Vapor-Liquid Interfaces, Bubbles and Droplets: Fundamentals and Applications

Heat and Mass Transfer Series Editors: D. Mewes and F. Mayinger

For further volumes: http://www.springer.com/series/4247

Shigeo Fujikawa · Takeru Yano · Masao Watanabe

Vapor-Liquid Interfaces, Bubbles and Droplets Fundamentals and Applications

With 74 Figures

123

Prof. Shigeo Fujikawa Hokkaido University Dept. Mechanical & Space Engineering Kita 13, Nishi 8 Sapporo 060-8628 Japan [email protected]

Prof. Takeru Yano Osaka University Dept. Mechanical Engineering Yamada-oka 2-1 Suita 565-0871 Japan [email protected]

Prof. Masao Watanabe Hokkaido University Dept. Mechanical & Space Engineering Kita 13, Nishi 8 Sapporo 060-8628 Japan [email protected]

ISSN 1860-4846 e-ISSN 1860-4854 ISBN 978-3-642-18037-8 e-ISBN 978-3-642-18038-5 DOI 10.1007/978-3-642-18038-5 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011923076 c Springer-Verlag Berlin Heidelberg 2011  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar S.L., Heidelberg Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This book is the outgrowth of work done over 30 years by the first author’s group in the departments of Mechanical Engineering at Kyoto University, Mechanical Systems Engineering at Toyama Prefectural University, and Mechanical and Space Engineering at Hokkaido University. The work is concerned with basics of evaporation and condensation at the vapor–liquid interface where the bulk vapor phase and the bulk liquid phase of the same molecules coexist side by side. It focuses on physical understanding and mathematical description of interfacial phenomena in length scales ranging from a molecular size to a usual fluid-dynamic one, such as kinetic and fluid-dynamic boundary conditions including the evaporation and condensation coefficients, vapor pressure and surface tension for nanodroplets, and applications of fluid-dynamic boundary conditions to vapor bubble dynamics. The meaning and significance of subjects to be discussed in the book are described in some detail in Chap. 1. It is needless to say that the evaporation and condensation are of paramount importance in various fields of engineering, physics, chemistry, meteorology, and oceanography. As examples of current topics related to the evaporation and condensation, we can refer to flows around aircraft in clouds, bubble formation in liquid fuels of rockets, vapor explosion in nuclear reactors and volcanoes, vapor bubble formation in LNG transport process, heterogeneous reaction on droplet and aerosol surfaces in the atmosphere, and so on. The crucial point in these problems can be attributed to boundary conditions at the interface for both the Boltzmann equation and the set of Navier–Stokes equations. It was 2005 when a kinetic boundary condition (KBC) for the Boltzmann equation was formulated in a physically correct form. However, accurate values of the evaporation and condensation coefficients for any vapors have not been determined up to now, and therefore, we can not obtain still now physically correct solutions to these problems in theoretical and numerical ways. Historically, since the end of nineteen century, it has been known that the evaporation or condensation process requires the kinetic theory of gases for its analysis, and numerous investigations have been made by the kinetic approach, resulting in various fruits. However, in 1990s, it has been recognized that the kinetic theory of gases on the evaporation or condensation further needs microscopic information of molecules at the interface, e.g., correct KBC, exact values of the evaporation and condensation coefficients included in the KBC. Since then, molecular dynamics (MD) has received much v

vi

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attention for simulation of the evaporation and condensation, and become a powerful tool to get microscopic information of the interface at atomic and molecular levels. The authors have engaged in investigation of the evaporation and condensation at the interface by using their unique methodology based on MD, molecular gas dynamics, and shock wave. Using MD, they have made numerical simulations of molecular motions in domains consisting of the bulk vapor of argon, its liquid, and the planar interface between them, and thereby formulated the physically correct KBC. Furthermore, using shock waves, they have made experiments of condensation for methanol and water vapors in nanometer and microsecond scales and deduced values of the evaporation and condensation coefficients of these materials by the aid of the polyatomic version of the Gaussian–BGK Boltzmann equation, a governing equation in molecular gas dynamics. The authors try to describe contents dealt with in this book as precisely as possible by restricting them to only their own work and to connect tightly them ranging from the microscopic to macroscopic scales. The evaporation or condensation phenomenon in the three space domains with utterly different length scales is analyzed by means of MD, the Gaussian–BGK Boltzmann equation, and the set of Navier– Stokes equations. Matching methods among the domains or the different governing equations are presented, and the reasonable matching between the microscopic and macroscopic scales is carried out to give the closed forms of the boundary conditions for both the Gaussian–BGK Boltzmann equation and the set of Navier–Stokes equations. A set of boundary conditions for the latter is applied to dynamics of a single vapor bubble in liquids as an application. However, the authors must say that they had to restrict the problems on the boundary conditions and the evaporation and condensation coefficients to only a single-component vapor–liquid two-phase system and to weak evaporation or condensation because of overwhelming difficulties of the problems. A two-phase system consisting of a liquid and its vapor-noncondensable gas mixture is of importance in engineering applications. However, the derivation of physically correct kinetic and fluid-dynamic boundary conditions have not been accomplished and these are under development. Problems of such a system as well as strong evaporation or condensation are left as challenging subjects in the future. The contributions to the chapters of this book are as follows: S. Fujikawa to Chaps. 1, 3, and 4; T. Yano to Chap. 2, and Appendices A and B; M. Watanabe to Chap. 5 and Appendix C. In writing this book, the authors are indebted to the following colleagues, their former Ph. D students; Prof. T. Ishiyama has contributed to Chap. 2 as his Ph. D work, Prof. K. Kobayashi to Chap. 3 as his Ph. D work, Dr. H. Yaguchi to Chap. 4 as his Ph. D work, and Drs S. Nakamura and M. Inaba partly to Chap. 3 as their Ph. D works. Without their contributions, this book might not have been born. The authors would also like to appreciate helps of Mr. Y. Nozaki, the technician in the first author’s laboratory, for his fine technique in making the experimental apparatuses, Miss K. Itagaki for typing the manuscripts, and Mrs. Y. Fujikawa for making fine figures. Finally, the first author would like to express his deepest gratitude to Ministry of Education, Culture, Sports, Science

Preface

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and Technology-Japan and Japan Society for the Promotion of Science for their continuous financial supports to his work on the evaporation and condensation over 30 years. Thanks to the financial supports, he could continue to do such a challenging work and accomplish his mission. Hokkaido, Japan Osaka, Japan Hokkaido, Japan November, 2010

Shigeo Fujikawa Takeru Yano Masao Watanabe

Contents

1 Significance of Molecular and Fluid-Dynamic Approaches to Interface Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Vapor–Liquid Interface and Kinetic Boundary Condition (KBC) . . . 1.2 Why Are Measurements of αe and αc So Difficult? . . . . . . . . . . . . . . . 1.2.1 Unsteady Nonequilibrium Condensation Induced by Shock Wave Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Temporal Transition Phenomenon of Interface Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Mechanism of Temporal Transition Phenomenon . . . . . . . . . . 1.3 Realization of Nonequilibrium States . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Another Prerequisition and Shock Wave . . . . . . . . . . . . . . . . . 1.3.2 Previous Studies of Condensation by Shock Wave . . . . . . . . . 1.4 Constitution of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Kinetic Boundary Condition at the Interface . . . . . . . . . . . . . . . . . . . . . . 2.1 Microscopic Description of Molecular Systems . . . . . . . . . . . . . . . . . . 2.1.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Liouville Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Definitions of Macroscopic Variables and Equations in Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Molecular Dynamics Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Lennard-Jones Potential and Normalization of Variables . . . . 2.2.2 Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Example: Vapor–Liquid Equilibrium State . . . . . . . . . . . . . . . 2.3 Kinetic Theory of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Boundary Condition for the Boltzmann Equation . . . . . . . . . . 2.4 Kinetic Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Evaporation into Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Evaporation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 6 6 10 11 13 13 14 15 16

19 19 21 23 24 31 31 33 35 38 39 43 45 46 49

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2.4.3

Condensation Coefficient and KBC in Weak Condensation States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Asymptotic Analysis of Weak Condensation State of Methanol . . . . 2.5.1 Problem and Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Asymptotic Analysis for Small Knudsen Numbers . . . . . . . . . 2.5.3 Boundary Condition for the Equations in Fluid-Dynamics Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Condensation Coefficient as a Linear Function of Mass Flux 2.6 Criticism on Hertz–Knudsen–Langmuir and Schrage Formulas . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Methods for the Measurement of Evaporation and Condensation Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Review of αe , αc , KBC, and Gaussian–BGK Boltzmann Equation . . 3.1.1 Definitions of αe and αc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Extension of Monatomic Version of KBC to Polyatomic One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 KBC Expressed by Net Mass Flux Measured at the Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Gaussian–BGK Boltzmann Equation in Moving Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Shock Tube Method for Measurement of Condensation Coefficient . 3.2.1 Principle of Shock Tube Method . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Characteristics of Film Condensation at Endwall behind Reflected Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Mathematical Modeling of Film Condensation on Shock Tube Endwall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Boundary Condition at Infinity in Vapor . . . . . . . . . . . . . . . . . 3.2.5 Heat Conduction in Liquid Film and Shock Tube Endwall . . 3.2.6 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Shock Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Schematic and Performance of Shock Tube . . . . . . . . . . . . . . . 3.3.2 Effect of Noncondensable Gases on Liquid Film Growth . . . 3.3.3 Effect of Association of Molecules on Vapor State . . . . . . . . . 3.4 Optical Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Theory of Optical Interferometer . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Method of Optical Data Analysis . . . . . . . . . . . . . . . . . . . . . . . 3.5 Properties of Adsorbed Liquid Film on Optical Glass Surface . . . . . . 3.5.1 Treatment of Optical Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Thickness of Temporarily Adsorbed Liquid Film . . . . . . . . . . 3.5.3 Refractive Index of Initially Adsorbed Liquid Film . . . . . . . . 3.6 Deduction of Condensation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Typical Output Examples of Energy Reflectance . . . . . . . . . . 3.6.2 Time Changes of Liquid Film Thickness . . . . . . . . . . . . . . . . .

52 54 55 58 61 64 66 67

71 71 71 72 76 77 78 78 80 82 84 84 85 86 86 87 88 89 89 92 93 93 94 95 96 96 98

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3.6.3 3.6.4

Propagation Process of Shock Waves . . . . . . . . . . . . . . . . . . . . Time Changes of Macroscopic Quantities and Condensation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.5 Values of αe and αc for Water and Methanol . . . . . . . . . . . . . . 3.7 Sound Resonance Method for Measurement of Evaporation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Vapor Pressure, Surface Tension, and Evaporation Coefficient for Nanodroplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Significance of Molecular Dynamics Analysis for Nanodroplets . . . . 4.2 Method of MD Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Computational Method of Pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Equilibrium States of Nanodroplets and Planar Liquid Films . . . . . . . 4.4.1 General Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Density Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Pressure Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Differentiability of Normal Pressure with Respect to Radial Coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Laplace Equation and Surface Tension . . . . . . . . . . . . . . . . . . . 4.4.6 Kelvin Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.7 Tolman Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Mass Transport Across Nanodroplet Surface . . . . . . . . . . . . . . . . . . . . 4.5.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Evaporation and Condensation Coefficients, and Mass Transfer Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Vacuum Evaporation Simulations . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Mass Fluxes and Evaporation Coefficient . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Dynamics of Spherical Vapor Bubble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Fluid-dynamic Definition of Interface . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Kinematics of Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Interface Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Interface Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Time Variation of Area of Surface Element . . . . . . . . . . . . . . . 5.2.4 Surface Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Equilibrium Thermodynamics of the Interface . . . . . . . . . . . . 5.3 General Conservation Equation at Interface . . . . . . . . . . . . . . . . . . . . . 5.3.1 Conservation Equations in Bulk Fluids . . . . . . . . . . . . . . . . . . 5.3.2 Conservation Equation in Frame Moving with Interface . . . . 5.3.3 Integration Form of Conservation Equation . . . . . . . . . . . . . . . 5.3.4 Flux Balance on Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Conservation of Mass on Interface . . . . . . . . . . . . . . . . . . . . . .

100 101 103 106 108

111 111 113 115 116 116 116 120 123 124 126 129 130 130 131 132 133 140 143 143 145 145 145 147 150 152 153 153 154 155 156 157

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5.3.6 Conservation of Momentum on Interface . . . . . . . . . . . . . . . . . 5.3.7 Conservation of Energy on Interface . . . . . . . . . . . . . . . . . . . . 5.4 Spherical Vapor Bubble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Governing Equations for Spherical Bubble . . . . . . . . . . . . . . . 5.4.2 Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Practical Description of Bubble Motion . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Flow Fields in Liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Uniform Pressure in Bubble Interior . . . . . . . . . . . . . . . . . . . . . 5.5.3 Temperature, Pressure, and Velocity Fields . . . . . . . . . . . . . . . 5.5.4 Boundary Conditions of Temperature Field . . . . . . . . . . . . . . . 5.6 Temperature Field of Bubble Exterior . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Lagrangian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Transformation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Laplace Transform of Heat Equation . . . . . . . . . . . . . . . . . . . . 5.6.4 Inverse Laplace Transform of Heat Equation . . . . . . . . . . . . . 5.6.5 Liquid Temperature at Bubble Wall . . . . . . . . . . . . . . . . . . . . . 5.6.6 Gradient of Liquid Temperature at Bubble Wall . . . . . . . . . . . 5.7 Temperature Field of Bubble Interior . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Adiabatic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Lagrangian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.3 Boundary Layer Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.4 Solution of Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.5 Pressure and Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Structure of Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Bubble Expansion with Uniform Interior . . . . . . . . . . . . . . . . . . . . . . . 5.9.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.2 Governing Equations and Conditions . . . . . . . . . . . . . . . . . . . . 5.9.3 Heat Equation for Liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.4 Solution of Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.5 Asymptotic Growth of Vapor Bubble . . . . . . . . . . . . . . . . . . . . 5.9.6 Bubble Motion Coupled with Heat Conduction . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159 161 162 163 165 168 171 172 172 174 175 176 176 177 179 181 186 188 189 190 191 191 193 196 197 199 199 200 202 203 206 208 209

Appendix A Vectors, Tensors, and Their Notations . . . . . . . . . . . . . . . . . . 211 A.1 Scalar, Vector, and Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 A.2 Einstein Summation Convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Appendix B Equations in Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 215 B.1 Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 B.2 Conservation Equations in Component Forms . . . . . . . . . . . . . . . . . . . 218

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Appendix C Supplements to Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1 Generalized Stokes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Characteristic Time of Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . C.3 Abel’s Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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219 219 221 223

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Chapter 1

Significance of Molecular and Fluid-Dynamic Approaches to Interface Phenomena

Abstract In this chapter, we introduce the fundamentals of the planar vapor–liquid interface between the bulk vapor phase and the bulk liquid phase of the same molecules, stressing some key concepts such as the transition layer between them, the Knudsen layer near the interface in the vapor region, and the boundary conditions at the interface for the Boltzmann equation and the set of Navier–Stokes equation. The reason why measurements of the evaporation and condensation coefficients in the boundary conditions have been difficult is clarified in a theoretical way. The significance of the matching among different governing dynamics, i.e., molecular dynamics (MD), molecular gas dynamics, and fluid dynamics for vapor flows near the interface is discussed to make relations among the following chapters clear.

1.1 Vapor–Liquid Interface and Kinetic Boundary Condition (KBC) In fluid dynamics and molecular gas dynamics, boundary conditions are of paramount importance because they have relevance to the drag and lift exerted on bodies, and heat and mass transport across boundaries. Especially, the boundary conditions for the interface of the bulk vapor phase and the bulk liquid phase, at which evaporation or condensation occurs, involve some difficult problems [4, 11]. This is because the derivation of the boundary conditions requires detailed information of molecular phenomena at the interface, while the governing equations such as the set of Navier–Stokes equations in fluid dynamics and the Boltzmann equation in molecular gas dynamics can be derived from macroscopic and microscopic conservation laws, respectively.1 In fact, recent studies on the boundary conditions at the interface have made significant progress by molecular dynamics (MD) simulations [8, 12, 16, 18, 24–26].

1

The set of Navier–Stokes equations is summarized in Appendix B at the end of this book, and the Boltzmann equation is discussed in Sect. 2.3.

S. Fujikawa et al., Vapor-Liquid Interfaces, Bubbles and Droplets, Heat and Mass C Springer-Verlag Berlin Heidelberg 2011 Transfer, DOI 10.1007/978-3-642-18038-5_1, 

1

2

1 Significance of Molecular and Fluid-Dynamic Approaches

ρ (kg/m3)

1000

(a = 4, b = 4) (a = 2, b = 2)

100

(a = 1, b = 1)

10 1 −2

−1

0

1 z∗

2

3

4

Fig. 1.1 Profiles of averaged density for argon at 85 K for some cases of (a, b). The dashed line denotes the saturated vapor density (ρV = 4.59 kg/m3 ) at 85 K

When the evaporation or condensation exists in the interface, the vapor near the interface is in a nonequilibrium state in the sense that the velocity distribution function of molecules deviates from the Maxwellian (the Maxwell distribution function) prescribed by a temperature of the interface,2 as will be discussed in Chap. 2. Let us first discuss the interface in a molecular level. Figure 1.1 shows profiles of averaged density ρ numerically obtained from MD simulations of argon [18]. As can be seen, the density continuously varies between the bulk liquid density ρ L and the bulk vapor one ρV . The region where the density changes is called the (density) transition layer. The parameters a and b in the figure represent the deviation from the equilibrium state. The equilibrium state corresponds to a = b = 1, where ρ V = 4.59 kg/m3 (the saturated vapor density at 85 K), and ρ L = 1410 kg/m3 . The vacuum evaporation state [16] is realized when a = 0. For a = b = 2 and 4, the vapors have higher densities than the saturated vapor density and negative velocities, which means net condensation states. Note that the compression factor p/(ρ RT ) is confirmed to be nearly unity in all cases, and hence the vapor can be regarded as an ideal gas. When the net condensation occurs, the interface moves toward the vapor phase. We therefore introduce a moving coordinate system [16], z ∗ = [z − (Z m − vs t)]/δ and vs = Js /ρ L , where Z m and δ are respectively the center position on a fixed coordinate and the 10–90 thickness (= 0.63 nm) of the transition layer, vs is the speed of the moving coordinate, t is the time from the beginning of MD simulations, and Js is the nonaveraged net mass flux across the interface. We can see that the profiles are almost flat in the range 2 < z ∗ < 4 of width 2δ in spite of the fact that the vapor is not in a local equilibrium state. This suggests that molecular collisions√rarely happen there. In fact, the Knudsen number estimated by Kn = /(2δ) = 1/[ 2π dm2 (ρ/m)2δ] is large (dm is the diameter of a molecule, m 2

According to molecular gas dynamics [29], an equilibrium state between the bulk vapor phase and the bulk liquid phase is defined as the state in which the velocity distribution function f of vapor molecules is given by the stationary Maxwellian in the coordinate system fixed at the vapor–liquid interface. Details are discussed in Sect. 2.3.1.

1.1

Vapor–Liquid Interface and Kinetic Boundary Condition (KBC)

Molecular Dynamics

Molecular Gas Dynamics

Matching

3

Fluid Dynamics

Matching

Vapor

Liquid Vapor–Liquid Interface Kinetic Boundary Condition Evaporation Coefficient? Condensation Coefficient?

Mass Momentum Energy

Transport Process?

Boundary Conditions?

Fig. 1.2 The figure shows the whole space to be considered in this book. The space consists of the bulk liquid phase, the transition layer, the planar vapor–liquid interface, the Knudsen layer, and the bulk vapor phase of the same molecules in turn from the left-hand side. The space can be classified into three regions as follows: the transition region, the nonequilibrium region, and the local equilibrium region. The three regions obey different governing equations, i.e., molecular dynamics (MD) in the transition region, the Boltzmann equation in the nonequilibrium region, and the set of Navier–Stokes equations in the local equilibrium region. Open circles represent molecules, but they are not figured in the local equilibrium region because the fluid is assumed to be there continuum. The figure is symbolically depicted in largely different scales for the three regions

is the mass of a molecule, and  is the mean free path of vapor molecules);3 if dm is replaced by the parameter σ (= 0.341 nm for argon) in the Lennard-Jones 12-6 potential,4 Kn = 20.9, 13.5, and 7.3 for a = b = 1, 2, and 4, respectively. Since the thickness δ of the transition layer is regarded as zero in the kinetic theory and the change in the vapor condition in the range 2 < z ∗ < 4 is negligible, the interface may be defined at an arbitrary position in this range. That is, the interface locates in the vapor phase adjacent to the vapor-side edge of the transition layer. We call it the kinetic interface. Hereafter, the kinetic interface will be called just the interface. As shown in Fig. 1.2, there exists a nonequilibrium region in the neighborhood of the interface in the vapor region and it is called the Knudsen layer. The extent of this layer is of the order of the mean free path of vapor molecules. The nonequilibrium behavior of the vapor in the Knudsen layer plays an important role in the evaporation or condensation. Vapor flows accompanied with the evaporation or condensation across the interface should therefore be treated by molecular gas dynamics based on the Boltzmann equation [4, 28, 29]. The Boltzmann equation then requires the kinetic boundary condition (KBC) which prescribes the velocity distribution of molecules leaving the interface for the vapor phase. When the evaporation or condensation across the interface is weak, the KBC is expressed by the product of the two-dimensional Gauss distribution with mean zero and variance RTT for the tangential components of molecular velocity and the one-dimensional Gauss distribution with mean zero and variance RTL , as will be 3

√ The mean free path is given by  = m/( 2π dm2 ρ), Eq. (2.72) in Sect. 2.3.1.

4

See Sect. 2.2.1.

4

1 Significance of Molecular and Fluid-Dynamic Approaches

explained in Chap. 2 [18]; the temperature TT is a linear function of energy flux across the interface and TL is the temperature of liquid. For the weak evaporation or condensation, TT can be regarded to be approximately equal to TL . The KBC has also a factor including the well-defined evaporation coefficient αe and condensation coefficient αc ; αe is identical with αc in the equilibrium state. This KBC reduces to the conventional KBC in the limit of the equilibrium state, i.e., TT = TL , but it does not contain any arbitrary parameter unlike the conventional KBC. The authors should note that any KBC for an arbitrarily strong evaporation or condensation has not been derived so far and its formulation is a challenging future work. There has been a long history over αe and αc since pioneering studies of Hertz [15] and Kundsen [19]. For reference, values of αe and αc of water vapor are shown in Table 1.1 for the αe -values and Table 1.2 for the αc -values; these tables are reproduced on the basis of data of αe and αc reported in Marek and Straub’s paper [23]. The recent MD simulation has succeeded in the determination of

Year

Table 1.1 The evaporation coefficient αe of water Author(s) αe

Temperature(◦ C)

1925 1931 1931 1933 1935 1939 1940 1953 1954 1954 1955 1959 1964 1964 1965 1967 1969 1971 1971 1971 1971 1973 1975 1975 1975 1975 1976

Rideal Alty Alty and Nicoll Alty Alty and Mackay Baranaev Pr¨uger Hammecke and Kappler Hickman Hickman and Torpey Kappler Fuchs Campbell Delaney et al. Mendelson and Yerazunis Maa Maa Cammenga et al. Cammenga et al. Duguid and Stampfer Tamir and Hasson Levine Davies et al. Kochurova et al. Narusawa and Springer Narusawa and Springer Bonacci et al.

25–30 5.9–32 12.1 −7.5–25 10.3–32.6 10–50 100 20 5.9–7.3 1.2 3.8–20.2 20 44.6–83.0 −0.8–4.1 38.9–78.3 0.05 0.8 24–30 18 25–35 42–50 20–28 4–19.5 25.5–34.5 18–27 18–27 2.1–8.7

1978 1987 1989 2005

Barnes ˘ Cukanov Hagen et al. Ishiyama et al. (MD)

0.0037–0.0042 0.0083–0.0155 0.0156 0.0289–0.0584 0.0061–0.0392 0.033–0.034 0.02 0.045 0.254–0.532 0.0047 0.0992–0.1015 0.03–0.034 0.0014–0.0122 0.0336–0.0545 0.0008–0.0038 1 1 0.002 0.248–0.380 0.5–1 0.10–0.30 1 1 0.050–0.065 0.038 0.19 0.065–0.665 (avg.0.54) 0.0002 0.008–0.034 0.13 1

25 39.8 16 ≈ 36

1.1

Vapor–Liquid Interface and Kinetic Boundary Condition (KBC)

5

Year

Table 1.2 The condensation coefficient αc of water vapor Author(s) αc

Temperature(◦ C)

1961 1963 1963 1964 1964 1965 1965 1967 1968 1969 1969 1971 1971 1973 1974 1974 1975 1975 1976 1976 1976 1978 1986 1987 1989 2010

Berman Nabavian and Bromley Wakeshima and Takata Goldstein Jamieson Jamieson Tanner et al. Mills and Seban Tanner et al. Maa Wenzel Magal Tamir and Hasson Vietti and Schuster Chodes et al. Gollub et al. Sinnarwalla et al. Vietti and Fastook Vietti and Fastook Bonacci et al. Finkelstein and Tamir Neizvestnyj et al. Hatamiya and Tanaka Garnier et al. Hagen et al. Fujikawa et al.

