Vapor-Liquid Interfaces, Bubbles and Droplets (eBook)

Preface 5
Contents 8
1 Significance of Molecular and Fluid-Dynamic Approaches to Interface Phenomena 13
1.1 Vapor--Liquid Interface and Kinetic Boundary Condition (KBC) 13
1.2 Why Are Measurements of ae and ac So Difficult? 18
1.2.1 Unsteady Nonequilibrium Condensation Induced by Shock Wave Reflection 18
1.2.2 Temporal Transition Phenomenon of InterfaceDisplacement 22
1.2.3 Mechanism of Temporal Transition Phenomenon 23
1.3 Realization of Nonequilibrium States 25
1.3.1 Another Prerequisition and Shock Wave 25
1.3.2 Previous Studies of Condensation by Shock Wave 26
1.4 Constitution of This Book 27
References 28
2 Kinetic Boundary Condition at the Interface 30
2.1 Microscopic Description of Molecular Systems 30
2.1.1 Equation of Motion 32
2.1.2 Liouville Equation 34
2.1.3 Definitions of Macroscopic Variables and Equations in Fluid Dynamics 35
2.2 Molecular Dynamics Simulation 42
2.2.1 Lennard-Jones Potential and Normalization of Variables 42
2.2.2 Finite Difference Method 44
2.2.3 Example: Vapor--Liquid Equilibrium State 46
2.3 Kinetic Theory of Gases 49
2.3.1 Boltzmann Equation 50
2.3.2 Boundary Condition for the Boltzmann Equation 54
2.4 Kinetic Boundary Condition 56
2.4.1 Evaporation into Vacuum 57
2.4.2 Evaporation Coefficient 60
2.4.3 Condensation Coefficient and KBC in Weak CondensationStates 63
2.5 Asymptotic Analysis of Weak Condensation State of Methanol 65
2.5.1 Problem and Formulation 66
2.5.2 Asymptotic Analysis for Small Knudsen Numbers 69
2.5.3 Boundary Condition for the Equations in Fluid-DynamicsRegion 72
2.5.4 Condensation Coefficient as a Linear Function of Mass Flux 75
2.6 Criticism on Hertz--Knudsen--Langmuir and Schrage Formulas 77
References 78
3 Methods for the Measurement of Evaporation and Condensation Coefficients 81
3.1 Review of ae, ac, KBC, and Gaussian–BGK Boltzmann Equation 81
3.1.1 Definitions of ae and ac 81
3.1.2 Extension of Monatomic Version of KBCto Polyatomic One 82
3.1.3 KBC Expressed by Net Mass Flux Measured at theInterface 86
3.1.4 Gaussian--BGK Boltzmann Equation in Moving Coordinate System 87
3.2 Shock Tube Method for Measurement of Condensation Coefficient 88
3.2.1 Principle of Shock Tube Method 88
3.2.2 Characteristics of Film Condensation at Endwall behind Reflected Shock Wave 90
3.2.3 Mathematical Modeling of Film Condensation on Shock Tube Endwall 92
3.2.4 Boundary Condition at Infinity in Vapor 94
3.2.5 Heat Conduction in Liquid Film and Shock Tube Endwall 94
3.2.6 Initial Conditions 95
3.3 Shock Tube 96
3.3.1 Schematic and Performance of Shock Tube 96
3.3.2 Effect of Noncondensable Gases on Liquid Film Growth 97
3.3.3 Effect of Association of Molecules on Vapor State 98
3.4 Optical Interferometer 99
3.4.1 Theory of Optical Interferometer 99
3.4.2 Method of Optical Data Analysis 102
3.5 Properties of Adsorbed Liquid Film on Optical Glass Surface 103
3.5.1 Treatment of Optical Glass 103
3.5.2 Thickness of Temporarily Adsorbed Liquid Film 104
3.5.3 Refractive Index of Initially Adsorbed Liquid Film 105
3.6 Deduction of Condensation Coefficient 106
3.6.1 Typical Output Examples of Energy Reflectance 106
3.6.2 Time Changes of Liquid Film Thickness 108
3.6.3 Propagation Process of Shock Waves 110
3.6.4 Time Changes of Macroscopic Quantities and Condensation Coefficient 111
