Fluid Mechanics

Preface xi

Chapter 1. Thermodynamics of Discrete Systems 1

1.1. The representational bases of a material system 1

1.1.1. Introduction 1

1.1.2. Systems analysis and thermodynamics 8

1.1.3. The notion of state 11

1.1.4. Processes and systems 13

1.2. Axioms of thermostatics 15

1.2.1. Introduction 15

1.2.2. Extensive quantities 16

1.2.3. Energy, work and heat 20

1.3. Consequences of the axioms of thermostatics 21

1.3.1. Intensive variables 21

1.3.2. Thermodynamic potentials 23

1.4. Out-of-equilibrium states 29

1.4.1. Introduction 29

1.4.2. Discontinuous systems 30

1.4.3. Application to heat engines 45

Chapter 2. Thermodynamics of Continuous Media 47

2.1. Thermostatics of continuous media 47

2.1.1. Reduced extensive quantities 47

2.1.2. Local thermodynamic equilibrium 48

2.1.3. Flux of extensive quantities 50

2.1.4. Balance equations in continuous media 54

2.1.5. Phenomenological laws 57

2.2. Fluid statics 63

2.2.1. General equations of fluid statics 63

2.2.2. Pressure forces on solid boundaries 68

2.3. Heat conduction 72

2.3.1. The heat equation 72

2.3.2. Thermal boundary conditions 72

2.4. Diffusion 73

2.4.1. Introduction 73

2.4.2. Molar and mass fluxes 77

2.4.3. Choice of reference frame 80

2.4.4. Binary isothermal mixture 85

2.4.5. Coupled phenomena with diffusion 97

2.4.6. Boundary conditions 99

Chapter 3. Physics of Energetic Systems in Flow 101

3.1. Dynamics of a material point 101

3.1.1. Galilean reference frames in traditional mechanics 101

3.1.2. Isolated mechanical system and momentum 102

3.1.3. Momentum and velocity 103

3.1.4. Definition of force 104

3.1.5. The fundamental law of dynamics (closed systems) 106

3.1.6. Kinetic energy 106

3.2. Mechanical material system 107

3.2.1. Dynamic properties of a material system 107

3.2.2. Kinetic energy of a material system 109

3.2.3. Mechanical system in thermodynamic equilibrium the rigid solid 111

3.2.4. The open mechanical system 112

3.2.5. Thermodynamics of a system in motion 116

3.3. Kinematics of continuous media 119

3.3.1. Lagrangian and Eulerian variables 119

3.3.2. Trajectories, streamlines, streaklines 121

3.3.3. Material (or Lagrangian) derivative 122

3.3.4. Deformation rate tensors 129

3.4. Phenomenological laws of viscosity 132

3.4.1. Definition of a fluid 132

3.4.2. Viscometric flows 135

3.4.3. The Newtonian fluid 146

Chapter 4. Fluid Dynamics Equations 151

4.1. Local balance equations 151

4.1.1. Balance of an extensive quantity G 151

4.1.2. Interpretation of an equation in terms of the balance equation 153

4.2. Mass balance 154

4.2.1. Conservation of mass and its consequences 154

4.2.2. Volume conservation 160

4.3. Balance of mechanical and thermodynamic quantities 160

4.3.1. Momentum balance 160

4.3.2. Kinetic energy theorem 164

4.3.3. The vorticity equation 171

4.3.4. The energy equation 172

4.3.5. Balance of chemical species 177

4.4. Boundary conditions 178

4.4.1. General considerations 178

4.4.2. Geometric boundary conditions 179

4.4.3. Initial conditions 181

4.5. Global form of the balance equations 182

4.5.1. The interest of the global form of a balance 182

4.5.2. Equation of mass conservation 184

4.5.3. Volume balance 184

4.5.4. The momentum flux theorem 184

4.5.5. Kinetic energy theorem 186

4.5.6. The energy equation 187

4.5.7. The balance equation for chemical species 188

4.6. Similarity and non-dimensional parameters 189

4.6.1. Principles 189

Chapter 5. Transport and Propagation 199

5.1. General considerations 199

5.1.1. Differential equations 199

5.1.2. The Cauchy problem for differential equations 202

5.2. First order quasi-linear partial differential equations 203

5.2.1. Introduction 203

5.2.2. Geometric interpretation of the solutions 204

5.2.3. Comments 206

5.2.4. The Cauchy problem for partial differential equations 206

5.3. Systems of first order partial differential equations 207

5.3.1. The Cauchy problem for n unknowns and two variables 207

5.3.2. Applications in fluid mechanics 210

5.3.3. Cauchy problem with n unknowns and p variables 216

5.3.4. Partial differential equations of order n 218

5.3.5. Applications 220

5.3.6. Physical interpretation of propagation 223

5.4. Second order partial differential equations 225

5.4.1. Introduction 225

5.4.2. Characteristic curves of hyperbolic equations 226

5.4.3. Reduced form of the second order quasi-linear partial differential equation 229