10 7–50 −16.1–5.1 25–30 0–70 – 100 7.6–10.2 22–46 0.8–8.2 22–46 25.9–82.8 48.5–105.5 – 23.9–24.9 11.4–17.5 22.5–25.7 20.8–23.2 20 5.5–7.0 60–99 20 6.9–26.9 ≈ 20–25 16 17–20

1 0.35–1 0.015–0.020 ≈ 0.1 0.305 0.35 > 0.08 0.45–1 > 0.1 1 1.0 0.040–0.044 0.09–0.35 0.21 0.031–0.037 0.010–0.012 0.021–0.032 1 0.1–1 0.417–0.693 0.006–0.060 0.3–1 0.2–0.6 0.01 0.01 1

αe -values for water [17] as shown in Table 1.1, although the simulation result is not yet verified by experiments. Concerning αc , it had not been determined accurately before the authors’ values in Table 1.2.5 We can see that the values of αe and αc largely scatter in the range of more than one hundred times. In the next section, we will consider theoretically the reason why the determination of the values of αe and αc has been so difficult. Without reliable αe and αc -values, the KBC remains open for ever. The establishment of the KBC allows us to derive a set of boundary conditions for the set of Navier–Stokes equations, i.e., a set of continuity, momentum, and energy equations in the local equilibrium region, fluid-dynamics region outside the Knudsen layer, by theoretical analysis of the Knudsen layer based on the Gaussian– BGK Boltzmann equation [1], as will be discussed in Chap. 2. The Gaussian–BGK Boltzmann equation is the only polyatomic version of the Boltzmann equation satisfying the H-theorem. For resolving the above-mentioned problems of αe , αc , and the boundary conditions for the set of Navier–Stokes equations, we have at present no any consistent law or any consistent system of governing equation. We have

5

Fujikawa et al.’s experimental values are given in Fig. 3.24 in Sect. 3.6.5.

6

1 Significance of Molecular and Fluid-Dynamic Approaches

the only theoretical method and two kinds of governing equations, i.e., MD, the Gaussian–BGK Boltzmann equation, and the set of Navier–Stokes equations. In this book, matching among MD, the Gaussian–BGK Boltzmann equation, and the set of Navier–Stokes equations will be consistently done.

1.2 Why Are Measurements of αe and αc So Difficult? In this section, we demonstrate that the difficulty in the measurement of αe and αc lies in the existence of different time scales essential for the phase change phenomena. This difficulty can be overcome by conducting the measurement of the condensation induced by an abrupt pressure elevation caused by the reflection of shock wave at the interface. We therefore start with the discussion of a shock reflection phenomenon.

1.2.1 Unsteady Nonequilibrium Condensation Induced by Shock Wave Reflection As mentioned in Sect. 1.1, the αe and αc -values measured in the past largely scatter in the range of more than one hundred times. We will here discuss the reason why these values are so different in such a wide range. The determination of αe or αc must be made through the measurement of a small amount of net mass flux of the evaporation or condensation at the interface in a nonequilibrium state; the measurement is not feasible at the equilibrium state. Such a nonequilibrium state can be realized by the following way. Let us consider the situation where the half-infinite extent of a vapor is in contact with the half-infinite extent of the liquid phase of the vapor, and these are facing each other with the plane interface between and in an equilibrium state. As shown in Fig. 1.3, a shock wave advancing from the right-hand side in the vapor collides with the interface and it is reflected, and propagating in the right-hand direction as the time elapses; the time is running upward. Just at the instant when the shock wave is reflected at the interface, the pressure, temperature, and density of the vapor increase stepwise from the initially low state to a high one. The temperature of the vapor at the interface changes little because of the large difference in heat capacities of the vapor and liquid. The Knudsen layer is formed near the interface in the vapor, and the thermal boundary layer also develops outside the Knudsen layer with the lapse of time. The vapor pressure at the interface then becomes higher than the saturated vapor pressure at the interfacial liquid temperature. As a result, the vapor becomes supersaturated at the interface, consequently condensing, and the interface moves toward right-hand side with time. The net mass flux of condensation at the interface, which we need, can be obtained from the measurement of interface movement. This problem has been solved by Fujikawa et al. [10] for the system of shock tube endwall, liquid film, and vapor on the basis of the method of matched asymptotic expansions, as will be mentioned in Sect. 3.2.2;

1.2

Why Are Measurements of αe and αc So Difficult?

7

Thermal Boundary Layer

Liquid

Vapor Flow

Interface

Time

Particle Path Trajectory of Reflected Shock Wave

O

Trajectory of Incident Shock Wave Vapor

Fig. 1.3 The propagation process of the shock wave in the vapor advancing toward and reflecting from the liquid surface. The time is running upward

the reflection of a shock wave at the shock tube endwall in a noncondensable gas has been analyzed by Clarke [5]. The problem shown in Fig. 1.3 is a simplified version of Ref. [10] and the result of its analysis can be summarized as follows. For simplicity, we will assume α = αe = αc and adopt a set of fluid-dynamic boundary conditions at the interface for the set of Navier–Stokes equations as follows [1, 28, 29]6 : p − p∗ 1 u = , √ √ ∗ ∗ p 2RTL C4 − 2 π 1−α α

(1.1)

T − TL u = d4∗ √ , TL 2RTL

(1.2)

where TL is the liquid temperature at the interface, T is the vapor temperature at the interface, p ∗ is the saturated vapor pressure at TL , p is the vapor pressure at the interface, u is the vapor velocity at the interface, R is the gas constant per unit mass, and C4∗ = −2.13204 and d4∗ = −0.44675 for Boltzmann–Krook–Welander (BKW) model; C4∗ and d4∗ are slightly different among models such as BKW and

6 This assumption holds for the weak evaporation or condensation which takes place near the equilibrium state, as will be discussed in Chaps. 2 and 3. Equations (1.1) and (1.2) are respectively Eqs. (2.138) and (2.139) in Sect. 2.5.3 for the case that the vapor flow is one-dimensional and the flow velocity is much larger than the moving velocity of the interface. And, also see Footnote 20 in Sect. 2.5.3.

8

1 Significance of Molecular and Fluid-Dynamic Approaches

hard-sphere models.7 The more general boundary conditions for polyatomic gases will be given in Chap. 2 in this book, but Eqs. (1.1) and (1.2) suffice for the present purpose. Fujikawa et al. have solved the set of one-dimensional Navier–Stokes equations of vapor and the heat conduction equation of lquid together with Eqs. (1.1) and (1.2) by the method of matched asymptotic expansions [9, 10]. The time-dependent position δ(t) of the interface from its initial one is then described at the level of the first approximation by dδ(t) = dt

√

2RTL (t) ρ∞ T∞ p∞ − p∗ (TL ) , φ(α) ρ L TL (t) p ∗ (TL )

ρL L TL (t) = T0 + kL



DL π

 0

t

dδ(t˜)/dt˜  dt˜. t − t˜

(1.3)

(1.4)

Here, t is the time measured from the instant of the step change of the state, T0 is the initial temperature of the vapor and liquid, ρ∞ and T∞ are the vapor density and temperature far from the interface behind the reflected shock wave, ρ L is the liquid density, D L is the thermal diffusivity of liquid, k L is the thermal conductivity of liquid, L is the latent heat of condensation, and φ(α) and p∗ (TL ) are respectively given by √ 1−α , φ(α) = −C4∗ + 2 π α ∗

p (TL ) =

p0∗



(1.5)

 A 1 − A + TL (t) , T0

(1.6)

where p0∗ = p ∗ (T0 ) and A = bT0 /(c + T0 )2 in which b and c are given later [Eqs. (1.9) and (1.10)]; ρ L , D L , k L , and L are constant values. Equation (1.6) is obtained from Antoine’s equation [30] by Taylor’s expansion, and p∗ (T0 ) = p0∗ as it should be so; Antoine’s equation is given below in Eq. (1.7). The saturated vapor pressure p ∗ and the saturated vapor density ρ ∗ can be obtained by Antoine’s equation and the state equation for ideal gases:  p = exp a − ∗

ρ∗ =

p∗ , RTL

b c + TL

 ,

(1.7)

(1.8)

7 The values of C ∗ and d ∗ are given in Eq. (2.134) for hard-sphere gas and Eq. (2.136) for the 4 4 BKW model.

1.2

Why Are Measurements of αe and αc So Difficult?

9

where the units of TL and p∗ are respectively (K) and (Pa), and the constant values a, b, and c are for methanol vapor a = 23.4803,

b = 3626.55,

c = −34.29,

(1.9)

b = 3816.44,

c = −46.13.

(1.10)

and for water vapor a = 23.1964,

From Eqs. (1.3), (1.4), and (1.6), we obtain a Volterra integral equation of the second kind on the displacement speed of the interface as follows: dδ(t) β2 = β1 − √ dt π



t

0

dδ(t˜)/dt˜  dt˜, t − t˜

(1.11)

where p∞ ( p∞ − p0∗ ) β1 = φ(α)ρ L p0∗



⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

2 , RT0

p∞ ( p∞ − p0∗ + 2Ap∞ )L β2 = φ(α)k L p0∗



⎪ ⎪ ⎪ ⎪ . ⎪ 3 ⎭ 2RT

(1.12)

DL

0

The solution of Eq. (1.11) can be obtained as [6] √ dδ(t) = β1 exp(β22 t)erfc(β2 t), dt

(1.13)

where √ 2 erfc(β2 t) = √ π



∞ √

β2 t

e−x dx. 2

(1.14)

Integrating Eq. (1.13) with respect to the time t leads to   √ √ β1 2 2 δ(t) = 2 exp(β2 t)erfc(β2 t) + √ β2 t − 1 . π β2

(1.15)

where δ(0) = 0. From Eqs. (1.4), (1.11), and (1.13), we obtain TL (t) = T0 +

√ √  β1 ρ L L D L  1 − exp(β22 t)erfc(β2 t) . β2 k L

(1.16)

where TL (0) = T0 . The position and temperature of the interface are respectively described by Eqs. (1.15) and (1.16).

10

1 Significance of Molecular and Fluid-Dynamic Approaches

1.2.2 Temporal Transition Phenomenon of Interface Displacement Equations (1.15) √ and (1.16) can be classified into two cases depending on values of the variable β2 t as follows: √ (1) for β2 t  18 ; δ(t) = β1 t,

(1.17)

2β1 ρ L L TL (t) = T0 + kL



DL t, π

(1.18)

√ (2) for β2 t 19 ; β1 2 β1 √ t − 2, δ(t) = √ π β2 β2

(1.19)

√ ρ L L D L β1 (= const.), TL = T0 + kL β2

(1.20)

where we should notice, for the following discussion, that β1 and β1 /β22 are dependent on φ(α), i.e., α, while β1 /β2 is independent of φ(α). The displacement of the interface is drastically influenced by√φ(α) and the time measured from the instant of in the step change of the state. For β2 t  1, the position of the interface changes √ proportion to the time and the interface speed depends on φ(α), while for β2 t 1 the position changes in proportion to the square root of the time, and its change gradually becomes independent of φ(α) as the time lapse and becomes strongly dependent on thermophysical properties of the vapor and liquid. This suggests that the position of the interface should be measured in early time stages just after the

8

For small values of x, erfc x can be expressed as follows [3]: ∞ 2  (−1)n x 2n+1 , erfc x = 1 − erf x = 1 − √ (2n + 1)n! π n=0

where erf x is the error function. 9

For large values of x, erfc x can be expressed as follows [3]: √   ∞ 1 1 1 1.3 π 1.3 . . . (2n − 3) 2 2 erfc x = e−ξ dξ = e−x − 3 + 2 5 − · · · + (−1)n−1 2 2 x 2x 2 x 2n−1 x 2n−1 x

+ (−1)n

1.3 . . . (2n − 1) 2n

∞ x

e−ξ dξ . ξ 2n 2

1.2

Why Are Measurements of αe and αc So Difficult?

11

step change caused by the shock reflection, when we determine α through the measurement of the interface displacement. It is quite natural to notice, in the above discussion, that there exists √ a transition time between the t-proportion displacement of the interface and the t-proportion √ one and that this time can be deduced from the relation β2 t = O(1). Defining the √ transition time as τt when β2 τt = 1, we obtain 2RT03 τt = DL



φ(α)k L p0∗ p∞ ( p∞ − p0∗ + 2Ap∞ )L

2 .

(1.21)

The transition time τt is in proportion to [φ(α)]2 . Generally, φ(α) approaches −C4∗ (= 2.13204 for hard-sphere gas) as α does unity, and on the other hand, φ(α) approaches infinity as α does zero. Evaluating τt for small pressure changes by 5 % from the saturated vapor pressures at 290 K for methanol and water vapors, we obtain, e.g., for α = 1, τt ∼ = 0.1 μs for methanol vapor and τt ∼ = 7 μs for water vapor, respectively. In both cases, the transition times are very short. The reason why the transition time of methanol vapor is shorter than that of water vapor is principally because the saturated vapor pressure of methanol is six times higher than that of water at 290 K; the vapor with the higher saturated pressure causes the more mass flux and the more rapid temperature rise of liquid, thereby resulting in the shorter transition time. If the liquid is a very thin film and it is on a solid wall with a thermal conductivity higher than that of the liquid, the transition time becomes a little longer (see Sect. 3.2.2). Although the restriction of transition time for the measurement of interface displacement is not so strict, we can understand that the measurement should be carried out in the time scale of microseconds, not in time scale of milliseconds or seconds. All values of αe and αc shown in Tables 1.1 and 1.2 have been measured in the past through the displacement of plane or curved liquid surface directly, or other indirect ways. However, there was no recognition of the existence of the temporal transition phenomenon in the past measurements. This is one of reasons why the values of αe and αc measured have been largely different.

1.2.3 Mechanism of Temporal Transition Phenomenon As clarified in the preceding subsection, the displacement of the interface greatly depends on the change of the saturated vapor pressure at the interfacial liquid temperature. The time evolution of the temperature is given by Eq. (1.18) for t  τt and by Eq. (1.20) for t τt . The temperature at the time τt is given by √ β1 ρ L L D L TL = T0 + 0.5724 . β2 k L

(1.22)

12

1 Significance of Molecular and Fluid-Dynamic Approaches

At this time stage, the temperature rise is about √ 57% of the temperature variation from its initial value to the asymptote [= β1 ρ L L D L /(β2 k L )]. Therefore, we can understand that the transition time is the characteristic time when the temperature approaches the asymptote over the time. Now, let us consider the balance of heat fluxes at the interface in order to understand the mechanism of the temporal transition phenomenon. The balance equation of heat fluxes per unit time and unit interface area is given by ρL L

⎫ dδ(t) ∼ ⎪ = heat conduction into liquid ⎪ ⎪ ⎬ dt ⎪ ⎪ ⎪ ⎭

TL (t) − T0 √ , ∝ t

√ ρLL DL β1 kL β2

TL

(1.23)

T0

Interface Time Lapse Liquid Space Coordinate

Vapor O

(a) Before Transition Time

√ ρLL DL β1 kL β2

TL

T0

Interface Time Lapse

Liquid Space Coordinate

Vapor O

(b) After Transition Time

Fig. 1.4 Temperature changes of liquid surface and liquid interior: (a) before transition time and (b) after transition time

1.3

Realization of Nonequilibrium States

13

√ because the thermal boundary layer in the liquid develops according to t; the heat flux term due to heat conduction in the vapor in the left-hand side of Eq. (1.23) is neglected because it is very small compared with that due to condensation, ρ L Ldδ(t)/dt [10]. For t  τt , from Eqs. (1.18) and (1.23), we obtain dδ(t)/dt = const., i.e., δ(t) ∝ t. It can therefore be concluded that the thermal process before τt is the one in which the interface temperature increases by condensation heat and the heat diffuses into the liquid interior, as shown in Fig. 1.4a. After τt , on the other hand, the interface temperature approaches the asymptote, and in consequence, from √ √ Eqs. (1.20) and (1.23), we obtain dδ(t)/dt ∝ 1/ t, i.e., δ(t) ∝ t. At this time stage, the temperature approaches the asymptote, and the heat diffuses into the liquid interior, as shown in Fig. 1.4b. Such a process of condensation after τt may be called a thermal diffusion-controlled condensation. If the measurement of α is conducted based upon the time history of delta for a long range of thermal diffusion-controlled condensation, the determined α-values easily become inaccurate. Furthermore, the determination of α in the time scale of τt is extremely difficult because of the lack of a suitable method for realization of nonequilibrium state of a vapor at the interface, except for our shock wave method. The lack of both recognition of the existence of transition time and suitable method for realization of the nonequilibrium state is the reason why the values of αe and αc have largely scattered as shown in Tables 1.1 and 1.2.

1.3 Realization of Nonequilibrium States 1.3.1 Another Prerequisition and Shock Wave In the preceding section, it has been elucidated that the displacement of the interface should be measured in the time scale of the transition time defined by Eq. (1.21), when the evaporation or condensation coefficient, i.e., αe or αc , is determined from the measurement of the net mass flux of evaporation or condensation. This is one prerequisition for determination of αe and αc . There is another prerequisition for preparation of a nonequilibrium state of a vapor. The nonequilibrium state must be realized in a time scale much shorter than the transition time, and the realization in such a short time is almost impossible by any mechanical tools. The method by shock wave discussed in the preceding section may also be the best one from the following view points: (i) The nonequilibrium state of a vapor can be realized at a vapor–liquid interface in the time scale of the mean free time of vapor molecules, when the shock wave advancing toward the liquid surface in the vapor is reflected at the surface. (ii) The nonequilibirium state can be held for a sufficiently long time compared with the transition time.

14

1 Significance of Molecular and Fluid-Dynamic Approaches

The mean free time τm is given by 1 τm = 4π(ρ/m)dm2



πm , kT

(1.24)

where k is the Boltzmann constant (k = 1.3806504 × 10−23 J/K) and T is the vapor temperature. Evaluating the values of τm for the saturated methanol and water vapors at 290 K, we obtain τm ∼ = 2 ns for methanol and τm ∼ = 8 ns for water; the values of dm for the vapors are replaced by σ00 in Ref. [17]. These values are much smaller than those of τt for both vapors, as estimated in Sect. 1.2.2. Let us here summarize the prerequisitions important for the determination of αe and αc : (i) The nonequilibrium state of the vapor must be realized in the time scale of nanoseconds. (ii) The determination of αe and αc must be performed in the time scale of microseconds. The shock wave is the useful tool satisfying these prerequisitions. Technical details of the method based on shock wave will be given in Chap. 3.

1.3.2 Previous Studies of Condensation by Shock Wave Goldstein [13] first made an experiment on condensation of water vapor on the sidewall of a shock tube and tried to measure the condensation coefficient. The condensation was produced in the following way. Behind an incident shock wave, the vapor is compressed and heated, but it is rapidly cooled because of the large difference in heat capacities between the vapor and the shock tube sidewall. An unsteady thermal boundary layer forms on the wall and it develops toward the vapor region. Under an appropriate initial condition, the vapor becomes supersaturated on the wall surface and it begins to condense on the wall. The experiment showed that filmwise or dropwise condensation took place on the sidewall depending upon the nature of wall surface. Following Goldstein, Grosse and Smith [14] also confirmed the condensation of Freon-11 vapor on the sidewall of a shock tube. The growth of a liquid film formed on the wall was, however, found to be disturbed by the presence of the viscous boundary layer on the wall behind the incident shock wave. Smith [27] improved Goldstein’s method by paying attention to film condensation on the endwall of a shock tube behind a reflected shock wave. The experiment demonstrated that a liquid film formed uniformly on the endwall and that its growth behavior might be analyzed in terms of two different models, one for time less than 10 µs and one for longer times. Except for the early stage of about 10 µs after the reflection of the shock wave, the liquid film grew in proportion to the square root of the time for time

1.4

Constitution of This Book

15

intervals of the order of milliseconds. Unfortunately, Smith was not able to deduce the condensation coefficient from the data obtained because he had no theoretical tool for its deduction. Maerefat et al. [21, 22] directed by Fujikawa followed the experimental method which Smith proposed [27], and succeeded in determining the condensation coefficients of methanol, water, and carbon tetrachloride vapors, by combining both experimental data of the liquid film growth and their theoretical analysis of the set of Naiver–Stokes equations. Later, Fujikawa et al. [7, 11] and Kobayashi et al. [20] adopted the Gaussian–BGK Boltzmann equation instead of the set of Navier–Stokes equations to analyze the reflection process of the shock wave at the shock tube endwall and the subsequent growth of the liquid film. The Gaussian–BGK Boltzmann equation was solved in a fully numerical way under the new KBC formulated with MD simulations [18]; in the analysis, the growth rate of the liquid film formed on the shock tube endwall was incorporated into the KBC, and this made it possible to fuse the experimental data and numerical ones and to deduce further microscopic information on the condensation. The most recent results of the evaporation and condensation coefficients for methanol and water vapors will be presented in Chap. 3.

1.4 Constitution of This Book In Chap. 2, we shall formulate the physically correct KBC for the Boltzmann equation on the basis of MD simulations. We demonstrate that when the evaporation or condensation across the interface is weak, the KBC is expressed by the product of a three-dimensional Gauss distribution function and a factor including the evaporation coefficient αe and condensation coefficient αc defined well. The Gaussian–BGK Boltzmann equation and the theoretical analysis of the Knudsen layer near the interface will be treated. We will derive the boundary conditions for the set of Navier– Stokes equations, i.e., the continuity, momentum, and energy equations in the fluid dynamics region outside the Knudsen layer based on the Gaussian–BGK Boltzmann equation and the KBC. In Chap. 3, the KBC formulated in Chap. 2 is extended to the case of polyatomic molecules by inclusion of the distribution function for energy associated with the internal structure of molecules. We determine αe and αc for methanol and water vapors by making use of carefully prepared shock tube experiments and precise nonequilibrium molecular gas dynamics simulations. This will lead to the substantial termination of the long lasting controversy over αe and αc . A method based on sound resonance is also presented for exact determination of αe as a complementary tool to the shock tube. In Chap. 4, we shall further extend the MD simulations conducted in Chap. 2 to nanodroplets and elucidate effects of droplet size on the surface tension, the equilibrium vapor pressure, and αe of the nanodroplets, and also clarify the applicable limit of thermodynamics in the nanodroplets. In Chap. 5, we shall derive a set of equations describing the dynamics of a spherical vapor bubble accompanied by the evaporation or condensation at the interface as an example of the application of the physically correct sets of equations

16

1 Significance of Molecular and Fluid-Dynamic Approaches

to practically important problems. The analytical method presented enables us to solve in a mathematically rigorous manner the flow fields of both internal and external the bubble, and the dynamics at the bubble wall by taking temperature and density distributions inside the bubble into account under the assumption of uniform pressure.