3.6.5 Values of ae and ac for Water and Methanol 113
3.7 Sound Resonance Method for Measurement of Evaporation Coefficient 116
References 118
4 Vapor Pressure, Surface Tension, and Evaporation Coefficient for Nanodroplets 120
4.1 Significance of Molecular Dynamics Analysis for Nanodroplets 120
4.2 Method of MD Simulations 122
4.3 Computational Method of Pressures 124
4.4 Equilibrium States of Nanodroplets and Planar Liquid Films 125
4.4.1 General Explanation 125
4.4.2 Density Distributions 125
4.4.3 Pressure Distributions 129
4.4.4 Differentiability of Normal Pressure with Respectto Radial Coordinate 132
4.4.5 Laplace Equation and Surface Tension 133
4.4.6 Kelvin Equation 135
4.4.7 Tolman Equation 138
4.5 Mass Transport Across Nanodroplet Surface 139
4.5.1 Problem Statement 139
4.5.2 Evaporation and Condensation Coefficients, and Mass Transfer Rate 140
4.5.3 Vacuum Evaporation Simulations 141
4.5.4 Mass Fluxes and Evaporation Coefficient 142
References 149
5 Dynamics of Spherical Vapor Bubble 151
5.1 Fluid-dynamic Definition of Interface 151
5.2 Kinematics of Interface 153
5.2.1 Interface Velocity 153
5.2.2 Interface Curvature 153
5.2.3 Time Variation of Area of Surface Element 155
5.2.4 Surface Divergence 158
5.2.5 Equilibrium Thermodynamics of the Interface 160
5.3 General Conservation Equation at Interface 161
5.3.1 Conservation Equations in Bulk Fluids 161
5.3.2 Conservation Equation in Frame Moving with Interface 162
5.3.3 Integration Form of Conservation Equation 163
5.3.4 Flux Balance on Interface 164
5.3.5 Conservation of Mass on Interface 165
5.3.6 Conservation of Momentum on Interface 167
5.3.7 Conservation of Energy on Interface 169
5.4 Spherical Vapor Bubble 170
5.4.1 Governing Equations for Spherical Bubble 171
5.4.2 Simplification 173
5.4.3 Boundary Conditions 176
5.5 Practical Description of Bubble Motion 179
5.5.1 Flow Fields in Liquid 180
5.5.2 Uniform Pressure in Bubble Interior 180
5.5.3 Temperature, Pressure, and Velocity Fields 182
5.5.4 Boundary Conditions of Temperature Field 183
5.6 Temperature Field of Bubble Exterior 184
5.6.1 Lagrangian Formulation 184
5.6.2 Transformation of Variables 185
5.6.3 Laplace Transform of Heat Equation 187
5.6.4 Inverse Laplace Transform of Heat Equation 189
5.6.5 Liquid Temperature at Bubble Wall 194
5.6.6 Gradient of Liquid Temperature at Bubble Wall 196
5.7 Temperature Field of Bubble Interior 197
5.7.1 Adiabatic Solution 198
5.7.2 Lagrangian Formulation 199
5.7.3 Boundary Layer Solution 199
5.7.4 Solution of Heat Equation 201
5.7.5 Pressure and Velocity 204
5.8 Structure of Mathematical Model 205
5.9 Bubble Expansion with Uniform Interior 207
5.9.1 Assumptions 207
5.9.2 Governing Equations and Conditions 208
5.9.3 Heat Equation for Liquid 210
5.9.4 Solution of Heat Equation 211
5.9.5 Asymptotic Growth of Vapor Bubble 214
5.9.6 Bubble Motion Coupled with Heat Conduction 216
References 217
Appendix A Vectors, Tensors, and Their Notations 219
A.1 Scalar, Vector, and Tensor 219
A.2 Einstein Summation Convention 220
Appendix B Equations in Fluid Dynamics 223
B.1 Conservation Equations 223
B.2 Conservation Equations in Component Forms 226
Appendix C Supplements to Chapter 5 227
C.1 Generalized Stokes Theorem 227
C.2 Characteristic Time of Heat Conduction 229
C.3 Abel's Integral Equation 231
Index 233