Fluid Mechanics at Interfaces 3

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 ANANIChapter 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 MBAY1.1. Introduction 11.2. Behavior modeling. 21.3. Evolution of the gas plume concentration 81.3.1. Definition of the variational problem 91.3.2. Definitions of initial and boundary conditions 91.3.3. Problem data 101.4. Results and interpretation 101.5. Conclusion and perspective 131.6. References 13Chapter 2. Drop Vaporization Frequency Response: An Approximate Analytical Solution for Mixed Injection Regimes 15Kwassi ANANI, Roger PRUD'HOMME and Mahouton Norbert HOUNKONNOU2.1. Introduction 152.2. Stabilized state description 162.2.1. General assumptions 162.2.2. Characteristic times 182.2.3. Unperturbed state equations 182.3. Linear analyses for small perturbations 202.3.1. Linear analysis of the liquid-phase equations 202.3.2. Gas-phase linearized equations 232.3.3. Mass response factor 242.4. Results and discussion 242.5. Conclusion 282.6. References 28Chapter 3. Considerations about the Hill Vortex 31Roger PRUD'HOMME3.1. Introduction 333.2. Spherical liquid drop subjected to a uniform external flow at infinity 343.2.1. Incompressible fluids in spherical coordinates 353.2.2. Flow inside the sphere 403.3. Flows of a spherical liquid drop subjected to an axial thermal gradient 423.3.1. Presentation of the problem 423.3.2. Thermo-capillary Hill vortex 453.3.3. Other thermo-capillary flows 473.4. Conclusion 493.5. Appendices 503.5.1. Appendix 1: reminders on 3D irrotational flows of perfect incompressible fluids: source, sink, doublet, revolution flows 503.5.2. Stationary irrotational revolution flows 523.5.3. Examples 533.5.4. Appendix 2: rotational flows of perfect incompressible fluids 553.5.5. Appendix 3: details of calculations valid inside the sphere 573.5.6. Appendix 4: Legendre polynomials and spherical harmonics 593.6. References 62Chapter 4. The Fire of Weapons 65Roger PRUD'HOMME4.1. Introduction 654.2. The "flame" weapon 674.2.1. The flamethrower 674.2.2. The thermobaric weapon 684.3. The bombs 694.3.1. Napalm 694.3.2. Explosive devices 714.3.3. Phosphorus bombs 724.4. Missiles 734.5. The Space War 734.6. Weapons and pollution 744.7. Conclusion 754.8. Appendices 754.8.1. Appendix 1: reflections on the context and meaning of this chapter 754.8.2. Appendix 2: additional information 76Chapter 5. Shifting Interface in Spacetime 81Roger PRUD'HOMME5.1. Introduction 865.2. Reminder concerning interfaces in 3D coordinates 875.2.1. Volume balances 875.2.2. Balances at the interfaces 885.3. Minkowski volume balance equations in spacetime 905.3.1. General (special relativity, spacetime) 905.3.2. 4D balances in the volume for mass-type quantities 915.3.3. Electromagnetic quad tensors in the volume 925.3.4. 4D momentum-energy balance in the volume 945.4. The balance equations at the interface in the Minkowski spacetime 945.4.1. Electromagnetic four-tensors 955.4.2. Momentum-energy balance at the interface 975.5. Conclusion 985.6. Appendices 985.6.1. Appendix 1: presentation of the Lorentz transformation 985.6.2. Appendix 2: application of the Lorentz transformation into electromagnetic quantities 1005.6.3. Appendix 3: the laws of behavior at the interface 1005.6.4. Appendix 4: application examples 1025.7. References 102Chapter 6. The Interest of Microgravity for the Study of Fluid Interfaces 105Roger PRUD'HOMME and Kwassi ANANI6.1. Introduction 1066.2. Instabilities between two bunk fluids in the presence of hair tension 1076.2.1. Presentation of the problem 1076.2.2. Rayleigh-Taylor instability 1096.2.3. Kelvin-Helmholtz instability 1106.2.4. The general case 1126.3. Interest of the microgravity in the presence of capillarity: influence of gravity on the structure and dynamics of the foam 1146.4. Influence of gravity