References 1. P. Andries, P.L. Tallec, J.P. Perlat, B. Perthame, The Gaussian-BGK model of Boltzmann equation with small Prandtl number. Eur. J. Mech. B-Fluids 19, 813–830 (2000) 2. K. Aoki, Y. Sone, Gas flows around the condensed phase with strong evaporation or condensation: Fluid dynamic equation and its boundary condition on the interface and their application, in Advances in Kinetic Theory and Continuum Mechanics, eds. by R. Gatignol, Soubbaramayer (Springer, Berlin, 1991), pp. 43–54 3. H.S. Carslaw, J.C.Jaeger, Conduction of Heat in Solids, 2nd edn. (Oxford University Press, Oxford, 1959) 4. C. Cercignani, Rarefied Gas Dynamics (Cambridge University Press, New York, NY 2000) 5. J.F. Clarke, The reflexion of a plane shockwave from a heat-conducting wall. Proc. R. Soc. Lond. A 299, 221–237 (1967) 6. A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Tables of Integral Transforms, Vol. I. (McGraw-Hill, New York, NY, 1954) 7. S. Fujikawa, Molecular transport phenomena and the kinetic boundary condition at the vaporliquid interface, in Proceedings of 7th World Conference Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, eds. by J.S. Szmyd, J. Spalek, T.A. Kowalewski (Jagiellonian University, Krakow, 2009), pp. 81–97 8. A. Frezzotti, P. Grosfils, S. Toxvaerd, Evidence of an inverted temperature gradient during evaporation/condensation of a Lennard-Jones fluid. Phys. Fluids 15, 2837–2842 (2003) 9. S. Fujikawa, M. Okuda, T. Akamatsu, T. Goto, Non-equilibrium vapour condensation on a shock-tube endwall behind a reflected shock wave. J. Fluid Mech. 183, 293–324 (1987) 10. S. Fujikawa, M. Kotani, N. Takasugi, Theory of film condensation on shock-tube endwall behind reflected shock wave: Theoretical basis for determination of condensation coefficient. JSME Int. J. 40, 159–165 (1997) 11. S. Fujikawa, Molecular gas dynamics applied to phase change processes at a vapor-liquid interface: Shock-tube experiment and MGD computation for methanol, in Proceedings of the 7th World Conference Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, eds. by J.S. Szmyd, J. Spalek, T.A. Kowalewski (Jagiellonian University, Krakow, 2009), pp. 81–97 12. A. Gilde, N. Siladke, C.P. Lawrence, Molecular dynamics simulations of water transport through butanol films. J. Phys. Chem. A 113, 8586–8590 (2009) 13. R. Goldstein, Study of water vapor condensation on shock-tube walls. J. Chem. Phys. 40, 2793–2799 (1964) 14. F.A. Grosse, W.R. Smith, Vapor condensation in a shock tube: Electrostatic effects. Phys. Fluids 11, 735–739 (1968) 15. H. Hertz, Ueber die Verdunstung der Flüssigkeiten, insbesondere des Quecksilbers, im luftleeren Raume. Ann. Phys. Chemie 17, 177–200 (1882) 16. T. Ishiyama, T. Yano, S. Fujikawa, Molecular dynamics study of kinetic boundary condition at an interface between argon vapor and its condensed phase. Phys. Fluids 16, 2899–2906 (2004) 17. T. Ishiyama, T. Yano, S. Fujikawa, Molecular dynamics study of kinetic boundary condition at an interface between polyatomic vapor and its condensed phase. Phys. Fluids 16, 4713–4726 (2004)

References

17

18. T. Ishiyama, T. Yano, S. Fujikawa, Kinetic boundary condition at a vapor-liquid interface. Phys. Rev. Lett. 95, 084504 (2005) 19. M. Knudsen, Die MaximaleVerdampfunggeschwindigkeit des Quecksilbers. Ann. Phys. Chemie 47, 697–708 (1915) 20. K. Kobayashi, S. Watanabe, D. Yamano, T. Yano, S. Fujikawa, Condensation coefficient of water in a weak condensation state. Fluid Dyn. Res. 40, 585–596 (2008) 21. M. Maerefat, T. Akamatsu, S. Fujikawa, Non-equilibrium condensation of water and carbontetrachloride vapour in a shock-tube. Exp. Fluids 9, 345–351 (1990) 22. M. Maerefat, S. Fujikawa, T. Akamatsu, T. Goto, T. Mizutani, An experimental study of non-equilibrium vapour condensation in a shock-tube. Exp. Fluids 7, 513–520 (1989) 23. R. Marek, J. Straub, Analysis of the evaporation coefficient and the condensation coefficient of water. Int. J. Heat Mass Transf. 44, 39–53 (2001) 24. R. Meland, A. Frezzotti, T. Ytrehus, B. Hafskjold, Nonequilibrium molecular-dynamics simulation of net evaporation and net condensation, and evaluation of the gas-kinetic boundary condition at the interface. Phys. Fluids 16, 223–243 (2004) 25. A. Morita, M. Sugiyama, H. Kameda, S. Koda, D.R. Hanson, Mass accommodation coefficient of water: Molecular dynamics simulation and revised analysis of droplet train/flow reactor experiment. J. Phys. Chem. B 108, 9111–9120 (2004) 26. G. Nagayama, T. Tsuruta, A general expression for the condensation coefficient based on transition state theory and molecular dynamics simulation. J. Chem. Phys. 118, 1392–1399 (2003) 27. W.R. Smith, Vapor-liquid condensation in a shock tube, in Proceedings of the 9th International. Shock Tube Symposium, eds. by B. Bershader, W. Griffith (Stanford University, Stanford, CA, 1973), pp. 785–792 28. Y. Sone, Kinetic Theory and Fluid Dynamics (Birkhäuser, Boston, MA, 2002) 29. Y. Sone, Molecular Gas Dynamics (Birkhäuser, Boston, MA, 2007) 30. The Society of Chemical Engineers of Japan (ed.), Handbook of Chemical Engineering (Maruzen, Tokyo, 1988), pp. 18–27

Chapter 2

Kinetic Boundary Condition at the Interface

Abstract The vapor–liquid interface can exist only where the bulk vapor phase and the bulk liquid phase of the same molecules coexist side by side. Therefore, all the properties of the interface are inevitably affected by the bulk liquid and vapor phases, and vice versa. The relation among these three constituents still remains unresolved in general nonequilibrium states. However, at least in a weak nonequilibrium state, the relations can be simplified and reformulated into a form of Kinetic Boundary Condition (KBC) at the vapor–liquid interface. In this chapter, from the microscopic point of view, we explain how the two bulk phases of vapor and liquid are connected via the KBC at the interface. The main tools used here are the nonequilibrium molecular dynamics simulation of vapor–liquid two-phase system and the Boltzmann equation for vapor. Our aims in this chapter are to establish the KBC at the interface by the molecular dynamics simulation and to reduce it into the boundary condition for the vapor flows in the fluid-dynamics region outside the Knudsen layer on the interface by the asymptotic analysis of the boundary-value problem of the Boltzmann equation for small Knudsen numbers.

2.1 Microscopic Description of Molecular Systems The physical properties of materials consisted of a large number of molecules are resulted from some kinds of averages over a large number of molecules, because our utilization activities of materials are usually carried out in some scales considerably large compared with molecular scales. For example, the density of a fluid is always evaluated as an averaged mass of a number of molecules in a volume divided by the volume. Consider the measurement of the temperature of a fluid by a thermometer. The motion of molecules forming the thermometer is in an equilibrium state as a result of the energy exchange with molecules in the fluid that contacts the thermometer. The total kinetic energy of molecules forming the thermometer is then translated into the temperature of thermometer, which is equal to the temperature of the fluid in the equilibrium state. In translating the kinetic energy to the temperature, we employ a fundamental relation in statistical mechanics [30]

S. Fujikawa et al., Vapor-Liquid Interfaces, Bubbles and Droplets, Heat and Mass C Springer-Verlag Berlin Heidelberg 2011 Transfer, DOI 10.1007/978-3-642-18038-5_2, 

19

20

2 Kinetic Boundary Condition at the Interface

  3 1 kT = m(ξx2 + ξ y2 + ξz2 ) , 2 2

(2.1)

where k is the Boltzmann constant, T is the temperature in the equilibrium state, m is the mass of a molecule, and (ξx , ξ y , ξz ) is the molecular velocity.1 For the case of temperature measurement, the angle brackets · · · in the right-hand side of Eq. (2.1) means the average over the molecules forming the thermometer and also a time average for some time duration of reading the scale of the thermometer. Thus, the macroscopic variables, e.g., T in the left-hand side of Eq. (2.1), are defined by some kinds of averages of microscopic variables, e.g., the molecular velocity (ξx , ξ y , ξz ) in the right-hand side of Eq. (2.1). When the temperature and pressure of a material (liquids or gases) concerned are uniform over its some volume and the materials is at rest, the relations connecting the temperature, pressure, density, internal energy, and so on can be described without using the microscopic information. In the last century, such relations have been compiled and integrated into thermodynamics, which explains the relations among macroscopic variables of materials in equilibrium states and does not contain the microscopic information at least apparently. Although fluid dynamics can discuss behaviors of nonuniform and flowing liquids and gases, its foundation is supported by thermodynamics under the assumption of the local equilibrium, which requires that the fluid in a sufficiently small volume2 is locally in an equilibrium state,3 even if the temperature and pressure are not uniform and the fluid is flowing over large scales. The interface, however, is a thin layer by its definition, and thermodynamics (and fluid dynamics) in such a thin layer has not been established yet or may not be expected to be established. Therefore, we have to consider the vapor–liquid interface and its neighborhood from the microscopic point of view. For example, in Sect. 4.4, using molecular dynamics simulations, we demonstrate that when a spherical nanodroplet and the surrounding vapor are in an equilibrium state in the sense of statistical mechanics, the thermodynamical vapor–liquid equilibrium condition (an equality of chemical potentials of vapor and liquid) does not hold. We start with a brief explanation of a general microscopic description of molecular systems.

1 For

simplicity, the fluid and the thermometer are assumed to be composed of monatomic molecules and at rest in the macroscopic sense. The formula (m/2) ξi2 = kT /2 (i = x, y, z) is called the equipartition theorem or the law of equipartition of energy.

2 The

sufficiently small volume in fluid dynamics is sufficiently large in molecular scales so that it may contain a number of molecules. Thus, the macroscopic variables defined by some kinds of averages can be regarded as continuous functions of the space coordinates and the time. If the fluid is an ideal gas in the standard state, the number of molecules in a cube with a side-length 1 µm is 2.6867774×107 . The number of molecules per unit volume is called the Loschmidt constant.

3 The

actualization of local equilibrium requires a sufficient number of molecular interactions (intermolecular collisions).

2.1

Microscopic Description of Molecular Systems

21

2.1.1 Equation of Motion For simplicity, we deal with a single component system consisted of a large number of monatomic molecules, assuming that the motion of each molecule is determined by classical mechanics (without any quantum effects), the molecules are electrically neutral, and any types of association and dissociation do not occur. Then, Newton’s equation of motion is the starting point: m

d2 x (i) = f (i) , dt 2

(i = 1, 2, . . . , N ),

(2.2)

where m is the mass of a molecule, N is the total number of molecules in the system concerned, x (i) is the position vector of the ith molecule, t is the time, and f (i) is the force exerted on the ith molecule. Equation (2.2) can be written as (i)

m (i)

d2 x j

(i)

where x j and f j

dt 2

(i)

= fj ,

(i = 1, 2, . . . , N ; j = 1, 2, 3),

(2.3)

are the jth components of vectors x (i) and f (i) , respectively.

Once given an explicit form of the force f (i) and initial positions and velocities of all molecules, Newton’s equation of motion (2.2) can in principle be solved, e.g., numercally. All the macroscopic properties of materials can then be determined through their definitions in terms of some kinds of averages of the microscopic quantities, i.e., x (i) , dx (i) /dt, and d2 x (i) /dt 2 (or f (i) ) for all molecules; the definitions of macroscopic variables, such as the temperature, density, velocity, pressure, internal energy, and so on, are shown in Sects. 2.1.3, 2.3.1, and 2.5.1. In general, the force f (i) includes external forces such as the gravity. The magnitude of acceleration due to gravity is, however, negligibly small compared with a typical intermolecular force in molecular scales in time and space, and hence we only consider f (i) in Eq. (2.2) as the force acting between molecules, i.e., the intermolecular force.4 Then, it seems to be plausible to assume that the force f (i) is a conservative force, and its potential U is a function of intermolecular distances only. We further limit ourselves to the case that the potential U may be approximated as a pairwise and additive one, i.e., 1  φ(rik ), 2 N

U=

N

(2.4)

i=1 k=1 k=i

  where rik =  x (i) − x (k)  is the distance between the ith molecule and the kth molecule (i = k) and φ is a function of intermolecular distance rik . The factor 1/2

4 Forces

acting between atoms making up a molecule (covalent bonds, ionic bonds, and metallic bonds) are called the intramolecular forces.

22

2 Kinetic Boundary Condition at the Interface

in the right-hand side of Eq. (2.4) is introduced since the double sum with respect to i and k counts each pair twice. The force exerted on the ith molecule can then be defined by f (i) = −

∂U , ∂ x (i)

(2.5)

and expressed in a component form as f j(i)

=−

N  ∂rik (i)

k=1 k=i

∂x j



φ (rik ) = −

N x (i) − x (k)  j j

rik

k=1 k=i

φ  (rik ),

(2.6)

or in a vector form as f (i) = −

N  x (i) − x (k)  φ (rik ), rik

(2.7)

k=1 k=i

where φ  (rik ) denotes the derivative of φ with respect to rik . The pairwise and additive potentials are widely used in molecular dynamics (MD) simulations of liquids and gases [1, 11], although the complete validation has not been given. On the other hand, multi-body potentials are used for various MD simulations of crystalline materials such as graphite, diamond, and carbon nanotube [6]. (i) Introducing the generalized momenta p j conjugate to the generalized coordi(i)

nates q j , Newton’s equation of motion (2.3) with Eq. (2.6) can be reformulated to the equations of motion in Hamilton’s form, or Hamilton’s canonical equations of motion [22]: dq (i) j dt

=

∂H

d p(i) j

∂pj

dt

, (i)

=−

∂H (i)

∂q j

Here,

,

(i = 1, 2, . . . , N ; j = 1, 2, 3).

(i)

(i)

pj = m

dx j dt

,

(i)

(i)

qj = xj ,

H=

n=1

⎢1 ⎢ ⎣2

(2.9) ⎤

⎡ N 

(2.8)

3  k=1

(n) (n) pk pk

m

+

1 2

N  k=1 k=n

⎥ φ(rnk )⎥ ⎦,

(2.10)

where H is the Hamiltonian of the whole system. Note that we confine ourselves to the molecular system of N monatomic molecules, and hence the kinetic energy included in the Hamiltonian H is that of translational motions only [the inclusion of

2.1

Microscopic Description of Molecular Systems

23

the internal rotational motions of a polyatomic molecule into Eqs. (2.8), (2.9), and (2.10) is straightforward]. Since the Hamiltonian H defined by Eq. (2.10) does not depend explicitly on the time t, it is a global constant of the motion and we write H = E, where E is the total energy of the system.

2.1.2 Liouville Equation An arbitrary state of the N molecular system can be specified by using the 6N dimensional phase space (q, p), where q = (q1(1) , q2(1), q3(1), q1(2), . . . , q2(N ), q3(N ) ) and p = ( p1(1), p2(1), p3(1), p1(2), . . . , p2(N ), p3(N ) ) [22, 30, 32]. The results derived from the concept of the phase space should be the same as those obtained from the solutions of Newton’s equation of motion (2.3) with Eq. (2.6), while the former is much more suitable for the deduction of macroscopic properties from the microscopic information, because the macroscopic properties are associated with some kinds of averages over a number of molecules. The 6N -dimensional phase space is sometimes called the Γ -space. Specifying an arbitrary point (q, p) in the 6N -dimensional phase space at a given time t is equivalent to specifying a set of initial conditions of Newton’s equations of motion for N molecules. From the existence and uniqueness of the solution of initial-value problem of a set of Newton’s equations of motion (a set of ordinary differential equations), there exists a trajectory of solution that passes through the specified arbitrary point (q, p) in the phase space at the time. That is, the phase space is filled with the trajectories of solutions of a set of Newton’s equations of motion and a point in the phase space represents a state of the N molecular system. The density distribution function (probability density function) F(q, p, t) in the phase space is now defined by dP = F(q, p, t) dq d p,

(2.11)

where dP is the probability that a point (q, p), moving in the phase space according to Newton’s equation of motion, lies in a 6N -dimensional volume element dq d p = (1) (1) (N ) (1) (1) (N ) dq1 dq2 · · · dq3 d p1 d p2 · · · d p3 at a time t. Thus, the probability of finding the system of N molecules in a region χ in the phase space is  P(χ ) =

χ

F(q, p, t) dq d p,

(2.12)

where the integration is taken over the region χ . If χ is the whole 6N -dimensional space, then P(χ ) = 1. The expected value of an arbitrary function G(q, p) is given by  G(q, p)F(q, p, t) dq d p.

(2.13)

24

2 Kinetic Boundary Condition at the Interface

From the conservation of probability, in the same manner as the derivation of equation of mass continuity in fluid dynamics, we have [22, 30, 32]      N 3 dq (i) d p (i) ∂ dF ∂F  ∂ j j F = + + (i) F = 0, (i) dt ∂t dt dt ∂pj i=1 j=1 ∂q j

(2.14)

where dF/dt is the total time derivative of F. Expanding the derivatives with respect (i) to q (i) j and p j in Eq. (2.14) and applying the equations of motion in Hamilton’s form (2.8), we can transform Eq. (2.14) into  (i)  3 N d p (i) ∂ F   dq j ∂ F ∂F j + + = 0. ∂t dt ∂q (i) dt ∂ p (i) i=1 j=1 j j

(2.15)

Equation (2.15) is called the Liouville equation. Using Eq. (2.9) and Newton’s equation of motion (2.3) in Eq. (2.15), we have ∂F  + ∂t N

3



(i)

pj

∂F

m ∂q (i) j

i=1 j=1

+

f j(i)



∂F ∂ p (i) j

= 0.

(2.16)

Equation (2.16) is also called the Liouville equation. If an isolated system is in an equilibrium state, then the probability density F must be time independent and we have N  3  i=1 j=1



(i)

pj

∂F

m ∂q (i) j

+

(i) fj

∂F (i)

∂pj

 = 0.

(2.17)

This means that the flow in the phase space is steady and incompressible in the sense of fluid dynamics [22, 30, 32]. In general nonequilibrium states, Eq. (2.17) does not hold and we have to return to the Liouville equation (2.16).

2.1.3 Definitions of Macroscopic Variables and Equations in Fluid Dynamics The formal relations between microscopic and macroscopic variables can readily be derived from the Liouville equation (2.16) by the method of Irving and Kirkwood [15]. The method is superior in that the definitions of macroscopic variables can be applied to those in nonequilibrium states for both liquids and gases, and the resulting definitions are suitable for the use in the analysis of the data obtained by MD simulations. Note that if we restrict ourselves to the case that the fluid is an ideal gas, the definitions of macroscopic variables have been established in the kinetic

2.1

Microscopic Description of Molecular Systems

25

theory of gases (molecular gas dynamics) in terms not of the probability density F but of the velocity distribution function of gas molecules [35]. The kinetic theory of gases are summarized in Sect. 2.3. In the following, we derive the equations in fluid dynamics (conservation equations of mass, momentum, and energy) from the Liouville equation (2.16) by the method of Irving and Kirkwood [15] with a small modification for the later use in Sect. 2.2. In the original paper [15], the Dirac delta function is used instead of a scalar function χ defined below. To begin with, we define an averaged fluid density as ρ(x, t) =

N  m  χ (q (i), x; h)F(q, p, t) dq d p, h3

(2.18)

i=1

where the integration is taken over the whole 6N -dimensional phase space and χ (q (i), x; h) =

⎧ ⎨1

   (i)   q j − x j   h/2 for j = 1, 2, 3,

⎩ 0 otherwise.

(2.19)

That is, if the ith molecule is in a cube with a side-length h centered at x, then χ = 1 and the integration yields the expected value that one will find the ith molecule in the cube (Fig. 2.1). Therefore, the right-hand side of Eq. (2.18) is the expected value of the number of molecules in the cube multiplied by m/ h 3 , which is the averaged fluid density ρ. Note that the introduction of the function χ is a coarse-graining process. Although the side-length h of the cube is arbitrary, it should be small compared with some length scale that is to be resolved. Similarly, an averaged fluid momentum per unit volume and an averaged total energy of fluid per unit volume are defined as

χ (q (i) , x ; h) ˙

.

˙

q (k) q (j)

x ˙ q ()

h

h

h

O Fig. 2.1 The value of the scalar function χ(q (i), x; h) is equal to unity if the ith molecule is contained in a cube with a side-length h centered at x

26

2 Kinetic Boundary Condition at the Interface

ρ(x, t)v j (x, t) =

N  1  χ (q (i), x, h) p (i) j F(q, p, t) dq d p, h3

(2.20)

i=1

N  1  ρ(x, t)E(x, t) = 3 χ (q (i), x, h)e(i) F(q, p, t) dq d p, h

(2.21)

i=1

e

(i)

(i) (i)

3 N 1 1  pj pj + = φ(rik ), 2 m 2 j=1

(2.22)

k=1 k=i

where Eq. (2.10) is used for the definition of e(i) , the sum of the kinetic and potential energies of the ith molecule.5 Thus, the fluid velocity v j and the total energy of fluid per unit mass E are defined.6 The conservation law of mass of a fluid is expressed in a partial differential equation, which can be derived from the Liouville equation (2.16) in the following manner. Multiplying Eq. (2.16) by χ (n) = χ (q (n), x, h), we have   (i) 3 N   p ∂F ∂ (n) (n) j (n) (i) ∂ F χ = 0. + χ fj (χ F) + (i) ∂t m ∂q (i) ∂p i=1 j=1

j

(2.23)

j

Integrating Eq. (2.23) over the whole phase space gives ∂ ∂t

 χ

(n)

F dq d p +

 N  3 

χ (n)

i=1 j=1

 +

p (i) ∂F j dq d p m ∂q (i) j

χ (n) f j(i)

∂F ∂ p (i) j



dq d p = 0.

(2.24)

With the aid of the Gauss divergence theorem, the following equations hold for an arbitrary function G, N  3   i=1 j=1 Ωq N  3   i=1 j=1 Ω p

G(q)

G( p)

∂F (i) ∂q j

∂F (i) ∂pj

dq = − dp = −

N  3   i=1 j=1 Ωq N  3   i=1 j=1 Ω p

∂G (i)

∂q j

∂G (i)

∂pj

F dq,

(2.25)

F d p,

(2.26)

5 The potential energy of the ith molecule is defined only formally. It is the total potential energy of all molecules that has the physical meaning. 6 In

Chap. 5 and in Appendix B, the internal energy per unit mass of fluid is denoted by e.

2.1

Microscopic Description of Molecular Systems

27

if the probability density F falls off rapidly outside a bounded region Ωq in the 3N -dimensional space of q in the case of Eq. (2.25) and outside a bounded region Ω p in the 3N -dimensional space of p in the case of Eq. (2.26). Using Eqs. (2.25) and (2.26), we can rewrite Eq. (2.24) into ∂ ∂t

 χ

(n)

F dq d p −

 3 N  

(i)

respect to

(i) pj

 (n)

(i) 

pj

F dq d p χ (i) m ∂q j   ∂ ! (n) (i) " + χ fj F dq d p = 0. (i) ∂pj

i=1 j=1

Since χ (n) and f j

∂

(i)

(2.27) (i)

are independent of p j , the differentiation of χ (n) f j

with

vanishes, i.e., !

∂ (i)

∂pj

(i)

χ (n) f j

"

= 0.

(2.28)

From the definition of the function χ , Eq. (2.19), we have N 

∂

i=1

∂q (i) j

 χ

(i) 

p (n) j

m

=

(n) 



∂

χ

∂q (n) j

p (n) j

m

∂ =− ∂x j

 χ

(n) 

p (n) j

m

,

(2.29)

where in the last equality we used ∂χ (n) (n)

∂q j

=−

∂χ (n) . ∂x j

(2.30)

Substituting Eqs. (2.28) and (2.29) into Eq. (2.27) and taking the sum over all n, we obtain the equation of mass continuity in fluid dynamics, ∂ρ  ∂ρv j + = 0, ∂t ∂x j 3

(2.31)

j=1

where the definitions of ρ and ρv j , Eqs. (2.18) and (2.20), are used. Equation (2.31) with the definitions of ρ and v j does not require the assumption of local equilibrium unlike in the case of fluid dynamics.7 only requirement is that ρ and v j are continuously differentiable functions of x and t. This will be satisfied by choosing h in the function χ so that the cube with a side-length h centered at x contains a large number of molecules. If the fluid is a gas, this does not warrant the local equilibrium, because the mean free path of gas molecules can be very large compared with h.

7 The

28

2 Kinetic Boundary Condition at the Interface

In the same manner as the derivation of Eq. (2.27), we can derive the equations of momentum conservation of nth molecule ∂ ∂t



(n) χ (n) pk F

 +

dq d p −

 N  3 

∂ (i)

∂q j

i=1 j=1

∂ ∂ p (i) j

 χ

p (n) p(i) j (n) k

! " χ (n) pk(n) f j(i) F dq d p



m

F dq d p

 = 0,

(k = 1, 2, 3),

(2.32)

where Eq. (2.20) is used. By making use of Eqs. (2.28) and (2.30), Eq. (2.32) can be rewritten into ∂ ∂t



χ (n) pk(n) F

(n) (n)  3  p pj ∂ (n) k F dq d p χ dq d p + ∂x j m j=1  (n) − χ (n) f k F dq d p = 0, (k = 1, 2, 3).

(2.33)

The third term in Eq. (2.33) can be transformed to yield8 ∂ ∂t



χ (n) pk(n) F

(n) (n)  3  p pj ∂ (n) k F dq d p χ dq d p + ∂x j m j=1

−

3  j=1

∂ ∂x j





1 h x j − q (n) j + 2 3



χ (n) f k(n) F dq d p = 0,

(k = 1, 2, 3). (2.34)

Taking the sum over n, we should have the conservation equation of the momentum of the fluid per unit volume, 3 $ ∂ρvk  ∂ # ρvk v j + Pk j = 0, + ∂t ∂x j

(k = 1, 2, 3),

(2.35)

j=1

where the microscopic definition of the stress tensor Pk j is given as

In the case that the fluid is a liquid, this may be a sufficient condition for the local equilibrium, because the mean free path of liquid molecules is usually comparable with or less than a typical diameter of a molecule. 8 The factor 1 in Eq. (2.34) is an ideal limit of N → ∞ in equilibrium states. The stress tensor in 3 the original paper [15] is different from this and much more cumbersome.

2.1

Microscopic Description of Molecular Systems N  1  Pk j = 3 h



   h 1 (n) (n) χ (n) F dq d p − xj − qj + fk 3 2

(n) (n)

pk p j m

n=1

29

− ρv j vk ,

( j, k = 1, 2, 3).

