
10.3 Transition to Turbulence: Scenarios, Routes to
Chaos ................................................. 398
10.3.1 The Landau-Hopf "Inadequate" Scenario ......... 399
10.3.2 The Ruelle-Takens-Newhouse Scenario ........... 399
10.3.3 The Feigenbaum Scenario ....................... 403
10.3.4 The Pomeau-Manneville Scenario ................ 406
10.3.5 Complementary Remarks ......................... 408
10.4 Strange Attractors for Various Fluid Flows ............ 414
10.4.1 Viscous Isochoric Wave Motions ................ 414
10.4.2 The Bénard-Marangoni Problem for
a Free-Falling Vertical Film: The Case of
Re = O(1) and the KS Equation ................. 417
10.4.3 The Bénard-Marangoni Problem for
a Free-Falling Vertical Film: The Case of
Re/ε = O(1) and the KS-KdV Equation ........... 424
10.4.4 Viscous and Thermal Effects in a Simple
Stratified Fluid Model ........................ 427
10.4.5 Obukhov Discrete Cascade Systems for
Developed Turbulence .......................... 435
10.4.6 Unpredictability in Viscous Fluid Flow
Between a Stationary and a Rotating Disk ...... 439
10.5 Some Comments and References .......................... 444
References .................................................... 449
Index ......................................................... 485
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