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Interfaces are present in most fluid mechanics problems. They not only denote phase separations and boundary conditions, but also thin flames and discontinuity waves. Fluid Mechanics at Interfaces 3 firstly positions models as relative to applications (i.e. pollution, drops for propulsion, wind power, etc.), then emphasizes the importance of social consequences.
Chapter 1 examines the questions raised by simulation of a pollutant's concentration degradation in permanent 2D flow using the finite element method. Chapter 2 considers an approximate analytical solution for mixed injection regimes, which acts on drop vaporization frequency response. Chapter 3 examines the case of an incompressible external flow of uniform speed at infinity, leading the liquid in the drop by friction. Chapter 4 gives a summary of combustion-based weapons and their effects. Chapter 5 then looks at the shifting interface in spacetime. Chapter 6 limits itself to two key concepts: the first is that of capillary interfaces where surface tension is present even at equilibrium, the second is that of thin flames which only exist outside of equilibrium, but which can be considered as generalized interfaces. Chapter 7 challenges the idea of constituents of matter, leading to radically transforming chemistry. Chapter 8 is concerned by the modeling of partial wetting by macroscopic approach in discrete mechanics. Chapter 9 states a numerical method of finished differences, making it possible to calculate the variables describing an average flow. Chapter 10 considers circulation in the vessels of the human body. Chapter 11 contributes by generalizing the classical series solution for initial boundary value problems of the 1D reaction-diffusion equations on any finite interval of the real line.
Roger Prud'homme is Emeritus Research Director at CNRS, France. His most recent research topics have included flames, two-phase flows and the modeling of fluid interfaces.
Stéphane Vincent is Professor at Gustave Eiffel University, France. He leads the Heat and Mass Transfer team of the MSME laboratory. His research focuses on models and numerical methods for multiphase flows.
Preface xiRoger PRUD'HOMME, Stéphane VINCENT, Christian CHAUVEAU, Mahouton Norbert HOUNKONNOU and Kwassi ANANI
Chapter 1. Simulation of a Pollutant's Concentration Degradation in Permanent Two-Dimensional Flow Using the Finite Element Method 1Deo MWELWA, David ILUNGA and Jean-Paul KATOND MBAY
1.1. Introduction 1
1.2. Behavior modeling. 2
1.3. Evolution of the gas plume concentration 8
1.3.1. Definition of the variational problem 9
1.3.2. Definitions of initial and boundary conditions 9
1.3.3. Problem data 10
1.4. Results and interpretation 10
1.5. Conclusion and perspective 13
1.6. References 13
Chapter 2. Drop Vaporization Frequency Response: An Approximate Analytical Solution for Mixed Injection Regimes 15Kwassi ANANI, Roger PRUD'HOMME and Mahouton Norbert HOUNKONNOU
2.1. Introduction 15
2.2. Stabilized state description 16
2.2.1. General assumptions 16
2.2.2. Characteristic times 18
2.2.3. Unperturbed state equations 18
2.3. Linear analyses for small perturbations 20
2.3.1. Linear analysis of the liquid-phase equations 20
2.3.2. Gas-phase linearized equations 23
2.3.3. Mass response factor 24
2.4. Results and discussion 24
2.5. Conclusion 28
2.6. References 28
Chapter 3. Considerations about the Hill Vortex 31Roger PRUD'HOMME
3.1. Introduction 33
3.2. Spherical liquid drop subjected to a uniform external flow at infinity 34
3.2.1. Incompressible fluids in spherical coordinates 35
3.2.2. Flow inside the sphere 40
3.3. Flows of a spherical liquid drop subjected to an axial thermal gradient 42
3.3.1. Presentation of the problem 42
3.3.2. Thermo-capillary Hill vortex 45
3.3.3. Other thermo-capillary flows 47
3.4. Conclusion 49
3.5. Appendices 50
3.5.1. Appendix 1: reminders on 3D irrotational flows of perfect incompressible fluids: source, sink, doublet, revolution flows 50
3.5.2. Stationary irrotational revolution flows 52
3.5.3. Examples 53
3.5.4. Appendix 2: rotational flows of perfect incompressible fluids 55
3.5.5. Appendix 3: details of calculations valid inside the sphere 57
3.5.6. Appendix 4: Legendre polynomials and spherical harmonics 59
3.6. References 62
Chapter 4. The Fire of Weapons 65Roger PRUD'HOMME
4.1. Introduction 65
4.2. The "flame" weapon 67
4.2.1. The flamethrower 67
4.2.2. The thermobaric weapon 68
4.3. The bombs 69
4.3.1. Napalm 69
4.3.2. Explosive devices 71
4.3.3. Phosphorus bombs 72
4.4. Missiles 73
4.5. The Space War 73
4.6. Weapons and pollution 74
4.7. Conclusion 75
4.8. Appendices 75
4.8.1. Appendix 1: reflections on the context and meaning of this chapter 75
4.8.2. Appendix 2: additional information 76
Chapter 5. Shifting Interface in Spacetime 81Roger PRUD'HOMME
5.1. Introduction 86
5.2. Reminder concerning interfaces in 3D coordinates 87
5.2.1. Volume balances 87
5.2.2. Balances at the interfaces 88
5.3. Minkowski volume balance equations in spacetime 90
5.3.1. General (special relativity, spacetime) 90
5.3.2. 4D balances in the volume for mass-type quantities 91
5.3.3. Electromagnetic quad tensors in the volume 92
5.3.4. 4D momentum-energy balance in the volume 94