(2.36)

If the fluid is in a uniform equilibrium state at rest, then v j = 0 and Pk j = Pδk j , where δk j is the Kronecker delta and pressure P is given by 3  N 1  P= 3 3h



n=1 k=1

   pk(n) pk(n) h (n) (n) χ (n) F dq d p. (2.37) − x k − qk + fk m 2

The energy conservation equation for the nth molecule, the counterpart of Eq. (2.34), is ∂ ∂t



χ (n) e(n) F dq d p +

(n)  3  pj ∂ F dq d p χ (n) e(n) ∂x j m j=1

−

 (n)   3  p ∂ h (n) (n) j x j − q (n) + χ f F dq d p = 0. j ∂x j 2 m j

(2.38)

j=1

After taking the sum over n, the conservation equation of the total energy of the fluid per unit volume and the microscopic definition of the heat flux Q j are obtained as ∂ρ E  ∂ + ∂t ∂x j 3

 ρ Ev j +

j=1

3 

 Pi j vi + Q j

= 0,

(2.39)

i=1

 (n)   N   pj 1  h (n) (n) (n) Qj = 3 χ (n) F dq d p fj e − xj − qj + h 2 m n=1

 − ρ Ev j +

3 

 Pi j vi ,

( j = 1, 2, 3).

(2.40)

i=1

It may be instructive to compare the above results with the definitions of macroscopic variables in the kinetic theory of gases [35]. The definitions of the stress tensor and the heat flux in the kinetic theory of gases are given as follows:  Pi j = =



(ξi − vi )(ξ j − v j ) f (x, ξ , t) dξ ξi ξ j f (x, ξ , t) dξ − ρvi v j ,

(i, j, = 1, 2, 3),

(2.41)

30

2 Kinetic Boundary Condition at the Interface



 1 (ξi − vi )2 f (x, ξ , t) dξ (ξ j − v j ) 2 i=1      3 3 3   3 1 2 1 2 = ξj ξi f (x, ξ , t) dξ − ρ vi v j + pi j vi , RT + 2 2 2 3

Qj =

i=1

i=1

i=1

( j = 1, 2, 3), (2.42) where ξi is the ith component of molecular velocity, T is the temperature, R = k/m is the gas constant (per unit mass), the integration is taken over the 3-dimensional space of molecular velocity ξ , and f (x, ξ , t) is the velocity distribution function of gas molecules, which gives the gas density  ρ=

f (x, ξ , t) dξ .

(2.43)

The precise definition of the velocity distribution function f (x, ξ , t) is given in Sect. 2.3.1. Equations (2.41) and (2.42) are also applicable to general nonequilibrium states with the definition of temperature 3 ρ RT = 2



1 2 1  2 ξi f dξ − ρ vi . 2 2 3

3

i=1

i=1

(2.44)

As compared with Eqs. (2.36) and (2.40), one can see that the contribution of intermolecular force is neglected in Eqs. (2.41) and (2.42). Thus, the microscopic definitions of the stress tensor and the heat flux are extended to nonequilibrium states in fluids including liquids where the intermolecular force cannot be neglected. They are used for the analysis of results from MD simulations. For the definition of temperature,  3 kT = 2 N

i=1



χ (q (i) − x; h)

(i) (i) 3 3 1  pj pj m 2 vi , (2.45) F(q, p, t) dq d p − 2 m 2 j=1

i=1

is used in MD simulations, instead of Eq. (2.44). We have derived the equations for conservation laws of macroscopic variables, Eqs. (2.31), (2.35), and (2.39), with the definitions of macroscopic variables, Eqs. (2.18), (2.20), (2.21), (2.36), (2.40), and (2.45). They can be applied to general nonequilibrium states of liquids and gases, for which we cannot expect (i) the thermodynamic relations (or the assumption of local equilibrium), (ii) the stress tensor of the Newtonian fluid, and (iii) the heat flux based on the Fourier law. For example, it is well known that the stress tensor for a slightly rarefied gas contains terms related to temperature gradient, called the thermal stress [35], which does not appear in equations of fluid dynamics (see Appendix B at the end of this book).

2.2

Molecular Dynamics Simulation

31

Since our target is the vapor–liquid interface, we have to know the behavior of molecules in the liquid phase. For this purpose, we use the MD simulations for the vapor–liquid two phase system.

2.2 Molecular Dynamics Simulation In the preceding subsection, we have given the definitions of macroscopic variables in terms of the microscopic information. They can be applied to liquids and gases in general nonequilibrium states, and expressed in appropriate forms for the analysis in MD simulation. Now, we move on to the explanation of the method of MD simulations. The standard method of MD simulation numerically solves Newton’s equation of motion (2.3) for a number of molecules confined in a simulation box fixed in a physical coordinate system. The total number of molecules in the box is unchanged during the simulation if an appropriate boundary condition on the surface of the box is imposed. Such a method of simulation is called N V E simulation because the number of molecules, N , the volume of the system, V , and the total energy of molecules, E, are constant in the simulation except for numerical errors (mainly truncation errors) in the total energy. It is possible and sometimes preferred to perform simulations with a constant temperature (N V T simulation) or those with constant temperature and pressure (N P T simulation) [1, 13]. However, N V T simulation and N P T simulation solve some dynamical systems different from Newton’s equation of motion (2.3) and the Liouville equation (2.16). The differences resulted from the differences from Newton’s equation of motion have not been figured out in nonequilibrium MD simulations.9 In this book, we concentrate on the dynamics of molecules based on Newton’s equation of motion (2.3), although not restricted to N V E simulations.

2.2.1 Lennard-Jones Potential and Normalization of Variables In MD simulations, the most widely used pairwise additive potential is the LennardJones 12-6 potential [23], 1  φ(rik ), U= 2 N



N

i=1 k=1 k=i

φ(rik ) = 4

σ rik

12

 −

σ rik

6  ,

(2.46)

where  (J) and σ (m) are the energy and length parameters of Lennard-Jones 12-6 potential. Using these two parameters and the molecular mass m, Newton’s equation of motion (2.3) can be nondimensionalized as 9 The

macroscopic properties in equilibrium states should not be different by the difference of simulation methods.

32

2 Kinetic Boundary Condition at the Interface (i)

d2 xˆ j

dtˆ2

(i) = fˆj ,

(2.47)

where xˆ (i) j

=

x (i) j σ

,

t tˆ = √ , σ m/

fˆj(i) =

f j(i) /σ

.

(2.48)

The nondimensional quantities are signified by ˆ. Then, from Eq. (2.6), the nondimensionalized intermolecular force fˆj(i) is given by (i) fˆj = −

N xˆ (i) − xˆ (k)  j j k=1 k=i

rˆik

φˆ  (ˆrik ),

(2.49)

where ˆ rik ) = 4 φ(ˆ



1 rˆik

12



1 − rˆik

6  ,

    rˆik =  xˆ (i) − xˆ (k) .

(2.50)

The macroscopic variables are also nondimensionalized as follows: σ 3 m 1/2 Q j .  3/2 (2.51) ˆ r ) and its In Fig. 2.2, the nondimensionalized Lennard-Jones 12-6 potential φ(ˆ derivative with respect to the argument, φˆ  (ˆr ), are shown in a solid curve and dashed ˆ r ) has a minimum at rˆ = 21/6 , at which a strong curve, respectively. The potential φ(ˆ 1/6 repulsive force (ˆr < 2 ) is switched to an attractive force (ˆr > 21/6 ). The repulsive force behaves like rˆ −13 and the asymptotic form of the tail of attractive force is rˆ −7 as rˆ → ∞. Figure 2.2 suggests that rˆ = 1 (r = σ ) is a reasonable measure of the diameter of the molecule modeled by the Lennard-Jones 12-6 potential. The depth ˆ corresponds to φ =  and indicates of potential well, i.e., the minimum of φ, the strength of molecular interaction. Thanks to a number of simulation studies up to now, the Lennard-Jones parameters  and σ suitable for modeling simple molecules such as Ar, Ne, Kr, N, O have been tabulated; for example, (/k, σ ) = (119.8 K, 0.341 nm) for Ar, (47.0 K, 0.272 nm) for Ne, (164.0 K, 0.383 nm) for Kr, (37.3 K, 0.331 nm) for N, (61.6 K, 0.295 nm) for O [1], where the Boltzmann constant k = 1.3806504 × 10−23 J/K. Before proceeding to subsections for numerical method, we note that the nondimensionalization of variables in Eqs. (2.48), (2.49), (2.50), and (2.51) are defined by the quantities in molecular scales. Effects of much larger scales in space and ρˆ =

σ 3ρ , m

kT Tˆ = , 

vj vˆ j = √ , /m

Pˆi j =

σ 3 Pi j , 

Qˆ j =

2.2

Molecular Dynamics Simulation

33

Normalized Potential & Force

3

ˆˆ dφ(r) ˆ dr

2

1

ˆˆ φ(r) 1

0

2

1/6

–1 0

1

2

3

Normalized Intermolecular Distance

ˆ r ) (solid curve) and its derivaFig. 2.2 The nondimensionalized Lennard-Jones 12-6 potential φ(ˆ tive with respect to the argument φˆ  (ˆr ) (dashed curve)

time may therefore slip through standard numerical methods usually used in MD simulations.10 We focus on molecular phenomena in molecular scales.

2.2.2 Finite Difference Method Newton’s equation of motion (2.47) is a set of ordinary differential equations. Although we have many sophisticated numerical techniques for solving ordinary differential equations [29], MD simulations usually use rather simple finite difference methods. This is because we have to deal with a number of molecules in the system N , and/or we have to continue computations for quite large steps M. The number of molecules N and the number of steps M sometimes exceed N = 106 and M = 108 . Large N and M directly result in the increase in the computational time. We therefore prefer numerical methods as simple as possible with less degradation in accuracy. The methods commonly used are the leap-frog scheme, the velocity Verlet scheme, and Gear’s predictor–corrector algorithms [1, 11]. We here explain the leap-frog scheme. In the leap-frog scheme, the nondimensionalized Newton’s equation of motion (2.47) is discretized as (i) ˆ ˆ ˆ ˆ (i) ˆ 1 ˆ ˆ3 xˆ (i) j (t + t ) = xˆ j (t ) + t vˆ j (t + 2 t ) + O(t ),

10 For

example, it is natural that large-scale fluid flows are affected by the gravity.

(2.52)

34

2 Kinetic Boundary Condition at the Interface (i) (i) (i) vˆ j (tˆ + 12 tˆ ) = vˆ j (tˆ − 12 tˆ ) + tˆ fˆj (tˆ ) + O(tˆ 3 ),

(2.53)

ˆ where vˆ (i) j is the jth component of velocity of the ith molecule and t is the time step. By using the Taylor expansions, it is easy to derive Eqs. (2.52) and (2.53), and (i) to confirm that the local truncation errors are of the order of tˆ 3 . Note that fˆj (tˆ) in the right-hand side of Eq. (2.53) is given by fˆj(i) (tˆ) = −

N xˆ (i) (tˆ )− xˆ (k) (tˆ )  j j k=1 k=i

rˆik (tˆ )

φˆ  (ˆrik (tˆ )),

    rˆik (tˆ ) =  xˆ (i) (tˆ )− xˆ (k) (tˆ ),

(2.54)

(2.55)

(i) (i) ˆ 1 ˆ ˆ 1 ˆ ˆ and hence vˆ (i) j (t + 2 t ) can be obtained by Eq. (2.53) if vˆ j (t − 2 t ) and xˆ j (t )

(i) 1 ˆ 1 ˆ ˆ ˆ are known. After obtaining vˆ (i) j (t + 2 t ), we can determine xˆ j (t + 2 t ) by Eq. (2.52), and the computation can be continued to the next time step. Thus, the time series of positions and velocities obtained by the leap-frog scheme (2.52) and (2.53) are shifted by the half of the time step. In many cases, it is important and useful to monitor the value of the total energy (total Hamiltonian) (2.10), which can be written in the nondimensional form as follows:

  (i) $ 1  # (i) ˆ tˆ ) = 1 φˆ rˆik (tˆ ) . vˆ j (tˆ )vˆ j (tˆ ) + H( 2 2 N

3

i=1 j=1

N

N

(2.56)

i=1 k=1 k=i

ˆ tˆ ), we can use For the evaluation of H( (i)

vˆ j (tˆ ) =

 1  (i) (i) vˆ j (tˆ + 12 tˆ ) + vˆ j (tˆ − 12 tˆ ) + O(tˆ 2 ). 2

(2.57)

ˆ tˆ ) by the use of Eq. (2.57) may not The error of the order of tˆ 2 brought into H( ˆ lead to serious problems, because Eq. (2.57) is used only in the evaluation of H, and significant errors are caused by the accumulation of local truncation errors for a large number of computational steps. Even if the number of molecules in the system is not so large, e.g., N = 104 , (i) the exact evaluation of fˆj is a hard task because of the long tail of attractive force, as shown in Fig. 2.2. Since the tail of attractive force decays as rˆ −7 , it seems to be reasonable to cut it off at some distance rˆcut from the center of the ith molecule. Many authors have so far used rˆcut = 2.5 in their MD simulations. However, the simulations of vapor–liquid interface and its neighborhood are strongly affected by the details of the numerical method, such as, the size of cut-off radius rˆcut , the thickness of liquid layer, and the area of the interface [12, 26, 40]. In particular, the

2.2

Molecular Dynamics Simulation

35

Density (kg/m3) & Temperature (K)

3000

1410 kg/m3

1000 Liquid

100

Temperature

84.6 K

Vapor

Vapor

Density

4.3 kg/m3

Interface

Interface

10

1 0

5

10 15 20 25 Space Coordinate (nm)

30

Fig. 2.3 The density and temperature in a vapor–liquid equilibrium state. In the figure, the density transition layer is shown as the interface

shorter is the cut-off radius, the lower the liquid density and the higher the vapor density. According to Refs. [12, 26, 40], to suppress artificial effects due to small rˆcut , it should be larger than or at least equal to 4.4 (it corresponds to 1.5 nm for the case of argon). Usually, the simulation is performed for a specified N molecules put in a simulation box fixed in the coordinate system. We therefore impose some boundary condition at the surface of the box. The most simple one is the periodic boundary condition [1, 11], and it enables us to avoid introducing artificial boundary conditions and to conserve the number of molecules. Needless to say, the use of the periodic boundary condition does not imply the simulation of infinitely large volume at all, even in the cases of equilibrium simulation. For example, the thickness of the vapor–liquid interface (density transition layer), as shown in Fig. 2.3 in Sect. 2.2.3 and illustrated in Fig. 2.7 in Sect. 2.4.1, is known to be a logarithmically increasing function of the area of the interface (the cross section of the simulation box) [12, 26, 40]. Furthermore, if the size of the box is not sufficiently large compared with the cut-off radius, the artificial effect of periodic boundary condition spoils the result of simulations [1, 11].

2.2.3 Example: Vapor–Liquid Equilibrium State As an example of MD simulations, we present the distributions of averaged density and temperature of a vapor–liquid coexistence system including two interfaces in Fig. 2.3, where a planar liquid layer of monatomic molecules exists between two vapor phases of the same molecules, and the system is in an equilibrium state.

36

2 Kinetic Boundary Condition at the Interface

Although the simulation is performed with nondimensional variables defined in previous subsections, the dimensional density and temperature are shown in Fig. 2.3 with the use of (/k, σ ) = (119.8 K, 0.341 nm) for argon. The temperature (84.6 K) is uniform,11 the liquid density is 1410 kg/m3 , and the vapor density is 4.3 kg/m3 , which are in agreement with known values [28] with errors less than 1%. The density profile in Fig. 2.3 shows that the vapor–liquid interface has a finite thickness. The dashed line in Fig. 2.3 denotes the edge of the bulk liquid region or bulk vapor region, where we tentatively use “bulk” to indicate that the averaged density is spatially uniform. The details of the simulation method are as follows: The simulation box is L 1 × L 2 × L 3 = 90 σ × 30 σ × 30 σ nm3 , and a planar liquid layer with thickness about 7 nm is set at around x 1 = L 1 /2 as the initial condition, and the total number of molecules is N = 17280. Under the periodic boundary condition, Newton’s equations of motion (2.47), (2.48), (2.49), and (2.50) for N molecules are numerically solved by √ using the leap-frog scheme (2.52) and (2.53) with the time step tˆ = 0.0005 (σ m/tˆ = 10−15 s) and the cut-off radius rˆ√ cut = 5 (σ rˆcut = 1.7 nm). The number of total simulation steps is M = 4 × 108 (σ m/ Mtˆ = 4 × 10−7 s). The first half of M simulation steps is dedicated to a relaxation process to a vapor– liquid equilibrium, and the results shown in Fig. 2.3 are evaluated from averages of samples obtained in the second half of M steps. The evaluation of density and temperature uses Eqs. (2.18) and (2.45). Since the phenomenon considered here is one-dimensional in the macroscopic sense, the function χ defined by Eq. (2.19) should be replaced by χ 1 (q (i) − x; h) =

⎧ ⎨1 ⎩0

    (i) q1 − x1   h/2,    (i)  q1 − x1  > h/2.

(2.58)

Accordingly, the volume of the cube h 3 in Eqs. (2.18) and (2.45) should also be replaced by h L 2 L 3 (see Fig. 2.4). The integration of any function multiplied by the probability density F(q, p, t) with respect to q and p is naturally replaced by the summation over a large number of samples obtained in the MD simulation. The errors introduced by the cut-off of the intermolecular force are illustrated    in   (i)   Fig. 2.5a, where the error is defined by the difference of  f  with rˆcut from  f (i)  with rˆcut = 10 for some ith molecule at some instant t. From the figure, we can approximately estimate that the error of molecules in the bulk liquid phase decreases with 10−3ˆrcut /4 as rˆcut increases, and that of molecules at the edge of the bulk liquid phase decreases more slowly with 10−ˆrcut /4 . The molecular motions in the vicinity of the interface is therefore affected by the size of cut-off radius of intermolecular potential strongly. The use of a small cut-off radius results in an equilibrium system with a high vapor density and a low liquid density [12, 26, 40].

11 The

triple point temperature is 83.8 K for argon [28].

2.2

Molecular Dynamics Simulation

37

χ1(q(i) –˙x ; h) .

˙

x3 L2

q (k) x2

q (j) L3

q () x1

O

h

Fig. 2.4 The function χ 1 for spatially one-dimensional problems in the macroscopic sense

Figure 2.5b illustrates how the system size (number of total molecules N ) affects ˆ Here, the error the total Hamiltonian Hˆ defined by (2.56) and numerical errors in H. ˆ In the figure, four systems is defined by three times the standard deviation of H. with different N (4320, 17280, 69588, 286487) are compared, for which we use the same leaf-frog scheme with the same cut-off radius rˆcut = 5 and the same time step tˆ = 0.0005, and different simulation boxes: (L 1 /σ, L 2 /σ, L 3 /σ ) = (90, 15, 15) for N = 4320, (90, 30, 30) for N = 17280, (150, 60, 60) for N = 69588, (390, 120, 120) for N = 286487. In our vapor–liquid two-phase systems, almost (b)

Bulk Liquid

10–5

10

Er

69588

17280 69588 4320 105

ia n

10–4

ro

106

H a × mi (– lt 1) on

10–3

286487

5

0

1 r×

17280

al

Edge of Bulk Liquid

To t

Total Hamiltonian & Error

Error of Intermolecular Force

(a) 10–2

4320 4

–6

10

2

4

6

ˆ rcut

8

103

104

105

Number of Molecules

Fig. 2.5 (a) The difference between the intermolecular force with various cut-off rˆcut and that with rˆcut = 10 is shown as a measure of error in the intermolecular force, where the open circle denotes the force on some molecule in the bulk liquid phase and the closed circle denotes that on some molecule at the edge of the bulk liquid phase. (b) The relation between the error in the total Hamiltonian and the number of molecules in the system. The open circles are Hˆ × (−1) and the closed circles are three times the standard deviation of Hˆ × 105 . The number near the symbol denotes the total number of molecules in the system, N . The solid line is a line of slope 1 and the dashed line slope 1/2

38

2 Kinetic Boundary Condition at the Interface

80% of Hˆ is the potential energy of molecules in liquid and hence Hˆ is substantially determined by the number of molecules in the liquid layer. This is the reason why Hˆ in Fig. 2.5b is proportional to √ N . The important conclusion from Fig. 2.5b is that if Hˆ ∝ N , then the error ∝ N . Theoretically, Hˆ should be constant in N V E simulations as well as in an isolated system, and therefore, the fluctuation in Hˆ is purely numerical. Figure 2.5b suggests that the numerical error in Hˆ also is governed by the central limit theorem.12

2.3 Kinetic Theory of Gases The kinetic theory is a microscopic theory of processes in systems not in statistical equilibrium [24]. A kinetic boundary condition means a boundary condition for a kinetic equation, the governing equation of a kinetic theory, such as the Boltzmann equation [35], the Vlasov equation for plasma [9], the Enskog equation for dense gases [31]. In contrast to the well-established kinetic theory of gases, the rigorous kinetic theory of liquids is considerably complicated [5], and its formidable mathematical difficulties impede reducing and interpreting the formal solutions. The origin of difficulties is clearly apparently random multi-body interactions of molecules in liquids. On the other hand, the kinetic theory of gases based on the Boltzmann equation, which is a mainstay of this book, deals with dilute gases in the sense that the three-body interaction of gas molecules does not occur. As a result, the kinetic theory of gases permits us to obtain a number of fruits from it, although the exact kinetic theory of gases is still mathematically difficult. In the context of this book, however, the random multi-body interactions of liquid molecules are not necessarily an obstacle for our purpose. It allows us to expect that the liquid is in a local equilibrium state everywhere except for the interface because of a sufficiently rapid relaxation to equilibrium due to frequent multi-body interactions of molecules in liquids. Once the local equilibrium in a liquid is admitted, we can focus our attention to the behavior of gas molecules under the given macroscopic condition of the liquid. Strictly speaking, if the evaporation or condensation at the interface is not so weak, the assumption of local equilibrium in the liquid may not be accepted uncritically. For example, if the heat flux across the interface due to the evaporation or condensation is fairly large, the liquid near the interface may deviate from a local equilibrium state. If the liquid near the interface is not in a local equilibrium state, we cannot specify the macroscopic condition of the liquid, which is necessary to analyze the vapor flow separately by the kinetic theory of gases. In this book, we therefore confine ourselves to the case of weak evaporation/condensation state, the precise definition of which is given in Sects. 2.5.1 and 2.5.2. Although we determine the molecular motions in the liquid in a rigorous way of the MD 12 It is easy to confirm that the probability density of numerically obtained H ˆ approaches a Gaussian with the increase in N .

2.3

Kinetic Theory of Gases

39

simulation, some constraints are still imposed on the bulk of the liquid, and we do not pursue the kinetic theory of liquids.

2.3.1 Boltzmann Equation For the present, we confine ourselves to the gases of monatomic molecules. In the kinetic theory of gases, the only unknown microscopic variable is the velocity distribution function of gas molecules, f (x, ξ , t), defined by m dN = f (x, ξ , t) dx dξ ,

(2.59)

where dN in the left-hand side denotes the number of molecules in a volume element dx dξ = dx1 dx 2 dx3 dξ1 dξ2 dξ3 centered at (x, ξ ) in the 6-dimensional space of the position x and the velocity ξ of a molecule [35] (N signifies the number of molecules). The governing equation for f is the Boltzmann equation, ∂f ∂f = J ( f ), + ξj ∂t ∂x j

(2.60)

where J ( f ) in the right-hand side represents the effects of the intermolecular collisions. Here and hereafter the Einstein summation convention is used, e.g.,  ∂f ∂f = ξj , ∂x j ∂x j 3

ξj

(2.61)

j=1

(see Appendix A at the end of this book). Furthermore, the notation of the jth component of a vector a, a j , will be used without notice. Once given the velocity distribution function f (x, ξ , t), the macroscopic variables are evaluated by    1 1 (2.62) ξi f dξ , T = (ξ j − v j )2 f dξ , ρ= f dξ , vi = ρ 3ρ R where ρ is the gas density, vi is the gas velocity, and T is the gas temperature. The definitions of stress tensor and heat flux have already been given by Eqs. (2.41) and (2.42) in Sect. 2.1.3. The gas pressure p and the internal energy e are given by p = ρ RT and e = (3R/2)T , respectively.13 The collision term J ( f ) for the molecules with a spherically symmetric intermolecular potential with a finite influence range dm is given by a five-fold integral of f [35], relations p = ρ RT and e = (3R/2)T do not imply that the gas is in a (local) equilibrium state. They are formal extensions to nonequilibrium states, as well as the definition of temperature T .

13 The

40

2 Kinetic Boundary Condition at the Interface

J( f ) =

1 m

 all αi , all ξi∗

( f  f ∗ − f f ∗ ) B dΩ(α) dξ ∗ ,

(2.63)

where  f = f (xi , ξi , t), f  = f (xi , ξi , t), f ∗ = f (xi , ξi∗ , t), f ∗ = f (xi , ξi∗ , t), (2.64)  = ξi∗ − αi α j (ξ j∗ − ξ j ), ξi = ξi + αi α j (ξ j∗ − ξ j ), ξi∗

 B=B

 α j (ξ j∗ − ξ j ) , |ξ ∗ − ξ | , |ξ ∗ − ξ |

(2.65)

(2.66)

and α is a unit vector expressing the variation of the direction of molecular velocity just before and after the collision, dΩ(α) is the solid-angle element in the direction of α, and the functional form of B is determined by the intermolecular force; for example, B = dm2 |α j (ξ j∗ − ξ j )|/2 for a gas consisting of hard-sphere molecules with diameter dm [35]. The mean free path of gas molecules, , is defined as the product of the molecular average speed ξ = (8RT /π )1/2 and the inverse of the mean collision frequency ν c [35], =

ξ , νc

(2.67)

where the mean collision frequency14 1 νc = ρm

 f (ξ ) f (ξ ∗ ) B dΩ(α) dξ dξ ∗ ,

(2.68)

is an average of the collision frequency of a molecule with velocity ξ νc (ξ ) =

1 m

 f (ξ ∗ ) B dΩ(α) dξ ∗ .