5.4.4. Second order partial differential equations in a finite domain 232

5.4.5. Second order partial differential equations and their boundary conditions 233

5.5. Discontinuities: shock waves 239

5.5.1. General considerations 239

5.5.2. Unsteady 1D flow of an inviscid compressible fluid 239

5.5.3. Plane steady supersonic flow 244

5.5.4. Flow in a nozzle 244

5.5.5. Separated shock wave 248

5.5.6. Other discontinuity categories 248

5.5.7. Balance equations across a discontinuity 249

5.6. Some comments on methods of numerical solution 250

5.6.1. Characteristic curves and numerical discretization schemes 250

5.6.2. A complex example 253

5.6.3. Boundary conditions of flow problems 255

Chapter 6. General Properties of Flows 257

6.1. Dynamics of vorticity 257

6.1.1. Kinematic properties of the rotation vector 257

6.1.2. Equation and properties of the rotation vector 261

6.2. Potential flows 269

6.2.1. Introduction 269

6.2.2. Bernoulli’s second theorem 269

6.2.3. Flow of compressible inviscid fluid 270

6.2.4. Nature of equations in inviscid flows 271

6.2.5. Elementary solutions in irrotational flows 273

6.2.6. Surface waves in shallow water 284

6.3. Orders of magnitude 288

6.3.1. Introduction and discussion of a simple example 288

6.3.2. Obtaining approximate values of a solution 291

6.4. Small parameters and perturbation phenomena 296

6.4.1. Introduction 296

6.4.2. Regular perturbation 296

6.4.3. Singular perturbations 305

6.5. Quasi-1D flows 309

6.5.1. General properties 309

6.5.2. Flows in pipes 314

6.5.3. The boundary layer in steady flow 319

6.6. Unsteady flows and steady flows 327

6.6.1. Introduction 327

6.6.2. The existence of steady flows 328

6.6.3. Transitional regime and permanent solution 330

6.6.4. Non-existence of a steady solution 334

Chapter 7. Measurement, Representation and Analysis of Temporal Signals 339

7.1. Introduction and position of the problem 339

7.2. Measurement and experimental data in flows 340

7.2.1. Introduction 340

7.2.2. Measurement of pressure 341

7.2.3. Anemometric measurements 342

7.2.4. Temperature measurements 346

7.2.5. Measurements of concentration 347

7.2.6. Fields of quantities and global measurements 347

7.2.7. Errors and uncertainties of measurements 351

7.3. Representation of signals 357

7.3.1. Objectives of continuous signal representation 357

7.3.2. Analytical representation 360

7.3.3. Signal decomposition on the basis of functions; series and elementary solutions 361

7.3.4. Integral transforms 363

7.3.5. Time-frequency (or timescale) representations 374

7.3.6. Discretized signals 381

7.3.7. Data compression 385

7.4. Choice of representation and obtaining pertinent information 389

7.4.1. Introduction 389

7.4.2. An example: analysis of sound 390

7.4.3. Analysis of musical signals 393

7.4.4. Signal analysis in aero-energetics 402

Chapter 8. Thermal Systems and Models 405

8.1. Overview of models 405

8.1.1. Introduction and definitions 405

8.1.2. Modeling by state representation and choice of variables 408

8.1.3. External representation 410

8.1.4. Command models 411

8.2. Thermodynamics and state representation 412

8.2.1. General principles of modeling 412

8.2.2. Linear time-invariant system (LTIS) 420

8.3. Modeling linear invariant thermal systems 422

8.3.1. Modeling discrete systems 422

8.3.2. Thermal models in continuous media 431

8.4. External representation of linear invariant systems 446

8.4.1. Overview 446

8.4.2. External description of linear invariant systems 446

8.5. Parametric models 451

8.5.1. Definition of model parameters 451

8.5.2. Established regimes of linear invariant systems 453

8.5.3. Established regimes in continuous media 458

8.6. Model reduction 465

8.6.1. Overview 465

8.6.2. Model reduction of discrete systems 466

8.7. Application in fluid mechanics and transfer in flows 474

Appendix 1. Laplace Transform 477

A1.1. Definition 477

A1.2. Properties 477

A1.3. Some Laplace transforms 478

A1.4. Application to the solution of constant coefficient differential equations 479

Appendix 2. Hilbert Transform 481

Appendix 3. Cepstral Analysis 483

A3.1. Introduction 483

A3.2. Definitions 483

A3.3. Example of echo suppression 484

A3.4. General case 485

Appendix 4. Eigenfunctions of an Operator 487

A4.1. Eigenfunctions of an operator 487

A4.2. Self-adjoint operator 487

A4.2.1. Eigenfunctions 487

A4.2.2. Expression of a function of f using an eigenfunction basis-set 488

Bibliography 489

Index 497