Fluid Mechanics

Jean-Laurent Puebe, University of Poitiers, France.

• Preface xiChapter 1. Thermodynamics of Discrete Systems 11.1. The representational bases of a material system 11.1.1. Introduction 11.1.2. Systems analysis and thermodynamics 81.1.3. The notion of state 111.1.4. Processes and systems 131.2. Axioms of thermostatics 151.2.1. Introduction 151.2.2. Extensive quantities 161.2.3. Energy, work and heat 201.3. Consequences of the axioms of thermostatics 211.3.1. Intensive variables 211.3.2. Thermodynamic potentials 231.4. Out-of-equilibrium states 291.4.1. Introduction 291.4.2. Discontinuous systems 301.4.3. Application to heat engines 45Chapter 2. Thermodynamics of Continuous Media 472.1. Thermostatics of continuous media 472.1.1. Reduced extensive quantities 472.1.2. Local thermodynamic equilibrium 482.1.3. Flux of extensive quantities 502.1.4. Balance equations in continuous media 542.1.5. Phenomenological laws 572.2. Fluid statics 632.2.1. General equations of fluid statics 632.2.2. Pressure forces on solid boundaries 682.3. Heat conduction 722.3.1. The heat equation 722.3.2. Thermal boundary conditions 722.4. Diffusion 732.4.1. Introduction 732.4.2. Molar and mass fluxes 772.4.3. Choice of reference frame 802.4.4. Binary isothermal mixture 852.4.5. Coupled phenomena with diffusion 972.4.6. Boundary conditions 99Chapter 3. Physics of Energetic Systems in Flow 1013.1. Dynamics of a material point 1013.1.1. Galilean reference frames in traditional mechanics 1013.1.2. Isolated mechanical system and momentum 1023.1.3. Momentum and velocity 1033.1.4. Definition of force 1043.1.5. The fundamental law of dynamics (closed systems) 1063.1.6. Kinetic energy 1063.2. Mechanical material system 1073.2.1. Dynamic properties of a material system 1073.2.2. Kinetic energy of a material system 1093.2.3. Mechanical system in thermodynamic equilibrium the rigid solid 1113.2.4. The open mechanical system 1123.2.5. Thermodynamics of a system in motion 1163.3. Kinematics of continuous media 1193.3.1. Lagrangian and Eulerian variables 1193.3.2. Trajectories, streamlines, streaklines 1213.3.3. Material (or Lagrangian) derivative 1223.3.4. Deformation rate tensors 1293.4. Phenomenological laws of viscosity 1323.4.1. Definition of a fluid 1323.4.2. Viscometric flows 1353.4.3. The Newtonian fluid 146Chapter 4. Fluid Dynamics Equations 1514.1. Local balance equations 1514.1.1. Balance of an extensive quantity G 1514.1.2. Interpretation of an equation in terms of the balance equation 1534.2. Mass balance 1544.2.1. Conservation of mass and its consequences 1544.2.2. Volume conservation 1604.3. Balance of mechanical and thermodynamic quantities 1604.3.1. Momentum balance 1604.3.2. Kinetic energy theorem 1644.3.3. The vorticity equation 1714.3.4. The energy equation 1724.3.5. Balance of chemical species 1774.4. Boundary conditions 1784.4.1. General considerations 1784.4.2. Geometric boundary conditions 1794.4.3. Initial conditions 1814.5. Global form of the balance equations 1824.5.1. The interest of the global form of a balance 1824.5.2. Equation of mass conservation 1844.5.3. Volume balance 1844.5.4. The momentum flux theorem 1844.5.5. Kinetic energy theorem 1864.5.6. The energy equation 1874.5.7. The balance equation for chemical species 1884.6. Similarity and non-dimensional parameters 1894.6.1. Principles 189Chapter 5. Transport and Propagation 1995.1. General considerations 1995.1.1. Differential equations 1995.1.2. The Cauchy problem for differential equations 2025.2. First order quasi-linear partial differential equations 2035.2.1. Introduction 2035.2.2. Geometric interpretation of the solutions 2045.2.3. Comments 2065.2.4. The Cauchy problem for partial differential equations 2065.3. Systems of first order partial differential equations 2075.3.1. The Cauchy problem for n unknowns and two variables 2075.3.2. Applications in fluid mechanics 2105.3.3. Cauchy problem with n unknowns and p variables 2165.3.4. Partial differential equations of order n 2185.3.5. Applications 2205.3.6. Physical interpretation of propagation 2235.4. Second order partial differential equations 2255.4.1. Introduction 2255.4.2. Characteristic curves of hyperbolic equations 2265.4.3. Reduced form of the second order quasi-linear partial differential equation 