(2.69)

The integrations are taken over 6-dimensional space of (ξ , ξ ∗ ) in Eq. (2.68) and over 3-dimensional space of ξ ∗ in Eq. (2.69). When the velocity distribution function f is the Maxwellian (the Maxwell distribution function) with constant ρ0 , T0 , and v0i ,

f M (ξ ) =

14 The

  ρ0 (ξi − v0i )2 , exp − 2RT0 (2π RT0 )3/2

inverse of the mean collision frequency is called the mean free time.

(2.70)

2.3

Kinetic Theory of Gases

41

the mean collision frequency ν c defined by Eq. (2.68) can be evaluated as ν c = 4dm2 (π RT0 )1/2

ρ0 , m

(2.71)

where ρ0 /m is the number density of molecules and, as mentioned earlier, dm is the radius of the influence range of the intermolecular force (not restricted to the diameter of a hard-sphere molecule) [35]. The mean free path is then given by 1 . = √ 2 2π dm (ρ0 /m)

(2.72)

Here, we should comment on the fundamental framework of the kinetic theory of gases based on the Boltzmann equation. The collision term of the Boltzmann equation only considers the binary collision of molecules. Mathematically, this is a situation where N → ∞ and dm → 0 with N dm2 fixed. This is called the Grad– Boltzmann limit [35]. In this limit, we have N dm3 → 0, and this means that the gas is an ideal gas. The N dm2 corresponds to the inverse of the mean free path of gas molecules,15 and it can be arbitrarily small or large as far as it is kept at a fixed value in the limiting process of N → ∞ and dm → 0. The wide applicability of the Boltzmann equation from atmospheric to very low pressures is therefore the consequence of the Grad–Boltzmann limit. In the right-hand side of Eq. (2.63), one can see a factor m −1 , which tends to infinity as m tends to zero. The collision term is, however, always bounded because the function B has a factor dm2 . In other words, it is implicitly assumed that the mass of a molecule m → 0 as N → ∞ with keeping N m fixed at a finite value, as it should be. This is the reason why we define the velocity distribution function by Eq. (2.59). In many literature, however, the velocity distribution function is defined by f /m, which is infinite in the Grad–Boltzmann limit. In addition to the hard-sphere model, the following model is widely used for the collision term of the Boltzmann equation (2.60): J ( f ) = Ac ρ( f e − f ),   (ξ j − v j )2 ρ exp − fe = , 2RT (2π RT )3/2

(2.73) (2.74)

where Ac is a constant, f e is the local Maxwellian with density ρ, velocity v j , and temperature T defined by Eq. (2.62). The Boltzmann equation with the collision term (2.73) with Eq. (2.74) is called the Boltzmann–Krook–Welander (BKW) equation [35]. Since ρ, v j , and T in Eqs. (2.73) and (2.74) are given by the integrals of

15 Precisely, the mean free path corresponds to 1/(n d 2 ), where n is a characteristic number 0 m 0 density of gas molecules, as shown by Eq. (2.72).

42

2 Kinetic Boundary Condition at the Interface

unknown function f as shown in Eq. (2.62), the BKW equation is a highly nonlinear integro-differential equation. The constant Ac in Eq. (2.73) is related to the mean free path  of molecules in the gas in an equilibrium state with density ρ and temperature T as =

(8RT /π )1/2 , Ac ρ

(2.75)

where Ac ρ is the collision frequency (νc ) of molecules described by the BKW equation. That is, the collision frequency of molecules described by the BKW equation is independent of the molecular velocity ξ [see Eqs. (2.68) and (2.69)], and hence νc = ν c . The BKW equation shares the important properties in the kinetic theory of gases with the standard Boltzmann equation with the collision term (2.63), (2.64), (2.65), and (2.66): (i) the Maxwellian (2.70) is the solution expressing the equilibrium state, where ρ0 , v0i , and T0 are constants, (ii) the same conservation equations for macroscopic variables can be derived as those from the standard Boltzmann equation, and (iii) the Boltzmann H-theorem16 holds [35]. By many theoretical and numerical studies, it has been confirmed that not only qualitatively but also quantitatively similar results are obtained for the BKW equation and the standard Boltzmann equation, except that the BKW equation gives the Prandtl number equal to unity [35]. Furthermore, the BKW equation has an advantage that two components of molecular velocity can be eliminated in spatially one-dimensional problems [35], and this considerably reduces the computational cost in simulation studies. The Gaussian– BGK Boltzmann equation [2] also has the same advantage, and as a result, we have obtained several substantial results in the analysis of shock-tube experiment for the condensation coefficients of water and methanol, as shown in Chap. 3.

16 The Boltzmann H-theorem corresponds to the entropy inequality extended to nonequilibrium states [35]. The theorem states that the H function or the integral H of H function over a domain D,  

H=

f ln( f /c) dξ

or

H=

H dx, D

never increases by an inequality     dH [1 + ln( f /c)]J ( f ) dξ dV  0, − (Hi − H vwi )n i dS = dt ∂D D if (Hi − H vwi )n i = 0 on the boundary ∂ D, where c is a constant to make f /c dimensionless and  Hi = ξi f ln( f /c) dξ .

2.3

Kinetic Theory of Gases

43

2.3.2 Boundary Condition for the Boltzmann Equation The gas molecules impinging on the surface of solid or liquid are scattered by some rule. That is, the velocity distribution of molecules leaving the boundary should be regulated by some rule other than the molecular interaction law in the gas. This is the boundary condition for the Boltzmann equation and called the kinetic boundary condition. In the case of a solid boundary [see Fig. 2.6a], the commonly used one is the diffuse-reflection condition [35], f (x, ξ , t) =

% & [ξi − vwi (x, t)]2 ρw exp − , 2RTw (x, t) [2π RTw (x, t)]3/2

(2.76)

for molecules leaving the boundary with the velocity ξ satisfying ξ · n(x, t) > v w (x, t) · n(x, t),

(2.77)

at a point x on the surface of the boundary and at a time t, where Tw (x, t) and v w (x, t) are respectively the temperature and velocity at the point on the surface of the boundary, n(x, t) is the unit vector normal to the surface and pointing to the gas phase, and  ρw = − ×

2π RTw (x, t) 

1/2

ξ j n j (x,t) 0, and the other to the region where Ξ < 0. It should be noted that such a hexahedron can be constructed in view of the assumed regularity of Ξ ; hence the four lateral faces of the hexahedron intersect the interface. We refer to these four lateral faces as A, B, C, and D, and to corresponding unit normals directed outward to be N A , N B , N C , and N D , respectively, as shown in Fig. 5.2. We introduce a local Cartesian-orthogonal-coordinates frame with the origin at P, and with axes ξ parallel to N C , η parallel to N D , and ζ parallel to n P . The lengths of sides of the hexahedron in the directions of ξ , η, and ζ are set as dξ , dη, and dζ , respectively. Note that the localization theorem [8]1 and the Gauss divergence theorem together yield the following interpretation of the divergence [8]: Let Φ be a continuous scalar or vector field on an open set R in E , where E is*a three-dimensional Euclidean point space. Then given any x0 ∈ E , Φ(x0 ) = limδ→0 (1/vol(Ωδ )) Ωδ Φd V , where Ωδ

1

146

5 Dynamics of Spherical Vapor Bubble nA A

P

nP

NA

B

C

D

nC NB NC dϑI

ND ζ

dϑI dϑI η

dϑII ξ

Fig. 5.2 The hexahedron with infinitesimal volume ΔV

1 ∇ · n = lim V →0 V

 ∂V

n · Nd (∂ V ) ,

(5.8)

where N is the unit normal directed outward on the boundary ∂ V of the faces considered, and n is naturally defined as the extension of Eq. (5.6) to the case of Ξ (x, t) = 0: n(x, t) = ∇Ξ (x, t)/ |∇Ξ (x, t)|

for any Ξ (x, t).

(5.9)

Now, we calculate the right-hand side of Eq. (5.8). We evaluate the area integration of the dot products of the outward directed unit normal N to the cube and the unit normal n to the interface for all six faces. The variation of the inner product n · N evaluated on a face can be neglected since we are considering the limit of the infinitesimally small area of the face; hence the unit normal n is constant on the surface. In the limit, the contributions of the faces perpendicular to n P cancel. From the faces parallel to the plane (η, ζ ) we have the contribution to the flux given by lim

V →0

nC · N C − n A · N A dηdζ. dξ dηdζ

(5.10)

Referring to Fig. 5.2, we find that n C · N C = −n A · N A = sin dϑ I ∼ = ϑ I , and I I I dξ = 2dϑ R where R is the radius of curvature of the trace of the interface on the (ξ, ζ ) plane passing through n P . The limit of Eq. (5.10) thus reduces to 1/R I . In the same way, from the two faces parallel to the plane (ξ, ζ ), we have

(δ > 0) is the closed ball of radius δ centered at x0 . Therefore, if ball Ω ⊂ R, then Φ = 0.

* Ω

Φd V = 0 for every closed

5.2

Kinematics of Interface

147

lim

V →0

n D · N D − nB · N B dξ dζ. dξ dηdζ

(5.11)

∼ ϑ II , and dη = 2dϑ II R II where We find that n D · N D = −n B · N B = sin dϑ II = II R is the radius of curvature of the trace of the interface on the (η, ζ ) plane passing through n P . The limit of Eq. (5.11) thus reduces to 1/R II . From Eqs. (5.10) and (5.11), we finally obtain the following relation: ∇·n=

1 1 + II . RI R

(5.12)

It should be noted that the orientation of the (ξ, η) axes in the tangent plane to the interface at the point P used here is arbitrary. Consequently, we obtain the important feature of Eq. (5.12) that (1/R I + 1/R II ) is indifferent to rotation of the hexahedron around n P , although values of R I and R II vary in general. The total curvature C of the interface at the point P is defined by C = 1/R + 1/R  ,

(5.13)

where R and R  are the radii of curvature of the section of the interface with any two planes orthogonal to each other and to the interface at P. The family of surfaces Ξ = constant defines, by means of Eq. (5.6), a vector field of normals; hence with the use of Eq. (5.12), the total curvature C can be written as C =∇·n

Ξ = 0.

on

(5.14)

5.2.3 Time Variation of Area of Surface Element The interface is a function of time t, as expressed by Eqs. (5.1) or (5.3) in the Lagrangian description. For further discussion of the interface motion, the increase in per unit surface area and unit time (1/Ξ )(DΞ/Dt) should be obtained. First, we consider a volume V(t), in general, bounded by a closed surface S(t). On the surface, the unit normal N is taken outward, and the velocity field both on the surface and inside the volume V(t) is vs (x, t). Now, we consider the volume that the closed surface S sweeps out during an infinitesimal time Δt. With the use of the Gauss divergence theorem, this volume can be calculated: 

 Δt



 S

vs · NdSdτ =

Δt

V

∇ · vs dVdτ.

(5.15)

We notice that the total volume V consists of the summation of each infinitesimally * small volume element dV i.e., V = V dV. We, here, consider infinitesimally small volume element V (t) bounded by a closed surface S(t), instead of dV. Then, Eq. (5.15) can be rewritten as

148

5 Dynamics of Spherical Vapor Bubble

1 dV (τ ) = ∇ · vs . V (τ ) dτ

(5.16)

The left-hand side of Eq. (5.16) is the increment of volume element due to the surface movement during an infinitesimally small time dt divided by V ; hence Eq. (5.16) can be rewritten as V (t + dt) = V (t) + ∇ · vs dt V (t) = (1 + ∇ · vs dt) V (t).

(5.17)

Let us consider the special case that the infinitesimal volume element V (t) is in a form of cylinder orthogonal to the interface and protruding equally on either side (interior and exterior) of the interface. Let an intersectional surface between V (t) and the interface be an infinitesimal surface element S(t). The peripheral edge of this surface element S(t) is assumed well-smooth-closed line and defined as C, and the origin of ζ -coordinate is located on the interface. It should be noticed here that this infinitesimal surface element S(t) is a function of time in the Lagrangian description as defined in Eq. (5.3). We notice that the closed surface enclosing the volume element V in the form of cylinder as shown in Fig. 5.3 can be divided into a lateral surface and a pair of upper and lower base-plane surfaces. The lateral surface of this cylinder is defined as S s . The upper surface of this cylinder in the external side is defined as S e , located on the coordinate ζ of ζ e , and similarly the lower surface is defined as S i , located on the coordinate ζ of ζ i , where the superscript e (for external side) denotes quantities evaluated on the positive side of Ξ and the superscript i (for internal side) denotes quantities evaluated on the negative side of Ξ . The unit normals to these bases are defined as N e = n |ζ =ζ e ,

and

N i = −n |ζ =ζ i ,

Ne Se | ζ e|

N S

h C

nC

| ζi | Si

ζ

η

Ni ξ

Fig. 5.3 The infinitesimal volume element

(5.18)

5.2

Kinematics of Interface

149

where n is defined by Eq. (5.9). The height of the cylinder, h, is given by h = ζe − ζi.

(5.19)

Now, the volume element V (t), as the one shown in Fig. 5.3, can be simply expressed by V (t) = h(t)S(t).

(5.20)

Substituting Eq. (5.20) into Eq. (5.17), we obtain S(t + dt) = h(t)S(t)(1 + ∇ · vs dt)/ h(t + dt).

(5.21)

We assume that the velocity vs is continuous and differentiable at the interface, then the following expansions are allowed: ∂(vs · n) e ζ , ∂ζ ∂(vs · n) i ζ , = vs · n |ζ =0 + ∂ζ

vs · n |ζ =ζ e = vs · n |ζ =0 +

(5.22)

vs · n |ζ =ζ i

(5.23)

where terms of O[(ζ e )2 ] and O[(ζ i )2 ] have been neglected and the similar approximations will be made hereafter. Then, with the use of Eqs. (5.22) and (5.23), we can write h(t + dt) as h(t + dt) = ζ e (t + dt) − ζ i (t + dt) ∂(vs · n) = h(t) + h(t)dt. ∂ζ

(5.24)

Substituting Eq. (5.24) into Eq. (5.21), we obtain   ∂ S(t + dt) − S(t) = ∇ · vs − (vs · n) S. dt ∂ζ

(5.25)

Here, it should be noted that S appearing in Eq. (5.25) is in the Lagrangian description as in Eq. (5.3). We had better rewrite Eq. (5.25) in the Eulerian description for convenience in deriving further equations; hence, in the limit dt → 0, the left-hand side of Eq. (5.25) is replaced by the convective derivative of Ξ following the motion of the interface, DΞ/Dt: 1 DΞ = ∇ · vs − (n · ∇)(vs · n), Ξ Dt

(5.26)

150

5 Dynamics of Spherical Vapor Bubble

with D/Dt = ∂/∂t + vs · ∇.

(5.27)

The fact that the differentiation of a scalar with respect to ζ is equivalent to the projection of the gradient of this scalar in the direction of the normal to the interface is used. For further investigation of the right-hand side of Eq. (5.26), the surface divergence should be discussed.

5.2.4 Surface Divergence Now, we consider a vector j which is continuous and differentiable at the interface, and apply the Gauss divergence theorem to this vector field to j :   j · NdS V = ∇ · j dV, (5.28) SV

V

where S V denotes the closed surface of a volume V and N is the unit normal directed outward at an arbitrary point on S V . Here, we use the same infinitesimal cylindrical volume element as that introduced in the previous section; then the surface can be divided into a lateral surface S s and a pair of upper and lower base-plane surfaces S e , S i . The unit normals to bases have already defined in Eq. (5.18), and the unit normal to the lateral surface is defined as N s . The left-hand side of Eq. (5.28) is rewritten as     V e e i i j · NdS = j · N dS + j · N dS + j · N s dS s SV Se Si Ss   . = j · nC dl + O(h 2 ), ( j · n)ζ =ζ e dS e − ( j · n)ζ =ζ i dS i + h Se

Si

C

(5.29) with the use of the definition of the height of the infinitesimal cylinder h (given by Eq. (5.19)), where nC is the unit normal directed outward to C, which has been already defined in the previous subsection as the peripheral edge of the intersectional surface between V (t) and the interface. Notice that since j is continuous and differentiable at the interface, the following expansions are allowed:  ∂( j · n)  ζ e, (5.30) ( j · n)ζ =ζ e = ( j · n)ζ =0 + ∂ζ ζ =0  ∂( j · n)  ζi. (5.31) ( j · n)ζ =ζ i = ( j · n)ζ =0 + ∂ζ  ζ =0

We first evaluate the integration of the first and second terms in the left-hand side of Eq. (5.29) by substituting Eqs. (5.30) and (5.31) into Eq. (5.29):

5.2

Kinematics of Interface

151



 ( j · n)ζ =ζ e dS e − ( j · n)ζ =ζ i dS i Se Si     ∂( j · n)  1 1 = h ( j · n)ζ =0 dS + h dS + O(h 2 ) +  R R ∂ζ ζ =0 S S     ∂( j · n)  dS + O(h 2 ), =h (5.32) ( j · n)ζ =0 (∇ · n) + ∂ζ ζ =0 S

where the change of the integration domains is obtained as follows. Suppose that (ξ, η) is a system of orthogonal coordinates on the interface, and (ξ e , ηe ) and (ξ i , ηi ) are similar coordinates on the surface S e and S i , respectively. Hence we have dS e = dξ e dηe = (1 + ζ e /R)(1 + ζ e /R  )dξ dη, dS i = dξ i dηi = (1 + ζ i /R)(1 + ζ i /R  )dξ dη,

(5.33) (5.34)

and then neglecting higher orders of h, with the help of Eq. (5.14), leads to Eq. (5.29), i.e.,    . ∂( j · n) ( j · n)(∇ · n) + j · nC dl + O(h 2 ), dS + h ∂ζ SV S C (5.35) where |ζ =0 is dropped. Notice that the right-hand side of Eq. (5.28) can be rewritten as 

j · NdS V = h



 ∇ · j dV = h V

∇ · j dS,

(5.36)

S

and the vector field j can be decomposed as j = j ⊥ + j ,

(5.37)

where j ⊥ and j  are the components of j normal and parallel to the interface, respectively. Equating the right-hand sides of Eqs. (5.36) and (5.35), dividing through by h, and taking the limit h → 0, we obtain .

 j · n dl = C

C

S

∇ S · j  dS,

(5.38)

where ∇ S · j  denotes the surface divergence of j . With this definition of the surface divergence, Eq. (5.38) is the analogue of the Gauss divergence theorem in three-dimensional space applied to the interface. Using Eq. (5.38), we can rewrite Eq. (5.26) by substituting vs into j :

152

5 Dynamics of Spherical Vapor Bubble

1 DΞ = ∇ · vs − (n · ∇)(vs · n) = (v s · n)(∇ · n) + ∇ S · vs . Ξ Dt

(5.39)

Finally, the surface divergence of a vector field vs0 which is constant in time can be obtained from Eq. (5.39) as ∇ S · vs0 = −(vs0 · n)(∇ · n).

(5.40)

5.2.5 Equilibrium Thermodynamics of the Interface We derive an expression for entropy of the interface by a simple argument based on Carnot’s cycles [11, 19]. It should be noted that we restrict our discussion only to a reversible process. Consider a system composed of two bulk fluids facing each other with an interface of area A between. Let this area increase reversibly by δ A, keeping the temperature T of the system constant. It should be noted that the interface considered here is that of either a drop or a bubble in a bulk fluid; hence, A can be increased by deforming this drop or bubble. We assume that the bulk fluids do no work during the surface increment, for simplicity. Next, we decrease the temperature of the system to T  = T − dT , with dT > 0, keeping the area constant; then, we decrease the area by the same amount δ A keeping the temperature T  constant. Finally, we increase the system temperature by dT , i.e., raise the temperature to the initial temperature T keeping the area A constant. These sequential processes complete a cycle. We can calculate an amount of work δW done by the interface during this cycle: δW = [−σ (T ) + σ (T  )]δ A = −(dσ/dT )δ AdT.

(5.41)

Now, we consider heat transport during this cycle. Notice that the net amount of heat absorbed by the bulk fluids is zero, but the interface itself absorbs and emits the net amount of heat. We define this absorbed heat amount δ Q at the temperature T , and the emitted heat amount δ Q  at the temperature T  . Using both the first and the second laws of thermodynamics, we have δW = δ Q − δ Q  and δ Q/T = δ Q  /T  = δS, where S is the surface entropy. These equations lead to δW = (1 − T  /T )δ Q = (dT /T )δ Q.

(5.42)

Equating Eqs. (5.41) and (5.42), we thus obtain δS = −(dσ/dT )δ A.

(5.43)

Similarly, change δE in internal energy E of the interface is given by δE = T δS − δW ; hence, using Eq. (5.43), we have δE = (σ − T (dσ/dT )) δ A.

(5.44)

5.3

General Conservation Equation at Interface

153

Hence, the entropy per unit area SS = δS/δ A and the energy per unit area e S = δE/δ A are obtained [13]. Finally, Helmholtz free energy per unit area f S of the interface is obtained as f S = e S − T SS = Fs /δ A = σ.

(5.45)

It should be reminded that the total amount of work performed by the system during the isothermal transformation is given by −δF. Equation (5.45) is consistent with this fact.

5.3 General Conservation Equation at Interface 5.3.1 Conservation Equations in Bulk Fluids The bulk fluid that is not on the interface is either a pure liquid or a pure vapor; hence we consider here conservation equations for a single-phase single-component pure fluid. As we have already defined in Sect. 5.2.3, the superscript e denotes the quantities evaluated in the external side of the interface, and i does those in the internal side of the interface. When we do not have to specify if a quantity is in either side, we write the quantity without superscript (Fig. 5.4). The conservation equations of mass, momentum, and energy, in general, are respectively written as [12] ∂ρ + ∇ · (ρv) = 0, ∂t ∂ (ρv) + ∇ · (ρvv) = −∇ p + ∇ · τ + ρb, ∂t ∂ (ρe) + ∇ · (ρev) = − p(∇ · v) + ε : τ − ∇ · q + ρ S, ∂t

(5.46) (5.47) (5.48)

where ρ is the mass density of the fluid, v is the velocity of the fluid, e is the internal energy per unit mass, p is the pressure, q is the heat flux vector, b is the body force per unit mass, S is the heat generated per unit time and unit volume. The viscous stress tensor τ in Eqs. (5.47) and (5.48) is given by Interface

Fluide

i

Fluid

Fig. 5.4 Definition of quantities evaluated in the external side of the interface and in the internal side of the interface

154

5 Dynamics of Spherical Vapor Bubble

  1 τi j = 2μ εi j − δi j εkk + μb εkk δi j 3

with εi j =

$ 1# vi, j + v j,i , 2

(5.49)

where μ is the shear viscosity, μb is the bulk viscosity, and here and hereafter, the summation convention [7] is used (see Appendices A and B at the end of this book). Carefully observing Eqs. (5.46), (5.47), and (5.48), we can find that these equations can be rewritten in a unified form as ∂ (ρ f ) = −∇ · (ρ f v + φ) + ρϑ. ∂t

(5.50)

We assume that Eq. (5.50) can apply to the interface itself with the expressions for the various quantities identical in form to those applicable in the bulk fluids. Based on this assumption, we find a certain relationship between normal components of the fluxes on the two sides of the interface.

5.3.2 Conservation Equation in Frame Moving with Interface Let ξ (t) be the instantaneous position of a point P on the interface in the laboratory frame F, and v 0 = ξ˙ be its velocity, where the overdot indicates the derivative with respect to time. Let us express an arbitrary point in F. We introduce a new reference frame F  with coordinates (x  , t  ). The frame F  is moving with P with axes parallel to those of F such that P is instantaneously at rest in F  . The relation between the frames F and F  are given as x = x − ξ ,

t  = t.

(5.51)

With the use of the chain rule, the differential operators ∂/∂t and ∇ are expressed in terms of (x  , t  ) by ∂ ∂ =  − v0 · ∇  , ∂t ∂t

∇ = ∇ .

(5.52)

In particular, the particle velocity v  in F is expressed by the particle velocity v in F as v = v − v 0 .

(5.53)

Notice that v 0 is the velocity of the specified point P, and hence it is a function of t alone. Substituting Eq. (5.52) into Eq. (5.50) gives ∂ (ρ f ) = −∇ · (φ + ρ f v  ) + ρϑ. ∂t 

(5.54)

Equation (5.54) represents the conservation equation in the moving frame F  .