5.4. The balance equations at the interface in the Minkowski spacetime 94
5.4.1. Electromagnetic four-tensors 95
5.4.2. Momentum-energy balance at the interface 97
5.5. Conclusion 98
5.6. Appendices 98
5.6.1. Appendix 1: presentation of the Lorentz transformation 98
5.6.2. Appendix 2: application of the Lorentz transformation into electromagnetic quantities 100
5.6.3. Appendix 3: the laws of behavior at the interface 100
5.6.4. Appendix 4: application examples 102
5.7. References 102
Chapter 6. The Interest of Microgravity for the Study of Fluid Interfaces 105Roger PRUD'HOMME and Kwassi ANANI
6.1. Introduction 106
6.2. Instabilities between two bunk fluids in the presence of hair tension 107
6.2.1. Presentation of the problem 107
6.2.2. Rayleigh-Taylor instability 109
6.2.3. Kelvin-Helmholtz instability 110
6.2.4. The general case 112
6.3. Interest of the microgravity in the presence of capillarity: influence of gravity on the structure and dynamics of the foam 114
6.4. Influence of gravity on flames 118
6.5. Conclusion 121
6.6. References 122
Chapter 7. Fire-Air: A Story of the Flame 125Roger PRUD'HOMME, Mahouton Norbert HOUNKONNOU and Guillaume LEGROS
7.1. Introduction 125
7.2. The phlogiston theory 127
7.3. The discovery of "air to fire" 129
7.3.1. Oxygen 130
7.4. From Lavoisier to Mendeleev 132
7.4.1. Thermodynamics 136
7.4.2. Mechanics 137
7.5. Conclusion 138
7.6. References 138
Chapter 8. Modelization of Partial Wetting by Macroscopic Approach in Discrete Mechanics 141Jean-Paul CALTAGIRONE and Roger PRUD'HOMME
8.1. Introduction 141
8.2. Framework of discrete mechanics 143
8.2.1. Maxwell's local frame of reference 143
8.2.2. Equivalence of conservation of acceleration and energy 146
8.2.3. Discrete law of motion 148
8.3. Concepts of capillary motion 150
8.3.1. Capillary motion law 150
8.3.2. Modelization of partial wetting 153
8.4. Analyses of some capillary flows 156
8.4.1. Spreading and equilibrium of a drop 156
8.4.2. Capillary rise between two vertical planes 159
8.5. Conclusions 161
8.6. References 162
Chapter 9. Numerical Simulation of an Average Flow in an Inter-Blade Channel of a Horizontal Axis Wind Mill 165Delphin TOMBORAVO, Roger VONY, Francis RAVELOSON and Tsialefitry ALY SAANDY
9.1. Introduction 165
9.2. Mathematical fluid motion modeling 166
9.2.1. Simplifying assumptions 166
9.2.2. Writing equations of motion in a moving coordinate system (E) 167
9.3. Definition of average flow 169
9.3.1. Definitions of the averages used 170
9.4. Writing the equations of motion of a mean flow in an inter-blade channel 172
9.4.1. Expressions of the equations of motion of the mean flows on the hub and on the crankcase 175
9.4.2. Expressions of the equations of motion of the mean flows on the upper surface and on the lower surface 179
9.4.3. Expressions of the equations of motion of flows on the corners 184
9.5. Calculating the drive torque of the wheel of a wind turbine 186
9.6. Different power expression and efficiency calculation 190
9.6.1. Available power 190
9.6.2. Starting power 190
9.6.3. Mechanical power 190
9.6.4. Performance 190
9.7. Numerical resolution method used 191
9.7.1. Mesh 191
9.7.2. Discretization scheme 191
9.7.3. Spatial discretization scheme 191
9.7.4. Synthesis 192
9.8. Application of the method 195
9.8.1. System description 195
9.8.2. Calculation process 197
9.9. Results and interpretation 198
9.9.1. Initial flow velocity within the inter-blade channel 198
9.9.2. Convergence of calculations based on NCO 198
9.9.3. Performances 199
9.9.4. Interpretation of the results obtained 201
9.10. Conclusion and perspective 201
9.11. References 202
Chapter 10. Circulation in the Vessels of the Human Body 205Roger PRUD'HOMME, François BOUSTANI and Stéphane VINCENT
10.1. Introduction 207
10.2. Blood circulation 207
10.2.1. A brief history of blood circulation 207
10.3. The heart, engine of movement 211
10.3.1. Cardiac muscle 211
10.3.2. Nodal tissue 211
10.3.3. Coronaries 212
10.3.4. The cardiac cycle: details on diastoles and systoles 213
10.3.5. The electrocardiogram 214
10.4. The fluids 215
10.4.1. Blood 215
10.4.2. Other fluids 217
10.5. Blood vessels 217
10.5.1. Arteries and veins 218
10.5.2. Capillary vessels 219
10.6. Fluid mechanics and the human body 221
10.6.1. Basic equations 222
10.6.2. Bernoulli's equations 222
10.6.3. The one-dimensional approximation 224
10.6.4. Viscous fluid problems 225
10.6.5. Multiphase fluids 228
10.6.6. Recent developments, viscoelastic fluids 229
10.7. Resolutions 230
10.8. Conclusion 231
10.9. Acknowledgements 231
10.10. References 231
Chapter 11. Series Solutions to Boundary Value Problems for Reaction-Diffusion Equations on Finite Intervals: Application to Stefan Problems 235Edoh TOSSOU, Kwassi ANANI and Roger PRUD'HOMME
11.1. Introduction 235
11.2. The reaction-diffusion equation 236
11.3. The Fourier decomposition method 237
11.3.1. Homogeneous equation and boundary conditions 237
11.3.2. Non-homogeneous problem and auxiliary functions 241
11.4. Applications 246
11.4.1. Practical implementation of the method 246
11.4.2. A two-sided contracting Stefan problem 247
11.5. Conclusion 250
11.6. References 250
List of Authors 253
Index 255
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