2295.4.4. Second order partial differential equations in a finite domain 2325.4.5. Second order partial differential equations and their boundary conditions 2335.5. Discontinuities: shock waves 2395.5.1. General considerations 2395.5.2. Unsteady 1D flow of an inviscid compressible fluid 2395.5.3. Plane steady supersonic flow 2445.5.4. Flow in a nozzle 2445.5.5. Separated shock wave 2485.5.6. Other discontinuity categories 2485.5.7. Balance equations across a discontinuity 2495.6. Some comments on methods of numerical solution 2505.6.1. Characteristic curves and numerical discretization schemes 2505.6.2. A complex example 2535.6.3. Boundary conditions of flow problems 255Chapter 6. General Properties of Flows 2576.1. Dynamics of vorticity 2576.1.1. Kinematic properties of the rotation vector 2576.1.2. Equation and properties of the rotation vector 2616.2. Potential flows 2696.2.1. Introduction 2696.2.2. Bernoulli’s second theorem 2696.2.3. Flow of compressible inviscid fluid 2706.2.4. Nature of equations in inviscid flows 2716.2.5. Elementary solutions in irrotational flows 2736.2.6. Surface waves in shallow water 2846.3. Orders of magnitude 2886.3.1. Introduction and discussion of a simple example 2886.3.2. Obtaining approximate values of a solution 2916.4. Small parameters and perturbation phenomena 2966.4.1. Introduction 2966.4.2. Regular perturbation 2966.4.3. Singular perturbations 3056.5. Quasi-1D flows 3096.5.1. General properties 3096.5.2. Flows in pipes 3146.5.3. The boundary layer in steady flow 3196.6. Unsteady flows and steady flows 3276.6.1. Introduction 3276.6.2. The existence of steady flows 3286.6.3. Transitional regime and permanent solution 3306.6.4. Non-existence of a steady solution 334Chapter 7. Measurement, Representation and Analysis of Temporal Signals 3397.1. Introduction and position of the problem 3397.2. Measurement and experimental data in flows 3407.2.1. Introduction 3407.2.2. Measurement of pressure 3417.2.3. Anemometric measurements 3427.2.4. Temperature measurements 3467.2.5. Measurements of concentration 3477.2.6. Fields of quantities and global measurements 3477.2.7. Errors and uncertainties of measurements 3517.3. Representation of signals 3577.3.1. Objectives of continuous signal representation 3577.3.2. Analytical representation 3607.3.3. Signal decomposition on the basis of functions; series and elementary solutions 3617.3.4. Integral transforms 3637.3.5. Time-frequency (or timescale) representations 3747.3.6. Discretized signals 3817.3.7. Data compression 3857.4. Choice of representation and obtaining pertinent information 3897.4.1. Introduction 3897.4.2. An example: analysis of sound 3907.4.3. Analysis of musical signals 3937.4.4. Signal analysis in aero-energetics 402Chapter 8. Thermal Systems and Models 4058.1. Overview of models 4058.1.1. Introduction and definitions 4058.1.2. Modeling by state representation and choice of variables 4088.1.3. External representation 4108.1.4. Command models 4118.2. Thermodynamics and state representation 4128.2.1. General principles of modeling 4128.2.2. Linear time-invariant system (LTIS) 4208.3. Modeling linear invariant thermal systems 4228.3.1. Modeling discrete systems 4228.3.2. Thermal models in continuous media 4318.4. External representation of linear invariant systems 4468.4.1. Overview 4468.4.2. External description of linear invariant systems 4468.5. Parametric models 4518.5.1. Definition of model parameters 4518.5.2. Established regimes of linear invariant systems 4538.5.3. Established regimes in continuous media 4588.6. Model reduction 4658.6.1. Overview 4658.6.2. Model reduction of discrete systems 4668.7. Application in fluid mechanics and transfer in flows 474Appendix 1. Laplace Transform 477A1.1. Definition 477A1.2. Properties 477A1.3. Some Laplace transforms 478A1.4. Application to the solution of constant coefficient differential equations 479Appendix 2. Hilbert Transform 481Appendix 3. Cepstral Analysis 483A3.1. Introduction 483A3.2. Definitions 483A3.3. Example of echo suppression 484A3.4. General case 485Appendix 4. Eigenfunctions of an Operator 487A4.1. Eigenfunctions of an operator 487A4.2. Self-adjoint operator 487A4.2.1. Eigenfunctions 487A4.2.2. Expression of a function of f using an eigenfunction basis-set 488Bibliography 489Index 497