5.3

General Conservation Equation at Interface

155

5.3.3 Integration Form of Conservation Equation Consider an infinitesimal volume element V in the form of cylinder orthogonal to the interface and protruding equally to interior and exterior regions at the interface. This volume element cuts an infinitesimal surface element S from the interface, as seen in Sect. 5.2.3. Let P be a point on S. Integration of Eq. (5.54) over the volume of the cylinder in the frame F  moving with the interface can be carried out by noticing that the integration and differentiation are commutable since the infinitesimal volume element V can be regarded as constant in the moving frame: d dt 





∇ · (φ + ρ f v  )dV +

ρ f dV = − V



V

ρϑdV + Σ,

(5.55)

V

where the last term Σ accounts for a process corresponding to the contribution of a conserved quantity on or along the interface. The first integral in the left-hand side can be rewritten with the use of the Gauss divergence theorem as 

∇ · (φ + ρ f v  )dV = V

 SV

(φ + ρ f v  ) · NdS V ,

(5.56)

where S V denotes the closed surface of V and N is the unit normal directed outward at a point of S V . Substituting Eq. (5.56) into Eq. (5.55) leads to d dt 



 ρ f dV = − V

SV

(φ + ρ f v  ) · NdS V +

 ρϑdV + Σ.

(5.57)

V

In principle, defining s as the surface density of a conserved quantity, we have the following characteristics on s: (i) Accumulation of the conserved quantity on the interface at a rate −∂s/∂t  per unit area and unit time. The minus sign is because an increase in s decreases the outgoing flux; (ii) Generation of s on the interface at a rate χ per unit area and unit time; (iii) Convective transport of s along the interface with surface flux svs  where vs  denotes the tangential component of the surface velocity, v s in F  ; (iv) Transport of s along the interface by means of a surface flux j  of non-convective origin. Considering the above characteristics of s, we obtain the following explicit expression for Σ:  .   ∂s  −  + χ dS − ( j  + svs ) · nC dC Σ= ∂t S C    ∂s   −  + χ − ∇ S · ( j  + svs ) dS, = ∂t S

(5.58)

where C is the closed line enclosing the surface element S of the interface, ∇S is the surface divergence in the moving frame F  , and nC is the unit normal vector to C.

156

5 Dynamics of Spherical Vapor Bubble

Notice that the surface divergence theorem (5.38) and the relation of the differential operators between the laboratory and moving frames (5.52) are used.

5.3.4 Flux Balance on Interface We come back to Eq. (5.57) and take the height h of the circular cylinder to be an infinitesimal of a higher order than the diameter of S. We consider the limit of h → 0, noticing that the integrand of volume integrals, i.e, the term in the left-hand side and the second term in the right-hand side, in Eq. (5.57) are bounded. Then the volume integration in Eq. (5.57) can be eliminated but only the surface integration survives. i.e.,  0=−

SV

(φ + ρ f v  ) · NdS V + Σ.

(5.59)

Substituting Eq. (5.58) into Eq. (5.59) leads to 

 



SV

(φ + ρ f v ) · NdS = V

S

 ∂s   −  + χ − ∇ S · ( j  + svs ) dS. ∂t

(5.60)

In the limit of h → 0, the surface integral domain S V in the left-hand side, consists of only S e and S i which are defined in Sect. 5.2.3. Noticing that the areas of S, S e , and S i are the same, and defining n as the unit normal on the interface in the direction of the external region, we can rewrite Eq. (5.60) as 

 ( ' e e e e (φ + ρ f v ) · n dS − [(φ i + ρ i f i v i ) · n]dS S S    ∂s   −  + χ − ∇ S · ( j  + sv s ) dS. = ∂t S

(5.61)

Since S is arbitrary, we obtain the following relation at an arbitrary point P on the interface:   ∂s (φ e + ρ e f e v e ) − (φ i + ρ i f i v i ) · n = −  + χ − ∇ S · ( j  + sv s ). (5.62) ∂t The surface divergence of the last term in the right-hand side of Eq. (5.62) can be rewritten as ∇ S · (sv s ) = ∇ S · (sv s ) = v s · ∇ S s + s∇ S · (vs − v0 ) ( ' s DΞ , = s ∇ S · vs + (vs · n)(∇ · n) = Ξ Dt

(5.63)

5.3

General Conservation Equation at Interface

157

where Eqs. (5.39), (5.40), and (5.53) are used. It should be noted that v s vanishes at the point P by definition, while v s is in general nonzero at points in the neighborhood of P on the interface; hence, v0 = v s − v s = v s at the point P. Because s is defined only on the interface so that ∇s = ∇ S s, Eqs. (5.27) and (5.52) give ∂s = ∂t 



     ∂ ∂ ∂ Ds + v0 · ∇  s = + vs · ∇ s = + vs · ∇ S s = . (5.64) ∂t ∂t ∂t Dt

Therefore, Eq. (5.62) is rewritten as   (φ e + ρ e f e v e ) − (φ i + ρ i f i v i ) · n =−

( ' Ds − s −(vs · n)(∇ · n) + ∇ S · vs − ∇ S · j  + χ . Dt

(5.65)

This equation shows that the outgoing normal flux of a conserved quantity Q equals to the incoming normal flux: (i) Minus the total time variation of the surface density of Q (the first term in the right-hand side); (ii) Minus the amount of Q necessary to maintain at the level s which is the surface density of Q in the newly formed surface area (the second term); (iii) Plus the non-convective influx of Q from neighboring points on the interface (the third term); (vi) Plus the rate of production of Q on the surface (the fourth term).

5.3.5 Conservation of Mass on Interface We now consider conservation equations on the interface between the pure liquid and the pure vapor by applying the general conservation equation, Eq. (5.65), on the interface. We first study the conservation equation of mass. We have already obtained the equation of conservation of mass in bulk fluid, Eq. (5.46): ∂ρ + ∇ · (ρv) = 0. ∂t This equation has the general form of the conservation equation (5.50): ∂ (ρ f ) = −∇ · (ρ f v + φ) + ρϑ. ∂t by taking f = 1, ϑ = 0, and φ = 0. Notice that the surface discontinuity can be defined in such a way that no molecules can accumulate on it [19]; hence, s = 0, j  = 0, and χ = 0, then Eq. (5.65) reduces to ! " # $ ρ e v e − vs · n = ρ i v i − vs · n.

(5.66)

158

5 Dynamics of Spherical Vapor Bubble

With the use of Eq. (5.66), we can rewrite Eq. (5.65) as ' ( (φ e − φ i ) · n + ρ e ( f e − f i ) (v e − vs ) · n   = (φ e − φ i ) · n + ρ i ( f e − f i ) (vi − vs ) · n =−

( ' Ds − s −vs · n(∇ · n) + ∇ S · vs − ∇ S · j  + χ . Dt

(5.67)

Equation (5.66) simply expresses the mass flux balance on the interface: outgoing mass flux from one side of the interface should be equal to incoming mass flux to another side of the interface. It should be strongly emphasized that this equation provides no information regarding to the amount of mass flux and this amount must be evaluated at the molecular level according to the results of MD, molecular gas dynamics, and shocktube experiments presented in Chaps. 2 and 3. As discussed in Sects. 2.5 and 2.6 the physically correct boundary conditions for the velocities v V and temperature TV of the vapor at the interface to a Navier-Stokes set of equations are given by αe p∞ − p ∗ = ∗ p αc

  √ 1 − αc (vV i − vwi ) n i α e − αc ∗ + , C4 − 2 π √ αc αc 2Rc TL TV − TL (vV i − vwi ) n i = d4∗ √ , TL 2Rc TL

(5.68) (5.69)

where TL is the liquid temperature at the interface, vw is the interface velocity, p∗ is the saturated vapor pressure, p∞ is the vapor pressure outside Knudsen layer, Rc is the gas constant per unit mass, αe is the evaporation coefficient, αc is the condensation coefficient, C4∗ and d4∗ are given in Chap. 2; αe and αc are given in Chap. 3. From the above two equations and the equation of state ( p∞ = ρ Rc TV ), we can obtain the mass flux j = ρ(vV i − vwi )n i as follows: p∞ / p∗ ! √ " j= ∗ c αe 2 π 1−α − C 4 αc

2 Rc

  p∗ p∞ αe √ − αc √ , TL TL

(5.70)

√ where terms of O[(V / 2π TL )2 ] are dismissed. For αe = αc = α, Eq. (5.70) becomes p∞ / p ∗

j = √ 1−α 2 π α − C4∗

2 Rc



p∞ p∗ √ −√ TL TL

 .

(5.71)

5.3

General Conservation Equation at Interface

159

5.3.6 Conservation of Momentum on Interface The conservation equation of momentum in bulk fluid is given by Eq. (5.47): ∂ (ρv) + ∇ · (ρvv) = −∇ p + ∇ · τ + ρb. ∂t As we have already observed in Sect. 5.3.1 that Eq. (5.47) can be rewritten in the general form of the conservation equation (5.50), ∂ (ρ f ) = −∇ · (ρ f v + φ) + ρϑ, ∂t by f = v, ϑ = b, and φ = p I − τ where I is the three-dimensional identity tensor. No momentum can accumulate on the interface so that s = 0, since the interface has zero mass. We can also take the surface source χ to vanish, except in rather special circumstances (e.g., an electrically charged interface). On the other hand, the surface momentum flux exists and is connected to the existence of surface tension. Notice that flux j  in Eq. (5.67) to be considered here in this subsection has the form of the second order tensor. For f = v in Eq. (5.65), ρvv · n in the left-hand side of this equation is the momentum flux, i.e., force acting on a unit surface, so is ∇ S · j  ; hence the total force acting on the surface S as defined in Fig. 5.3 with a closed contour C which is the line around S: C = ∂ S, on the interface is given by  . . σ I S · nC dl, (5.72) F = − σ nC dl = ∇ S · j  dS = C

S

C

where σ is the surface tension, nC is the outward unit normal to C, and I S is the two-dimensional identity tensor on the tangent plane of the interface. Comparing Eq. (5.72) with the explicit form of the conserved quantity taking place on or along the interface, Eq. (5.58), gives the expression of j  : j  = −σ I S

or jin = −σ δin .

(5.73)

If the positive direction on the closed contour C is chosen as the fingers of the right hand when the thumb is in the direction of the normal n to the surface S, the outward unit normal nC to the closed contour C is related to the unit tangent t C to ∂ S by nC = t C × n

C or n C n = kmn tk n m .

For further consideration of Eq. (5.72), the Stokes theorem [22] is useful:  . u i tiC dl = n i i jk ∂ j u k dS, C

S

(5.74)

(5.75)

160

5 Dynamics of Spherical Vapor Bubble

where u i is an arbitrary vector. The right-hand side of Eq. (5.75) is an integration over the surface S, and the left-hand side is an integration over the closed contour C bounding S; hence the Stokes theorem is used to simplify line integrals. We take S as the surface S as defined in Fig. 5.3, and C as the closed contour which is the line around S. It turns the line integral of u along its boundary C of the closed surface S into the surface integral of the derivative of u (the curl) over the interior of S Simple expansion of Eq. (5.75) by replacing vector u i to any order of tensorial quantity (T . . .k ) provides the following generalized Stokes theorem [5] (see Appendix C at the end of this book) 

. C

T . . .k tkC dl

=

n i i jk ∂ j (T . . .k )dS.

(5.76)

S

Notice that the integrand of the most right-hand side of Eq. (5.72) can be rewritten by the use of Eqs. (5.73) and (5.74) as C σ I S · nC dl = σ δln n C n dl = σ δln kmn n m tk dl.

(5.77)

Substituting (T . . .k ) = (σ δln kmn n m tk ) into Eq. (5.76), and applying Eq. (A.22) in Appendix A, we can rewrite Eq. (5.72) as .



Fl = C

(σ δln kmn n m )tkC dl =

n i i jk ∂ j (σ δln kmn n m )dS S

[n m ∂l (σ n m ) − n l ∂m (σ n m )] dS.

=

(5.78)

S

Comparing the integrand of Eqs. (5.72) and (5.78), and observing that n · n = 1 and that n and ∇α = ∇ S α are mutually orthogonal, we obtain ∇ S · j  = −(σ/2)∇(n · n) + n(n · ∇σ ) + nσ (∇ · n) − ∇σ = nσ (∇ · n) − ∇σ.

(5.79)

With the use of Eqs. (5.47) and (5.79), we obtain ρ e v e (v e − vs ) · n + p e n − τ e · n = ρ i v i (v i − vs ) · n + pi n − τ i · n + ∇σ − nσ (∇ · n),

(5.80)

or, with the use of Eq. (5.67) we obtain the following form of conservation equation of momentum: ' ( ρ e (v e − v i ) (v e − vs ) · n = −( p e − pi )n + (τ e − τ i ) · n + ∇σ − nσ (∇ · n).

(5.81)

5.3

General Conservation Equation at Interface

161

5.3.7 Conservation of Energy on Interface Now, we consider the conservation equation of energy at the interface. The conservation equation of energy in bulk fluid is given by Eq. (5.48), i.e.,  ρ

 ∂e + (v · ∇)e + ∇ · q = − p(∇ · v) + ε : τ + ρ S. ∂t

(5.82)

We should recall that Eq. (5.48) can be rewritten in the general form of the conservation equation (5.50) ∂ (ρ f ) = −∇ · (ρ f v + φ) + ρϑ, ∂t with f = (1/2)v · v + e, ϑ = S, and the energy flux vector φ: φ = v · ( p I − τ ) + q,

(5.83)

substituted. In the local thermodynamic equilibrium, the internal energy per unit area e S of the interface, as obtained in Sect. 5.2.5, is written as e S = σ − T (dσ/dT ).

(5.84)

Substituting f = (1/2)v · v + e in Eq. (5.65), we obtain the term ρv · n that is the energy flux, in the left-hand side Eq. (5.65); hence j  is the surface energy flux of non-convective origin. This flux j  can be split into two terms accounting for the transport of mechanical energy and that of thermal energy. We consider the nonconvective transport of thermal energy across an element of area normal to the interface. We may assume that this transport vanishes as the element of area shrinks to a line lying on the interface. Thus j  consists of only transport of mechanical energy. Now, we consider first the rate at which the mechanical work done by the surface tension force P on the surface S, as defined in Fig. 5.3, with the closed contour C on the interface as written by . P= σ v · nC dC. (5.85) C

Comparing this equation with Eqs. (5.54) and (5.58), we obtain j  = −σ vs .

(5.86)

We may take the rate of surface energy production χ in Eq. (5.65) to vanish, as we have neglected the momentum surface source in our consideration on the conservation of the momentum. Substituting Eqs. (5.83), (5.84), and (5.86) into Eq. (5.67) leads to

162

5 Dynamics of Spherical Vapor Bubble

  ( ' 1 2 2 ρ e (v e − vs ) · n ee − ei + (v e − vi ) 2 # " ! $ + v e · ( pe I − τ e ) + q e − v i · ( pi I − τ i ) + q i · n   ( D dσ dσ ' =T v S · n(∇ · n) + ∇ S · vs − σ −T Dt dT dT +v s · ∇ S σ + σ ∇s · vs .

(5.87)

5.4 Spherical Vapor Bubble From this section, we consider a more practical situation, e.g., bubble. We assume that the center of a bubble is fixed at a point in the space and only its volume changes. Hereafter flow parameters of liquid (in Re ) will be denoted by subscript L, and the corresponding parameters of vapor (in Ri ) by the subscript V . In general, we all know most small bubbles in ordinary liquids such as water are spherical; hence we assume that the bubble is spherical [23]. Hereafter we refer to the interface as the bubble wall and assume that flow fields are spherically symmetric. In Sect. 5.1, we have discussed that the interface at a time t is given by the set of points x satisfying Eq. (5.1): Ξ (x, t) = 0, where Ξ is a scalar function of time t and position x. We assume every physical quantity depends on the space coordinates through r only, the radial distance from the bubble center. Then we can express the interface as Ξ (r, t) = 0.

(5.88)

The equation for the bubble wall, i.e., Eq. (5.88), can be given explicitly as Ξ (r ) = r − R(t) = 0,

(5.89)

where R(t) is the instantaneous radius of the bubble. From Eqs. (5.9), (5.13), and (5.14), we obtain n= where r is the radial vector.

r , r2

∇·n=

2 , r

(5.90)

5.4

Spherical Vapor Bubble

163

5.4.1 Governing Equations for Spherical Bubble Now, we consider the governing equations for bulk liquid phase and bulk vapor phase [9], without indicating the subscript L and V . The discussion in the present subsection applies to the both phases. Using vector analysis formulas [4], differential operations are simplified as ∇·v =

∂v 2v + , ∂r r

∇s =

∂s r . ∂r r

(5.91)

The conservation equations of mass, Eq. (5.46), momentum, Eq. (5.47), and energy, Eq. (5.48), have been already obtained in Sect. 5.3.1. They are rewritten in slightly different forms as ∂ρ + ∇ · (ρv) = 0,   ∂t ∂v + (v · ∇) v = −∇ p + ∇ · τ + ρb, ρ ∂t   ∂e ρ + (v · ∇) e = − p(∇ · v) + ε : τ − ∇ · q + ρ S. ∂t

(5.92) (5.93) (5.94)

With the use of Eq. (5.91), the conservation equation of mass, Eq. (5.92), is rewritten as ∂ρ ∂ 2ρv + (ρv) + = 0. ∂t ∂r r

(5.95)

The conservation equation of momentum, Eq. (5.93), is greatly simplified in the spherical coordinates. Remind that ∇ × (∇ × v) is rewritten as i jk  jlm vl,mk , then we obtain the following vector analysis relation: ∇ × (∇ × v) = ∇(∇ · v) − Δv.

(5.96)

Observing the fact that ∇ × v = 0 for spherically symmetric vector field v, we obtain Δv = ∇(∇ · v).

(5.97)

Tr(ε) = ∇ · v, 1 ∇ · ε = [∇(∇ · v) + Δv] . 2

(5.98)

Notice that

(5.99)

Then, with the use of Eq. (5.97), the radial component of the divergence of the second order tensor τ defined by Eq. (5.49) is given by

164

5 Dynamics of Spherical Vapor Bubble

 (∇ · τ )r =

4μ + μb 3



∂ ∂r



∂v 2v + ∂r r

 .

(5.100)

Substituting Eq. (5.100) into Eq. (5.93), we can obtain the conservation equation of the radial component of momentum: ∂v 1 ∂v +v = ∂t ∂r ρ



4μ + μb 3



∂ ∂r



∂v 2v + ∂r r

 + br −

1 ∂p , ρ ∂r

(5.101)

where v is the radial component of fluid velocity and br is the radial component of the external force. Now, we consider the conservation equation of energy. The second term in the right-hand side of Eq. (5.94) is the scalar product of two tensors:   1 ε : τ = 2μ εi2j − θ 2 + μb θ εi j δi j , 3

(5.102)

where θ is the divergence of velocity (∇ · v), and εi j is written as ⎛ ∂v

⎞ 0 0 ⎜ ∂r ⎟ v ⎟ ⎜ εi j = ⎜ 0 0 ⎟, ⎝ r v⎠ 0 0 r

(5.103)

for the spherically symmetric flow, and hence  (εi j )2 =

∂v ∂r

2 +2

! v "2 r

.

(5.104)

Equation (5.102) thus becomes 4μ ε:τ = 3



∂v v − ∂r r



2 + μb

∂v 2v + ∂r r

2 .

(5.105)

Substituting Eq. (5.105) into Eq. (5.94), with Eq. (5.91), we obtain the conservation equation of energy:  ρ

∂e ∂e +v ∂t ∂r





  ∂v 2v 4μ ∂v + − + ∂r r 3 ∂r   2v 2 ∂q 2q ∂v + − +μb − ∂r r ∂r r

= −p

where the heat flux vector q has the radial component q only.

v r

2

+ ρ S,

(5.106)

5.4

Spherical Vapor Bubble

165

5.4.2 Simplification For the ease of the discussion, we introduce further several simplifications in the following. We assume: (i) No body force is present, b = 0, (ii) Bulk viscosity coefficients of liquid and vapor are zero, μb = 0, (iii) The effect of the interaction between compressibility and viscosity is negligible since the liquid is essentially incompressible, (iv) Fourier’s Law, q = −λ∇T,

(5.107)

holds in both the liquid and vapor phases, where λ is the coefficient of the thermal conductivity. We assume that λ is constant for the liquid and λ is a function of temperature for the vapor, (v) No heat generation is present, S = 0. The normal components of velocities are rewritten as (v L · n)r =R(t) = w L ,

(v V · n)r =R(t) = wV ,

˙ (vs · n)r =R(t) = R,

(5.108)

where w L and wV are the liquid and vapor velocities at the bubble wall, respectively, and R˙ is the bubble wall velocity. 5.4.2.1 Conservation Equations for Bubble Exterior For the liquid in the exterior of the bubble, the conservation equation of mass, Eq. (5.95), is written as ρL ∂ ! 2 " ∂ρ L ∂ρ L + vL + 2 r v L = 0. ∂t ∂r r ∂r

(5.109)

Let us assume that the liquid is incompressible, ρ L = const., then the above equation can be further simplified: 1 ∂ ! 2 " r v L = 0. r 2 ∂r

(5.110)

Then, with the use of Eq. (5.108), the velocity is easily obtained: vL (r, t) =

R(t)2 w L (t). r2

(5.111)

The conservation equation of momentum is given by Eq. (5.101):     4μ L ∂ Dρ L 1 ∂ pL ∂v L ∂v L + vL = μbL + . + br − ∂t ∂r 3 ∂r Dt ρ L ∂r

(5.112)

The second term in the right-hand side is the body force term can be neglected by the assumption (i); b = 0. The first term in the right-hand side is apparently the

166

5 Dynamics of Spherical Vapor Bubble

interaction between compressibility (Dρ/Dt) and viscosity; hence this term can be neglected by the assumption (iii). Consequently, the following equation of motion ∂ pL = −ρ L ∂r



∂v L ∂v L + vL ∂t ∂r

 ,

(5.113)

is obtained. Substituting Eq. (5.111) into Eq. (5.113) and integrating with respect to r from r = ∞ to r , we obtain  p L (r, t) − p L (r = ∞, t) = ρ L

˙ L + R 2 w˙ L 2R Rw 1 R 4 w2L − r 2 r4

 .

(5.114)

Similarly, by neglecting the bulk viscosity and heat source [the assumption (ii)] and by substituting Eq. (5.107) [the assumption (iv)], we can simplify conservation equation of energy (5.106) as follows:  ρL

∂e L ∂e L + vL ∂t ∂r



4μ L = 3



∂v L vL − ∂r r

2

λL ∂ + 2 r ∂r

  2 ∂ TL r , ∂r

(5.115)

where we have used Eq. (5.110) to eliminate the first term in the right-hand side of Eq. (5.106). We assume that e L can be written as follows: e L = c L TL ,

(5.116)

where c L is the specific heat of liquid and assumed constant. We also assume that viscous heat dissipation, the second term in the right-hand side of Eq. (5.115), is negligible compared to heat conduction, the third term in the right-hand side of Eq. (5.115). With the use of Eq. (5.111), we obtain R 2 w L ∂ TL λL 1 ∂ ∂ TL + = ∂t ρ L c L r 2 ∂r r 2 ∂r

  ∂ TL r2 . ∂r

(5.117)

5.4.2.2 Conservation Equations for Bubble Interior The conservation equations of mass (5.95), momentum (5.101), and energy (5.106) for the vapor are respectively rewritten as " ∂ρV 1 ∂ ! 2 + 2 r vV ρV = 0, ∂t r ∂r ∂vV 1 ∂ pV 4 μV ∂ ∂vV + vV =− + ∂t ∂r ρV ∂r 3 ρV ∂r



∂vV 2vV + ∂r r

(5.118)  ,

(5.119)

5.4

Spherical Vapor Bubble

 ρV

∂eV ∂eV + vV ∂t ∂r

167





   ∂vV 2vV 4μV ∂vV vV 2 = − pV + + − ∂r r 3 ∂r r   ∂ T 1 ∂ V + 2 r 2 λV , (5.120) r ∂r ∂r

where the assumptions (i)–(v) except for (iii) have been used. We assume that eV and pV can be written as follows: # $ eV = cvV (TV )TV = c pV (TV )/γ0 TV , pV = ρ V Rc TV ,

(5.121) (5.122)

where Rc is the gas constant per unit mass, cvV is the specific heat of vapor at constant volume of vapor. The ratio of specific heats, γ0 in Eq. (5.121), is defined by γ0 = cpV (Tr )/cvV (Tr ) = c pV 0 /cvV 0 ,

(5.123)

where Tr is a reference temperature, i.e., constant. Mayer’s relation is written as Rc = cpV0 − cvV 0 = cvV (Tr )(γ0 − 1),

(5.124)

and hence the equation of state (5.122) becomes pV =

γ0 − 1 c p0 ρV TV . γ0

(5.125)

We rewrite the first term in the right-hand side of Eq. (5.120) with the use of Eqs. (5.118), (5.122), and (5.124):  2vV ∂vV + − pV ∂r r     ∂ pV ∂ pV ∂ TV ∂ TV = + vV + vV − ρV (c pV − cvV ) . ∂t ∂r ∂t ∂r 

(5.126)

We also assume that viscous heat dissipation, the second term in the right-hand side of Eq. (5.120), is negligible compared to heat conduction, the third term in the right-hand side of Eq. (5.120). Then, substituting Eq. (5.126) back into Eq. (5.120) leads to  ρV

   ∂(c pV TV ) ∂(c pV TV ) ∂ TV 1 ∂ d pV + vV + 2 = r 2 λV . (5.127) ∂t ∂r dt ∂r r ∂r

168

5 Dynamics of Spherical Vapor Bubble

5.4.3 Boundary Conditions With the use of Eq. (5.108), the conservation equation of mass at the bubble wall (5.66) becomes ˙ = ρ Lw (w L − R), ˙ ρV w (wV − R)

(5.128)

where ρV w and ρ Lw are densities of vapor and liquid at the bubble wall, respectively; however, we have already assumed that ρ L , μ L , and λ L are constant. Then ρ Lw = ρ L . The sign of Eq. (5.128) is positive for condensation and negative for evaporation. The molecular mass flux j [Eq. (5.70)] is defined to be positive for evaporation (negative for condensation). Therefore, Eq. (5.128) should be equal to − j. We discuss the above relation taking bubble expansion as an example. Suppose that a vapor bubble in an equilibrium state begins to expand by a decrease of the pressure in the ambient liquid, as shown in Fig. 5.5. As bubble grows, pressure inside bubble may decrease below saturated vapor pressure at the temperature of the liquid; hence evaporation occurs. This evaporation induces the difference between the vapor velocity at the interface and bubble wall velocity by an additional velocity associated with the mass flux, as shown in Fig. 5.5. Therefore we obtain ˙ = ρV w (wV − R) ˙ = − j. ρ L (w L − R)

(5.129)

From the above equation, the boundary condition of the bubble interior is given by wV = R˙ −

j , ρV w

Fig. 5.5 Velocity of bubble wall, and velocities of vapor and liquid at the bubble wall

(5.130)

5.4

Spherical Vapor Bubble

169

and that of the bubble exterior by j , w L = R˙ − ρL

(5.131)

where we can assume that the liquid density ρ at the bubble wall is the same as that in bulk fluid. We also have already known that the relative normal velocity of the vapor to the moving bubble wall is given by Eq. (5.70); hence Eqs. (5.130) and (5.131) can be explicitly written as p∞ / p ∗ ! √ " wV = R˙ − ∗ c αe ρV w 2 π 1−α αc − C 4 p∞ / p ∗ ! √ " w L = R˙ − ∗ c αe ρ Lw 2 π 1−α αc − C 4

2 Rc

  p∗ p∞ αe √ , (5.132) − αc √ TLw TLw

2 Rc

  p∗ p∞ αe √ , (5.133) − αc √ TLw TLw



where p ∗ is the saturated vapor pressure, and in the case of bubble, p ∗ is given by Kelvin equation [20]: ∗

p =

∗ p∞ exp

 −

2σ ρ L R c TL R

 ,

(5.134)

∗ is the saturated vapor pressure for the plain interface, and written as a where p∞ function of the liquid temperature at the bubble wall only: ∗ ∗ = p∞ p∞ (TLw ) .