on flames 1186.5. Conclusion 1216.6. References 122Chapter 7. Fire-Air: A Story of the Flame 125Roger PRUD'HOMME, Mahouton Norbert HOUNKONNOU and Guillaume LEGROS7.1. Introduction 1257.2. The phlogiston theory 1277.3. The discovery of "air to fire" 1297.3.1. Oxygen 1307.4. From Lavoisier to Mendeleev 1327.4.1. Thermodynamics 1367.4.2. Mechanics 1377.5. Conclusion 1387.6. References 138Chapter 8. Modelization of Partial Wetting by Macroscopic Approach in Discrete Mechanics 141Jean-Paul CALTAGIRONE and Roger PRUD'HOMME8.1. Introduction 1418.2. Framework of discrete mechanics 1438.2.1. Maxwell's local frame of reference 1438.2.2. Equivalence of conservation of acceleration and energy 1468.2.3. Discrete law of motion 1488.3. Concepts of capillary motion 1508.3.1. Capillary motion law 1508.3.2. Modelization of partial wetting 1538.4. Analyses of some capillary flows 1568.4.1. Spreading and equilibrium of a drop 1568.4.2. Capillary rise between two vertical planes 1598.5. Conclusions 1618.6. References 162Chapter 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 SAANDY9.1. Introduction 1659.2. Mathematical fluid motion modeling 1669.2.1. Simplifying assumptions 1669.2.2. Writing equations of motion in a moving coordinate system (E) 1679.3. Definition of average flow 1699.3.1. Definitions of the averages used 1709.4. Writing the equations of motion of a mean flow in an inter-blade channel 1729.4.1. Expressions of the equations of motion of the mean flows on the hub and on the crankcase 1759.4.2. Expressions of the equations of motion of the mean flows on the upper surface and on the lower surface 1799.4.3. Expressions of the equations of motion of flows on the corners 1849.5. Calculating the drive torque of the wheel of a wind turbine 1869.6. Different power expression and efficiency calculation 1909.6.1. Available power 1909.6.2. Starting power 1909.6.3. Mechanical power 1909.6.4. Performance 1909.7. Numerical resolution method used 1919.7.1. Mesh 1919.7.2. Discretization scheme 1919.7.3. Spatial discretization scheme 1919.7.4. Synthesis 1929.8. Application of the method 1959.8.1. System description 1959.8.2. Calculation process 1979.9. Results and interpretation 1989.9.1. Initial flow velocity within the inter-blade channel 1989.9.2. Convergence of calculations based on NCO 1989.9.3. Performances 1999.9.4. Interpretation of the results obtained 2019.10. Conclusion and perspective 2019.11. References 202Chapter 10. Circulation in the Vessels of the Human Body 205Roger PRUD'HOMME, François BOUSTANI and Stéphane VINCENT10.1. Introduction 20710.2. Blood circulation 20710.2.1. A brief history of blood circulation 20710.3. The heart, engine of movement 21110.3.1. Cardiac muscle 21110.3.2. Nodal tissue 21110.3.3. Coronaries 21210.3.4. The cardiac cycle: details on diastoles and systoles 21310.3.5. The electrocardiogram 21410.4. The fluids 21510.4.1. Blood 21510.4.2. Other fluids 21710.5. Blood vessels 21710.5.1. Arteries and veins 21810.5.2. Capillary vessels 21910.6. Fluid mechanics and the human body 22110.6.1. Basic equations 22210.6.2. Bernoulli's equations 22210.6.3. The one-dimensional approximation 22410.6.4. Viscous fluid problems 22510.6.5. Multiphase fluids 22810.6.6. Recent developments, viscoelastic fluids 22910.7. Resolutions 23010.8. Conclusion 23110.9. Acknowledgements 23110.10. References 231Chapter 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'HOMME11.1. Introduction 23511.2. The reaction-diffusion equation 23611.3. The Fourier decomposition method 23711.3.1. Homogeneous equation and boundary conditions 23711.3.2. Non-homogeneous problem and auxiliary functions 24111.4. Applications 24611.4.1. Practical implementation of the method 24611.4.2. A two-sided contracting Stefan problem 24711.5. Conclusion 25011.6. References 250List of Authors 253Index 255