(5.135)

Now, we consider the conservation equation of momentum at the bubble wall. The general equation (5.81) can be rewritten as ρ Lw (v L − v V ) [(v L − vs ) · n] = −( p Lw − pV w )n + (τ Lw − τ V w ) · n + ∇σ − nσ (∇ · n).

(5.136)

With the use of Eq. (5.129), Eq. (5.81) is also rewritten as ˙ ρV w (wV − R)(w V − w L ) = − j (wV − w L )    4μ L ∂v L  v L  = −( p L − pV w ) + −  3 ∂r r =R + r r =R +    2σ vV  4μV w ∂vV  − − , −   − 3 ∂r r =R − r r =R R or with the further use of Eqs. (5.130) and (5.131),

(5.137)

170

5 Dynamics of Spherical Vapor Bubble

  ρV w 2σ j2 1− p Lw + = pV w + R ρV w ρL         4μ L ∂v L  4μV w ∂vV  vL  vV  + − , (5.138) − −   3 ∂r r =R + r r =R + 3 ∂r r =R − r r =R − where R + is the radius of the bubble wall in the liquid and R − is that in the vapor. The conservation equation of energy (5.87):   1 2 2 ρ [(v L − vs ) · n] e Lw − eV w + (v L − v V ) 2 '# $ # $( + v L · ( p Lw I − τ Lw ) + q Lw − v V · ( pV w I − τ V w ) + q V w · n   ( dσ ' D dσ vs · n(∇ · n) + ∇ S · vs − σ −T =T Dt dT dT (5.139) + vs · ∇ S σ + σ ∇ S · vs , is rewritten as  1 2 2 − j e Lw − eV w + (w L − wV ) 2     4μ L ∂v L  v L  + w L p Lw − −  3 ∂r r =R + r r =R +     4μV ∂vV  vV  = wV pV w − −  3 ∂r r =R − r r =R −   ∂ TL  ∂ TV  2σ ˙ + λL − λV w − R, ∂r r =R + ∂r r =R − R 

(5.140)

where we assume that Fourier’s law (5.107) holds in both the liquid and vapor phases [the assumption (iv)], and that only σ vs · n(∇ · n) survives in the right-hand side of Eq. (5.139). It should be noted that there exists an ambiguity2 in definition of T appearing in Eq. (5.139); however, this ambiguity vanishes in the present study with the use of Eq. (5.66), and we obtain   ∂ TL  ∂ TV  − λ Vw ∂r r =R + ∂r r =R −  2  2  1 1 j j = j L− + 2 ρL 2 ρV w

λL

2 The ambiguity is a result of the difference between the vapor temperature and liquid temperature at the interface, as shown in Fig. 5.5 [see also Eq. (5.69)].

5.5

Practical Description of Bubble Motion

171

     ∂v L  4μV w ∂vV  v L  vV  − − −   ∂r r =R + r r =R + 3ρV w ∂r r =R − r r =R − 2σ ˙ R, (5.141) + w L p Lw − wV pV w + R +

4μ L 3ρ L



where L is the latent heat: L = eV w +

pV w p Lw − e Lw − . ρV w ρL

(5.142)

Noticing that the latent heat term usually dominates over the other terms in the right-hand side of Eq. (5.141), we obtain   ∂ TL  ∂ TV  − λV w = j L. λL ∂r r =R + ∂r r =R −

(5.143)

It should be reminded that there exists a temperature discontinuity at the bubble wall as already given by Eq. (5.69): TV w wV − R˙ = 1 + d4∗ √ . TLw 2Rc TLw

(5.144)

This discontinuity is also illustrated in Fig. 5.5.

5.5 Practical Description of Bubble Motion We now apply the formulations obtained in Sect. 5.4 to the growth and collapse of a vapor bubble. A bubble motion may start typically when the pressure in liquid increases or decreases, or when a bubble is placed in a super heated or sub cooled liquid. In the former case, the inertial effect may become the most dominant factor for the bubble motion, and on the other hand, in the later case the thermal effect may be prevailing. In the following, we derive the most general form of the solution which is applicable to both the inertially and thermally controlled bubble motion; however, the explicit form of the solution, in general, can only be obtained with the use of numerical analysis mainly because equations whose solutions should satisfy are integro-differential equations. We have derived all equations and conditions necessary for the analysis of the motion of a spherical bubble. Now, we further investigate the structure of the equation set.

172

5 Dynamics of Spherical Vapor Bubble

5.5.1 Flow Fields in Liquid First, we consider flow fields outside of a bubble, i.e., in liquid phase. Velocity field v L (r, t) and pressure field p L (r, t) are given by Eqs. (5.111) and (5.114), once R and w L and time derivatives of them are obtained. w L , in turn, can be obtained from Eqs. (5.130) and (5.131) with Eq. (5.70), with the use of the assumption that density ρ L is constant. Therefore with R obtained, v L and p L can be obtained. Now, we turn our attention to R. We derive a governing equation of bubble dynamics:  ρL

3 R w˙ L + w 2L 2



2σ + 2 jw L = pV (r = R − , t) − p L (r → ∞, t) − R       2 ρV w 4μ L ∂v L  vL  j 1− + − +  ρV ρL 3 ∂r r =R + r r =R +    4μV w ∂vV  vV  − , (5.145) −  3 ∂r r =R − r r =R −

where we have substituted Eq. (5.138) into Eq. (5.114) with r = R, with the use Eq. (5.131). Once the bubble wall velocity R˙ is obtained from Eq. (5.145), velocity fields inside and outside of the bubble can be determined from Eqs. (5.70), (5.132), and (5.133). Then, the conservation equation of energy inside the bubble, Eq. (5.127), should be solved to carry on our discussion further.

5.5.2 Uniform Pressure in Bubble Interior First, we assume that the pressure inside the bubble is uniform, and a function of time only: pV = pV (t).

(5.146)

The reason why this assumption is appropriate is discussed in the latter half of this subsection. Then, Eq. (5.145) can be further simplified by assuming that both the normal viscous stress and density in vapor phase are negligible compared to those in liquid phase:  ρL

3 R w˙ L + w 2L 2

 + 2 jw L = pV (t) − p∞ (t) −

2σ j2 wL , (5.147) − 4μ L + R R ρV

where p∞ = p L (r → ∞, t), and Eq. (5.111) is used. Notice that neglecting ˙ Then, the molecular mass flux j [Eq. (5.70)] in Eq. (5.131), we obtain w L = R. Eq. (5.147) becomes the well-known Rayleigh–Plesset equation [9].

5.5

Practical Description of Bubble Motion

173

We now verify the legitimacy of Eq. (5.146). It should be emphasized here that we do not suppose violent bubble motions such as rapid bubble collapse caused by a strong shock wave in the liquid, but rather mild bubble motions such that the bubble wall velocity can be assumed well below the speed of sound in the vapor. Recall that the local pressure disturbances in the bubble propagate with the speed of sound a in the vapor. If a is significantly larger than the characteristic speed of bubble motion, i.e., the speed of radial oscillation, VR , Eq. (5.146) can hold. Here, we consider the characteristic time tC , which can be defined by the characteristic length scale divided by the characteristic velocity. We use a bubble radius R0 as the characteristic length. Then, the characteristic time for pressure propagation tC p can be simply determined: tC p = R0 /a.

(5.148)

The characteristic time for bubble motion can be estimated as follows. First, we can assume the adiabatic process in the bubble caused by a moderately large VR , where the heat exchange across the bubble wall may not be completed during the characteristic time for bubble motion. We want to deduce such a smallest characteristic time for the discussion of appropriateness of Eq. (5.146). With the use of Eq. (5.121), the increment of internal energy deV can be written as deV = cvV 0 dTV ,

(5.149)

where cvV in Eq. (5.121) is assumed constant for simplicity. Total derivative of Eq. (5.125) becomes pV γ0 − 1 d pV − 2 d pV = c pV 0 dTV . ρV γ0 ρV

(5.150)

Substituting Eq. (5.149) and an adiabatic relation deV = −( pV /ρV2 )dρV into Eq. (5.150), we obtain γ0

dρV d pV = , ρV pV

(5.151)

where Mayer’s relation (5.124) is used. Integrating Eq. (5.151) leads to the following adiabatic relation: pV γ = const. ρV0

(5.152)

With the use of Eq. (5.152) and neglecting viscous effects, the characteristic time tCm that is defined as the inverse of the natural frequency of bubble oscillation is given by [15]

174

5 Dynamics of Spherical Vapor Bubble Table 5.1 Thermodynamic properties of saturated water at 300 K ρV 0 c pV γ0 a σ λV kg/m3 kJ/(kg·K) m/s mN/m mW/(m·K)

pV 0 kPa

ρL

3.534

996.62

0.02556

1.872

1.332

428.8

71.69

18.47

DV mm2 /s 386.0

Table 5.2 Characteristic times associated with transport phenomena in bubble motion (µs) tC p tCm tC T 2.33

648



tC m

pV 0 2σ = 2π 3γ0 − 2 ρ L R0 ρ L R03

1660

−1/2 ,

(5.153)

where pV 0 and R0 are undisturbed pressure and radius, respectively. For further discussion, it is recommended to refer to [9]. We also estimate the characteristic time of propagation of heat √ conduction, tC T . With the use of the combination of variables, we have χ = r/(2 DV t), where DV is the coefficient of thermal diffusivity and defined by DV = λV /(ρV c pV ),

(5.154)

(see Appendix C at the end of this book). Then, we can obtain the expression of tC T as follows: tC T = R02 /(4DV ).

(5.155)

Now, we can evaluate the characteristic times with the use of thermodynamic properties of vapor and water at the saturated condition at 300 K from a saturated water table [10], which are listed on Table 5.1. We suppose R0 is 1 mm, then we can calculate tC p , tCm , and tC T with the use of Eqs. (5.148), (5.153), and (5.155), respectively. The results are shown in Table 5.2. We found that tC p is significantly smaller than the other characteristic times associated with bubble motion and transport phenomena in a bubble; hence, the propagation of pressure disturbance is much faster than the bubble motion itself. This result supports the assumption of the uniform pressure inside bubble. In contrast, tC T is relatively large; then, we should take the temperature distribution for the bubble interior into consideration.

5.5.3 Temperature, Pressure, and Velocity Fields With the use of Eq. (5.146), the temperature field in vapor can be obtained by solving conservation equation of energy in vapor, Eq. (5.127). Solution inside the bubble,

5.5

Practical Description of Bubble Motion

175

i.e., temperature field in the vapor TV , is decomposed into two parts: isentropic temperature field Tis and perturbation temperature field Θ. The former is a function of time only and the latter is a function of both time and space. It should be emphasized here that a pair of temperature boundary conditions, Eqs. (5.143) and (5.144), is not a priori specified but it is a part of the solution. Finally, pressure pV (t) can be obtained from the conservation equation of mass, Eq. (5.118), the conservation equation of energy, Eq. (5.127), and the equation of state, Eq. (5.125), with the use of the fact that p V is independent of the space as shown in Eq. (5.146). Then, the velocity distribution in the bubble, vV , can also be obtained.

5.5.4 Boundary Conditions of Temperature Field We can obtain the temperature field in the liquid, R(t) < r < ∞, and that in the vapor, 0 < r < R(t), by solving Eqs. (5.117) and (5.127), respectively, with proper boundary conditions. The boundary conditions at infinity in the liquid and that on the center of the vapor bubble, respectively, are lim TL (r, t) = TL0 ,

r →∞

lim

r →0

∂ TV (r, t) (r, t) = 0, ∂r

(5.156)

(5.157)

where TL0 is the temperature of undisturbed liquid at the bubble wall in liquid. Notice that Eqs. (5.117) and (5.127) contain the second-order derivatives in space and the first-order derivatives in term. Therefore, the number of boundary conditions in space for each equation should be two, and consequently four boundary conditions are necessary. With the use of Eq. (5.71), the other two conditions are Eqs. (5.143) and (5.144):  ∂ TL  = λV w λL ∂r r =R +  ∂ TV  = λV w + ∂r r =R −

TV w

 ∂ TV  + jL ∂r r =R −  ∗  2 p p∞ / p ∗ p∞ L , (5.158) − √ √ √ 1−α TL TL 2 π α − C 4∗ Rc

 ˙  ∗ wV − R , = TLw 1 + d4 √ 2Rc TLw

(5.159)

where R + and R − are the radius of the bubble wall in the liquid and the radius of the bubble wall in the vapor, respectively.

176

5 Dynamics of Spherical Vapor Bubble

If the boundary conditions for the temperature at the bubble wall had been explicitly specified, we should have solved the Dirichlet problem. If the boundary conditions for the temperature gradient at the bubble wall had been explicitly specified, we should have solved the Neumann problem. However, the boundary conditions for the temperature and temperature gradient at the bubble wall are coupled through Eqs. (5.158) and (5.159); hence we have to solve the mixed boundary value problem. For the practical sake of solving the problem, we consider that Eq. (5.158) defines the boundary condition for the temperature gradient at the bubble wall in the liquid and that Eq. (5.159) defines the boundary condition for the temperature at the bubble wall in the vapor. Therefore we solve the Neumann problem for the temperature field in the liquid and the Dirichlet problem for the temperature field in the vapor.

5.6 Temperature Field of Bubble Exterior As we discussed in Sect. 5.5.4, with the use of Eqs. (5.156) and (5.158), the temperature field in the liquid, R(t) < r < ∞, can be determined with the following boundary and initial conditions: # $ ∂ TL  (r = R + , t) = TLw = λV w TV w + j L /λ L , ∂r lim TL (r, t) = TL0 , r →∞

TL (r, t = 0) = TL0 ,

(5.160) (5.161) (5.162)

 is the where the prime denotes the partial differentiation with respect to r and TLw temperature gradient in the direction of r at the bubble wall in the liquid, and R + is the radius of the bubble wall in the liquid. It should be emphasized here that  . The temperature at the bubble wall in the liquid Eq. (5.160) is the definition of TLw is denoted by TLw :

TL (r = R + , t) = TLw .

(5.163)

5.6.1 Lagrangian Formulation One of difficulties lying in studying thermo-fluid dynamic problems with interface is that the interface moves. This class of problems is categorized, in general, as “moving boundary problems”. It is often useful to study equations in Lagrangian coordinates for one-dimensional moving boundary problems. We introduce Lagrangian coordinates (z L , τ ) which are related to Eulerian coordinates (r, t) by the following relations [9]:  z L (r, t) =

r R(t)

ξ 2 dξ, τ = t.

(5.164)

5.6

Temperature Field of Bubble Exterior

177

Now, we consider the time derivative with z L fixed. With the use of Eq. (5.164), we obtain dz L =

∂z L ∂z L ˙ dr + dt = r 2 dr − R(t)2 R(t)dt = 0. ∂r ∂t

(5.165)

From Eq. (5.164), we have 1 ∂ ∂ = 2 . ∂z L r ∂r

(5.166)

  2 dr R ∂r  ˙ R(t), = =  ∂t z L dt r

(5.167)

Equation (5.165) leads to

where |z L is the operation with z L fixed, d/dt is the derivative with respect to t with z L fixed. Then, we take the derivative with respect to τ :         2 ∂  ∂  ∂r R ∂  ∂  ∂  ˙ R(t) = + = + , ∂τ z L ∂t r ∂t z L ∂r t ∂t r r ∂r t

(5.168)

where Eq. (5.167) is used. With the use of Eqs. (5.130) and (5.168), the left-hand side of Eq. (5.117) becomes     R 2 w L ∂ TL  ∂ TL  j R 2 ∂ TL  ∂ TL  + = − , ∂t r r2 ∂r t ∂τ z L ρ L r 2 ∂r t

(5.169)

and then, with the use of Eqs. (5.166) and (5.169), Eq. (5.117) is rewritten as   j R 2 ∂ TL ∂ TL λL ∂ 4 ∂ TL − r (z L , τ ) . = ∂τ ρ L ∂z L ρ L c L ∂z L ∂z L

(5.170)

In the following, the subscript L is dropped for simplicity.

5.6.2 Transformation of Variables Now we replace an independent variables τ and z with new variables s and η defined, respectively, by

178

5 Dynamics of Spherical Vapor Bubble



τ

s=

R 4 (ξ )dξ,

(5.171)

0

z . η= √ (λ/(ρc)

(5.172)

Then, Eq. (5.170) can be written as   ∂T ∂T ∂ ! r "4 ∂ T + vcL (s) = , ∂s ∂η ∂η R ∂η

(5.173)

where vcL (s) is defined by vcL (s) = −

j ρ R2



ρc , λ

(5.174)

and ∂s/∂τ = R(τ )4 is used. Subtracting ∂ 2 T /∂η2 from the both sides of Eq. (5.173), we have ∂T ∂ ∂2T ∂T = + vcL (s) − 2 ∂s ∂η ∂η ∂η



  ∂T r4 −1 , R4 ∂η

(5.175)

with boundary and initial conditions: 1 ∂ T (η = 0, s) = TLw,η = 2 ∂η R



$ 1 # λV w TV w + j L , λρc

(5.176)

lim T (η, s) = TL0 ,

(5.177)

T (η, 0) = TL0 .

(5.178)

η→∞

We note that T (η = 0, s) = TLw .

(5.179)

Notice now that ∂ T /∂η and ∂ 2 T /∂η2 should be small enough to neglect the righthand side of Eq. (5.175), except in the neighborhood of r = R, while (r 4 /R 4 − 1) should be extremely small there; hence Eq. (5.175) reduces to the heat equation with fixed boundary: ∂T ∂T ∂2T . + vcL (s) = ∂s ∂η ∂η2

(5.180)

5.6

Temperature Field of Bubble Exterior

179

5.6.3 Laplace Transform of Heat Equation Equation (5.180) with conditions (5.179)–(5.178) can be solved using Laplace transform. We use the two-sided Laplace transform (bilateral Laplace transform) [24]:  ϕ(ξ, s) = Lη {φ(η, s)} =

∞

−∞

e−ηξ φ(η, s)dη.

(5.181)

It should be noted that the lower limit of the integration in Eq. (5.181) is −∞. In contrast, the lower limit of the integration of one-sided Laplace transform, which is commonly referred to as the Laplace transform, is 0. The usefulness of using the two-sided Laplace transform will be revealed in Sect. 5.6.4.2. We notice that the domain of definition of T is 0  T < ∞; hence we need to extend the domain of definition of T to −∞ < T < ∞ to apply Eq. (5.181). We define T E whose the domain of definition is −∞ < T < ∞ and 0  s < ∞. We impose the following on T E and derivatives of T E with respect to η: for η > 0, T E (η, s) = T (η, s) ∂ T E (η, s) = ∂ T (η, s)/∂η for η > 0, ∂η ∂ 2 T E (η, s) = ∂ 2 T (η, s)/∂η2 for η > 0, ∂η2

(5.182) (5.183) (5.184)

and lim T E (η, s) = T (0, s),

(5.185)

lim ∂ T E (η, s)/∂η = ∂ T (0, s)/∂η,

(5.186)

lim ∂ 2 T E (η, s)/∂η2 = ∂ 2 T (0, s)/∂η2 .

(5.187)

η→+0 η→+0 η→+0

We also impose the restriction on T E at s = 0 as % T E (η, s = 0) =

T (η, s = 0) for η > 0, 0 for η < 0.

(5.188)

We consider the heat conduction equation (5.180). Noticing the derivatives with respect to η when η < 0 are physically undetermined, we rewrite the heat conduction equation for T E as ∂T E ∂2T E ∂T E , + H (η)vcL (s) = H (η) ∂s ∂η ∂η2 where H (η) is the Heaviside function:

(5.189)

180

5 Dynamics of Spherical Vapor Bubble

% H (η) =

1 for η > 0, 0 for η < 0.

(5.190)

Then, the two-sided Laplace transform of T E (η, s) can be properly defined as 5 6  ϕ(ξ, s) = Lη T E (η, s) =

∞

−∞

e−ηξ T E (η, s)dη =



∞

e−ηξ T (η, s)dη.

0

(5.191)

Similarly, each term in Eq. (5.189) can be transformed as % Lη

∂T E ∂s

& =

∂ϕ , ∂s

(5.192)

% & ∂T E = vcL (s) [−T (η = 0, s) + ξ ϕ(ξ, s)] , Lη vcL (s) ∂η % Lη

∂2T E ∂η2

& =−

(5.193)

∂T (η = 0, s) + ξ [−T (η = 0, s) + ξ ϕ(ξ, s)] . ∂η

(5.194)

Then, the Laplace transform of Eq. (5.189) becomes ∂ϕ (ξ, s) − vcL (s)T (η = 0, s) + ξ vcL (s)ϕ(ξ, s) ∂s ∂T =− (η = 0, s) − ξ T (η = 0, s) + ξ 2 ϕ(ξ, s). ∂η

(5.195)

With the use of Eqs. (5.160), (5.163), and (5.176), Eq. (5.195) can be rewritten as ∂ϕ + ξ(vcL − ξ )ϕ = TLw (vcL − ξ ) − TLw,η . ∂s

(5.196)

The method of variation of parameters [2] with Eqs. (5.178), (5.182), and (5.185) gives the solution of Eq. (5.196):    s vcL (ζ )dζ ϕ(ξ, s) = ϕ0 (ξ ) exp ξ 2 s − ξ 0   s  8 7 + TLw (ς ) [vcL (ς ) − ξ ] − TLw,η (ς ) · exp ξ 2 (s − ς ) − ξ 0

ς

s

 vcL dζ dς, (5.197)

5.6

Temperature Field of Bubble Exterior

181

where 5

6



∞

e−ηξ TL0 dη ϕ0 (ξ ) = Lη T (η, 0) = 0  ∞ = TL0 e−ηξ H (η)dη = TL0 Lη {H (η)} , E

(5.198)

−∞

with the definition of T E (η, 0): T E (η, 0) = TL0 H (η).

(5.199)

5.6.4 Inverse Laplace Transform of Heat Equation We evaluate inverse Laplace transform of ϕ(ξ, s) in Eq. (5.197), by calculating the inverse Laplace transform of the following four functions: (i) ϕ1 (ξ, s) = exp(ξ 2 s).

(5.200)

(ii)   ϕ10 (ξ, s) = ϕ0 (ξ ) exp ξ 2 s − ξ

s

 vcL (ζ )dζ .

(5.201)

0

(iii) 

s

ϕ20 (ξ, s) =

'

  ( TLw (ς )vcL (ς ) − TLw,η (ς ) exp ξ 2 (s − ς ) − ξ

ς

0

s

 vcL dζ dς. (5.202)

(iv) 

s

ϕ30 (ξ, s) = −

  TLw (ς )ξ exp ξ 2 (s − ς ) − ξ

0

s ς

 vcL dζ dς.

(5.203)

Then, the inverse Laplace transform of ϕ(ξ, s) in Eq. (5.197) is given by −1 φ(η, s) = L−1 η {ϕ(ξ, s)} = Lη {ϕ10 (ξ, s) + ϕ20 (ξ, s) + ϕ30 (ξ, s)} .

(5.204)

5.6.4.1 Inverse Laplace Transform of ϕ1 (ξ, s) First, we evaluate the inverse Laplace transform of ϕ1 (ξ, s), which can be calculated as follows:

182

5 Dynamics of Spherical Vapor Bubble Im(z) Re(z)

+A

–A O – ip – A

– ip

– ip + A

Fig. 5.6 The integral path used in evaluation of the inverse Laplace transform of eξ

2s

 σ +i∞ 5 2 6 1 2 ξ s e = φ1 (η, s) = = eξ s · eξ η dξ 2πi σ −i∞  2   −i p+∞ " ! 1 η (5.205) exp −z 2 dz, = √ exp − 4s 2π s −i p−∞ L−1 η {ϕ1 (ξ, s)}

L−1 η

where σ is a real number [24], and we substituted ξ = 2π λi with the use of the following definitions: √ η p=σ s+ √ . 2 s

√ iη z = 2π sλ − √ , 2 s

(5.206)

Now, we evaluate the integration in Eq. (5.205) with the use of the integral path shown in Fig. 5.6. The integration can be rewritten as  lim

−i p+A

A→∞ −i p−A

 exp(−z 2 )dz = lim

A→∞

−A

−i p−A

 +

A −A

 +

−i p+A 

exp(−z 2 )dz.

A

(5.207) The first and third integrations in Eq. (5.207) vanishes since exp(−z 2 ) is a rapidly decaying function as |z| → ∞; hence we obtain  2 1 η φ1 (η, s) = √ exp − . 4s 2 πs

(5.208)

5.6.4.2 Laplace Transform of Distribution In order to proceed further to evaluate the inverse Laplace transform of ϕ(ξ, s) in Eq. (5.197), we need the Laplace transform of distribution. The derivative of the Heaviside function in the sense of distribution [21] is given by dH (η) = δ(η), dη

(5.209)

where δ(η) is the Dirac delta function, or delta distribution, which satisfies for an arbitrary integrable and continuous function f (η),

5.6

Temperature Field of Bubble Exterior



∞

183

f (η)δ(η)dη = f (0).

−∞

(5.210)

Then, the two-sided Laplace transform of δ(η) is carried out with the use of integration by parts:  Lη {δ(η)} =

∞ −∞

e−ηξ δ(η)dη =



∞

−∞

e−ηξ

dH (η) dη = ξ dη



∞

e−ηξ dη = 1,(5.211)

0

where Eqs. (5.190) and (5.209) are used. Similarly, 5 6  (1) Lη δ (η) =

∞



∞ dδ(η) (η)dη = e−ηξ dη dη −∞ −∞  ∞ ' (∞ = e−ηξ δ(η) −∞ − (−ξ )e−ηξ δ(η)dη = ξ.

e

−ηξ (1)

δ

−∞

(5.212)

Since we used the two-sided, not one-sided, Laplace transform, we can escape from the evaluation of δ(η = 0) in Eq. (5.212) and H (η = 0) in Eq. (5.211). This is the main reason why we choose to use the two-sided Laplace transform. The following inverse Laplace transforms are useful: L−1 η {1} = δ(η),

(5.213)

(1) L−1 η {ξ } = δ (η).

(5.214)

5.6.4.3 Inverse Laplace Transform of ϕ10 (ξ, s) Next, we evaluate the inverse Laplace transform of ϕ10 (ξ, s) in Eq. (5.201). By the convolution theorem [24], φ10 (η, s) can be written as φ10 (η, s) = L−1 η {ϕ10 (ξ, s)} = TL0 H (η) ∗ φ11 (η, s),

(5.215)

where ∗ is the convolution, and the following definition of φ11 (η, s) is used: %   −1 2 {ϕ φ11 (η, s) = L−1 exp ξ (ξ, s)} = L s − ξ 11 η η

s

& vcL (ζ )dζ

.

(5.216)

0

With the use of Eqs. (5.205) and (5.208), Eq. (5.216) can be transformed to yield 9 2 :   s 1 1 vcL (ζ )dζ η− . φ11 (η, s) = √ exp − 4s 2 πs 0

(5.217)

By substituting Eqs. (5.204) and (5.217) into Eq. (5.215), and carrying out convolution, φ10 (η, s) is obtained:

184

5 Dynamics of Spherical Vapor Bubble

 φ10 (η, s) = TL0

∞

0

9 2 :   s 1 1 dς. (5.218) vcL (ζ )dζ − ς η− √ exp − 4s 2 πs 0

With the use of the following definition: v=

η−

*s 0

vcL (ζ )dζ − ς , √ 2 s

1 dv = − √ dς, 2 s

(5.219)

we obtain the following expression of φ10 (η, s)3 : TL0 φ10 (η, s) = 2

9

 1 + erf

η−

*s

vcL (ζ )dζ √ 2 s

0

: .

(5.220)

5.6.4.4 Inverse Laplace Transform of ϕ20 (ξ, s) We, then, evaluate the inverse Laplace transform of ϕ20 (ξ, s) in Eq. (5.202) as follows: φ20 (η, s) = L−1 η {ϕ20 (ξ, s)}  s ( ' 1 TLw (ς )vcL (ς ) − TLw,η (ς ) = √ 2 π 0 ⎡ ! "2 ⎤ *s η − v (ζ )dζ ς cL 1 ⎢ ⎥ exp ⎣− ×√ ⎦ dς, s−ς 4(s − ς )

(5.221)

where we have used that the integration and inverse Laplace transform are commutable. 5.6.4.5 Inverse Laplace Transform of ϕ30 (ξ, s) Finally, we evaluate the inverse Laplace transform of ϕ30 (ξ, s) in Eq. (5.203). φ30 (η, s) can be written as φ30 (η, s) = L−1 η {ϕ30 (ξ, s)}  %  s  2 = L−1 ξ T (ς ) exp ξ (s − ς ) − ξ − Lw η 0

ς

s

 & vcL (ζ )dζ dς . (5.222)

The (a.k.a. the Gauss error function) erf(x) is defined as erf(x) = * x error function 2 0 exp(−ξ )dξ , and the complementary error function erfc(x) is defined as erfc(x) = 1 − √ *∞ erf(x) = √2π x exp(−ξ 2 )dξ . It should be noted that limx→∞ erf(x) = √2π · 2π = 1.

3

√2 π

5.6

Temperature Field of Bubble Exterior

185

Now, we consider the inverse Laplace transform of the following function: %   −1 2 {ϕ φ31 (η, s) = L−1 ξ exp ξ (ξ, s)} = L (s − ς ) − ξ 31 η η

s

ς

& vcL (ζ )dζ

.

⎛  2 ⎞ *s η − v (ζ )dζ ς cL dδ(η) 1 ⎜ ⎟ exp ⎝− = ∗ √ ⎠ dη 4(s − ς ) 2 π(s − ς )  =

∞ −∞

⎛  2 ⎞ *s (η − y) − v (ζ )dζ cL ς dδ(y) 1 ⎟ ⎜ exp ⎝− √ ⎠ dy dy 2 π(s − ς ) 4(s − ς )

1 =− √ 2 π

η−

*s ς

vcL (ζ )dζ

2(s − ς )3/2

⎛  2 ⎞ *s η − ς vcL (ζ )dζ ⎜ ⎟ exp ⎝− ⎠, 4(s − ς )

(5.223)

where Eqs. (5.212) and (5.217), and the following inverse Laplace transform is used: %   2 L−1 exp ξ (s − ς ) − ξ η

s

ς

⎛  1 ⎜ exp ⎝− = √ 2 π(s − ς )

& vcL (ζ )dζ

η−

*s ς

vcL (ζ )dζ

2 ⎞

4(s − ς )

⎟ ⎠.

(5.224)

Substituting Eq. (5.223) into Eq. (5.222), noticing that the integration and inverse Laplace transform are commutable, we obtain 1 φ30 (η, s) = √ 2 π

 0

s

TLw (ς )

η−

*s ς

vcL (ζ )dζ

2(s − ς )3/2

⎛  2 ⎞ *s η − ς vcL (ζ )dζ ⎜ ⎟ × exp ⎝− ⎠ dς. 4(s − ς )

(5.225)

Putting subscript L back again in equations, and with the use of Eqs. (5.204), (5.220), (5.221), and (5.225), the temperature field is obtained:

186

5 Dynamics of Spherical Vapor Bubble

TL (η, s) = φ10 (η, s) + φ20 (η, s) + φ30 (η, s)    *s η − 0 vcL (ζ )dζ TL0 1 + erf = √ 2 2 s 1 + √ 2 π 1 + √ 2 π

 0



⎛ 

TLw (ς )vcL (ς ) − TLw,η (ς ) ⎜ exp ⎝− √ s−ς

s

s

TLw (ς )

0

η−

*s ς

vcL (ζ )dζ

2(s − ς )3/2

η−

*s ς

vcL (ζ )dζ

2 ⎞

4(s − ς )

⎟ ⎠ dς

⎛  2 ⎞ *s η − ς vcL (ζ )dζ ⎜ ⎟ exp ⎝− ⎠ dς. 4(s − ς ) (5.226)

The integrals in the right-hand side of Eq. (5.226) involve both TLw and TLw,η . TLw,η that is defined by Eq. (5.176) has been provided as Neumann boundary condition, as we discussed in Sect. 5.5.4. In contrast, TLw has not been specified; hence we proceed to determine TLw in the next subsection.

5.6.5 Liquid Temperature at Bubble Wall Our next task is to determine liquid temperature at bubble wall, TLw , by taking the limit of Eq. (5.226): TLw (s) = lim T (η, s) = lim [φ10 (η, s) + φ20 (η, s) + φ30 (η, s)] . η→0

η→0

(5.227)

Evaluation of the limit of the first and second terms in the right-hand side of Eq. (5.227) is rather simple4 : TL0 lim φ10 (η, s) = η→0 2

9

* s 1 − erf

0

vcL (ζ )dζ √ 2 s

: ,

(5.228)

lim φ20 (η, s)

η→0

1 = √ 2 π



s 0

⎧ * 2 ⎫ s ⎪ ⎪ ⎨ ⎬ v (ζ )dζ ς cL TLw (ς )vcL (ς ) − TLw,η (ς ) exp − dς. √ ⎪ ⎪ s−ς 4(s − ς ) ⎩ ⎭ (5.229)

4

erf(−x) =

√2 π

* −x 0

exp(−ξ 2 )dξ = − √2π

*x 0

exp(−ζ 2 )dζ = −erf(x), where ζ = −ξ .

5.6

Temperature Field of Bubble Exterior

187

In contrast, the evaluation of the third term in Eq. (5.227) requires an extra caution in taking the limit of the following function: lim φ32 (η, s)

η→0

1 = lim √ η→0 2 π 1 = lim √ η→0 2 π

⎧  2 ⎫ *s ⎪ ⎨ ⎬ s η − ς vcL (ζ )dζ ⎪ η − TLw (ς ) exp dς ⎪ ⎪ 2(s − ς )3/2 4(s − ς ) 0 ⎩ ⎭ 9 ' *s (2 :  ∞ η − s−τ vcL (ζ )dζ TLw (s − τ ) exp − (2dσ ), η 4η2 /(4σ 2 )) √ 

2 s

(5.230) √ where τ = s − ς and σ = η/(2 τ ). Equation (5.230) leads to lim φ32 (η, s)

η→0

9  2 :  ∞  s 1 1 dσ TLw (s − τ ) exp − σ − √ vcL (ζ )dζ = lim √ η η→0 π 2 τ s−τ √ 2 s  TLw (s) ∞ 1 = √ exp(−σ 2 )dσ = TLw (s), (5.231) 2 π 0

where we have evaluated the limit of the second term in the square bracket in exponential function in the second row of Eq. (5.231) as follows: 1 √ η, τ →0 2 τ



s

lim

  1 √ vcL (s) s − s + τ + O(τ 2 ) η, τ →0 2 τ vcL (s) √ = lim τ = 0. (5.232) η, τ →0 2

vcL (ζ )dζ = lim

s−τ

We, then, can evaluate the limit of the third term in the right-hand side of Eq. (5.227): 1 TLw (s) 2 ⎧ * 2 ⎫ *s s ⎪ ⎪  s ⎬ ⎨ v (ζ )dζ ς cL 1 ς vcL (ζ )dζ dς. (5.233) − TLw (ς ) exp − √ ⎪ ⎪ 4(s − ς ) 2(s − ς )3/2 2 π 0 ⎭ ⎩

lim φ30 (η, s) =

η→0

Substituting Eqs. (5.228), (5.229), and (5.233) into Eq. (5.227), we obtain

188

5 Dynamics of Spherical Vapor Bubble

TLw (s)

 s   TLw,η (ς ) 1 = TL0 − TL0 erf [Z L (s, 0)] − √ exp −Z L (s, ς )2 dς √ s−ς π 0    s   1 1 1 TLw (ς ) vcL (ς ) − vm L (s, ς ) √ +√ exp −Z L (s, ς )2 dς, 2 s−ς π 0 (5.234)

where *s vm L (s, ς ) =

ς

vcL (ζ )dζ s−ς

*s ,

Z L (s, ς ) =

vcL (ζ )dζ . √ 2 s−ς

ς

(5.235)

Solving above integral equation with the use of TLw,η defined by Eq. (5.176), we can obtain TLw . Then, we can properly determine T (η, s), by substituting TLw and TLw,η into Eq. (5.226).

5.6.6 Gradient of Liquid Temperature at Bubble Wall We further proceed to determine the gradient of liquid temperature at the bubble wall, TLw,η . The result obtained in this subsection will be useful in evaluating TV w,η in Sect. 5.7.4. First, we differentiate Eq. (5.226) with respect to η: ⎧  2 ⎫ *s ⎨ η − 0 vcL (ζ )dζ ⎬ TL0 ∂ TL (η, s) = √ exp − √ ⎭ ⎩ ∂η 2 πs 2 s *s  s ( η − ς vcL (ζ )dζ ' 1 − √ TLw (ς )vcL (ς ) − TLw,η (ς ) 2(s − ς )3/2 2 π 0 ⎧  ⎫  2 * ⎪ ⎨ η − ςs vcL (ζ )dζ ⎪ ⎬ × exp − dς ⎪ ⎪ 4(s − ς ) ⎩ ⎭ ⎧  2 ⎫ *s ⎪  s ⎨ ⎬ η − ς vcL (ζ )dζ ⎪ 1 1 TLw (ς ) − + √ 3/2 ⎪ ⎪ 2 π 0 4(s − ς )5/2 ⎩ 2(s − ς ) ⎭ ⎧  ⎫ 2 * ⎪ ⎨ η − ςs vcL (ζ )dζ ⎪ ⎬ × exp − dς, (5.236) ⎪ ⎪ 4(s − ς ) ⎩ ⎭ and then, we need to take the limit of Eq. (5.236) as η → 0. With the use of Eq. (5.233), the limit of Eq. (5.236) becomes,

5.7

Temperature Field of Bubble Interior

189

∂ TL (η, s) η→0 ∂η   2TL0 = √ exp −Z L (s, 0)2 − TLw (s)vcL (s) πs  s ∂ 7T (ς ) exp '−Z (s, ς )2 (8 Lw L 1 ∂ς −√ dς √ s−ς π 0  s   ' ( vm (s, ς ) 1 TLw (ς )vcL (ς ) − TLw,η (ς ) √ exp −Z L (s, ς )2 dς + √ 2 π 0 (s − ς )  s   vm (s, ς )2 1 (5.237) TLw (ς ) √ exp −Z L (s, ς )2 dς. − √ 2 s−ς 2 π 0

TLw,η (s) = lim

Substituting Eq. (5.234) into the second term of the right-hand side of Eq. (5.237), we finally obtain   2TL0 TLw,η (s) = √ exp −Z L (s, 0)2 − vcL (s)TL0 {1 − erf [Z L (s, 0)]} πs  s ∂ 7T (ς ) exp '−Z (s, ς )2 (8 Lw L 1 ∂ς dς −√ √ s−ς π 0   &  s% 1 1 + √ TLw (ς ) vcL (ς ) − vm L (ς ) − TLw,η (ς ) 2 2 π 0   vm L (s, ς ) − vcL (s) (5.238) · exp −Z L (s, ς )2 dς. √ (s − ς )

5.7 Temperature Field of Bubble Interior Now, we turn our attention to temperature field in vapor. As we discussed in Sect. 5.5.4, with the use of Eqs. (5.157) and (5.159), the temperature field in vapor in the region 0 < r < R(t), can be determined with the following boundary and initial conditions:  ˙  − ∗ wV − R , (5.239) TV (r = R , t) = TV w = TLw 1 + d4 √ 2Rc TLw ∂ TV (r, t) (r, t) = 0, (5.240) lim r →0 ∂r (5.241) TV (r, t = 0) = TV 0 , where TV 0 is the temperature of undisturbed vapor, TV w is the temperature of vapor at the bubble wall, the prime denotes the partial differentiation with respect to r , and R − is the radius of the bubble wall in the vapor. It should be emphasized here that Eq. (5.239) is the definition of TV w . The temperature gradient in the direction of r at the bubble wall in the vapor is denoted by TV w :

190

5 Dynamics of Spherical Vapor Bubble

∂ TV (r = R + , t) = TV w . ∂r

(5.242)

In the following, the subscript V is dropped for simplicity. Then, Eq. (5.127) becomes  ρ

   ∂(c pv T ) ∂(c pv T ) dp 1 ∂ ∂T +v = + 2 λr 2 . ∂t ∂r dt ∂r r ∂r

(5.243)

5.7.1 Adiabatic Solution We restrict the problem considered to the case that thermal boundary layer is relatively thin compared with the bubble radius. Then, the temperature field outside the thermal boundary layer is, with the neglect of the thermal conduction term in Eq. (5.243), given by ρ

$ dp d # = 0. cpT − dt dt

(5.244)

Substituting Eq. (5.125), we can rewrite Eq. (5.244) as $ γ0 − 1 1 d p d # . cpT = c p0 T dt γ0 p dt 1

(5.245)

It is known [14] that for an ideal gas, c p = c p0 νcp T νc ,

(5.246)

may hold for wide ranges of temperature and pressure, where νc is a constant. Substituting Eq. (5.246) into Eq. (5.245) leads to γ0 − 1 T d p d νc +1 )− (T = 0. dt νcp γ0 p dt

(5.247)

Imposing the condition that p = pV 0 at T = TV 0 , the integration of Eq. (5.245) with Eq. (5.246) leads to  Tis = TVνc0 +

γ0 − 1 νc ln νcp (νc + 1) γ0



p pV 0

1/νc

.

(5.248)

It should be noted that Tis in Eq. (5.248) is a solution of Eq. (5.245), not that of (5.243); hence Tis cannot satisfy the boundary conditions (5.239) and (5.242).

5.7

Temperature Field of Bubble Interior

191

5.7.2 Lagrangian Formulation As we have already used in Sect. 5.6.1, here, we also introduce the Lagrangian coordinates (z, τ ):  z(r, t) =

R(t)

ρ(ξ, t)ξ 2 dξ,

τ = t.

(5.249)

r

We should notice the difference of Eq. (5.249) from Eq. (5.164) in the integration domain and ρ in Eq. (5.249). With the use of Eq. (5.249), the total derivative is given by dz =

∂z ∂z ˙ dr + dt = −ρr 2 dr + ρ R(t)2 R(t)dt = 0, ∂r ∂t

(5.250)

for fixed z. Then, we obtain   2 R dr ∂r  ˙ = R(t), =  ∂t z dt r       j ∂ T  R 2 W ∂ T  ∂ T  R2 ˙ ∂ T  + 2 = + 2 R− ∂t r r ∂r t ∂t r r ρ ∂r t    2 ∂ T  jR ∂ T  = − . ∂τ z ρr 2 ∂r t

(5.251)

(5.252)

However, we should notice that 1 ∂ ∂ =− 2 , ∂z ρr ∂r

(5.253)

which is derived from Eq. (5.249), causes the significant difference in the governing equation. With the use of Eqs. (5.253) and (5.252), Eq. (5.243) is transformed to yield   ∂(c p T ) 1 dp ∂ 2 ∂(c p T ) 4 ∂T + jR = + λρr (z, τ ) . ∂τ ∂z ρ dτ ∂z ∂z

(5.254)

5.7.3 Boundary Layer Solution We assume that the difference between the temperature T inside the bubble and the isentropic temperature Tis which is given by Eq. (5.248) is small, i.e., T (r, t) = Tis (t) + Θ(r, t)

with ||  1,

(5.255)

192

5 Dynamics of Spherical Vapor Bubble

and hence, physical properties, c p and λ, which are dependent on temperature, in Eq. (5.243) can be written as functions of Tis alone. With T replaced with Tis , Eq. (5.246) is rewritten as c p = c p0 νcp Tisνc .

(5.256)

We recall the well-established method in compressible boundary layer theory called the Howarth–Illingworth–Stewartsort transformation [14] in which it is assumed that μ is independent of the pressure, and the following relation holds: μ = cμ Tis ,

(5.257)

where cμ is a constant. Notice that Pr =

μc p = const. λ

(5.258)

is valid for wide ranges of temperature and pressure. From Eqs. (5.246), (5.257), and (5.258), we obtain λ = cλV Tisνc +1 ,

(5.259)

where cλV is a constant. Then, with the use of Eqs. (5.256) and (5.259), Eq. (5.254) can be transformed to yield   νc ∂(Tisνc T ) γ0 − 1 T d p cλV ∂ 2 ∂(Tis T ) 4 νc +1 ∂ T + jR − = ρr Tis . (5.260) ∂τ ∂z νcp γ0 p dτ c p0 νcp ∂z ∂z Substituting Eq. (5.255) into Eq. (5.260) leads to  ν +1 ∂(Tisc ) γ0 − 1 T d p − ∂τ νcp γ0 p dτ    νc ∂(Tisνc Θ) ∂(Tisνc Θ) cλV ∂ 2 ∂(Tis Θ) 4 + jR − ρr Tis = 0. + ∂τ ∂z c p0 νcp ∂z ∂z (5.261)



We notice that terms in the first square bracket are exactly the same as the lefthand side of Eq. (5.247); hence these terms disappear. Thus, Eq. (5.247) is eventually a partial differential equation of Q with respect to τ and z as ∂Q cλV ∂ ∂Q + j R2 − ∂τ ∂z c p0 νcp ∂z

  ∂Q 4 ρr Tis = 0, ∂z

(5.262)

5.7

Temperature Field of Bubble Interior

193

where Q is defined by Q(r, t) = Tisνc (t)Θ(r, t).

(5.263)

5.7.4 Solution of Heat Equation Now, as in Sect. 5.6.2, we replace independent variables τ and z with new variables s and η defined respectively by 

τ

p(ξ )R 4 (ξ )dξ, νcp (γ0 − 1) η = c p0 z, cλV γ0 s=

(5.264)

0

(5.265)

where we should notice that Eq. (5.264) is different from Eq. (5.171) in having p in the integrand. Then, Eq. (5.254) can be written as   ∂ ! r "4 ∂ Q ∂Q ∂Q + vcV (s) = , ∂s ∂η ∂η R ∂η

(5.266)

where vcV is defined by vcV (s) = c p0

νcp (γ0 − 1) j . cλV γ0 p R2

(5.267)

We obtained Eq. (5.266) with the use of Eq. (5.125) and the followings: ∂s = p(τ )R(τ )4 , ∂τ ρTis γ0 γ0 ρTis = = (1 + O()). p c p0 (γ0 − 1) T c p0 (γ0 − 1)

(5.268)

(5.269)

The initial and boundary conditions for T , Eq (5.239) to Eq. (5.241), can be rewritten as T (η = 0, s) = TV w lim

η→∞

∂ T (η, s) = 0, ∂η T (η, 0) = TV 0 .

 ˙  ∗ wV − R , = TLw 1 + d4 √ 2Rc TLw

(5.270) (5.271) (5.272)

194

5 Dynamics of Spherical Vapor Bubble

The temperature gradient in the direction of η at η = 0 in the vapor is denoted by TV w,η : 1 1 ∂ T (η = 0, s) = TV w,η = 2 ∂η R c p0



cλV γ0 T . νcp (γ0 − 1) V w

(5.273)

Then, the initial and boundary conditions for Q are obtained as Q(η = 0, s) = Q w = Tisνc (TV w − Tis ), lim Q(η, s) = 0,

η→∞

Q(η, 0) = 0.

(5.274) (5.275) (5.276)

We note that ∂ Q(η = 0, s) = Q w,η = Tisνc TV w,η . ∂η

(5.277)

With the use of the same discussion in derivation of Eq. (5.180), Eq. (5.266) can be reduced to the heat equation with fixed boundary: ∂Q ∂Q ∂2 Q . + vcV (s) = ∂s ∂η ∂η2

(5.278)

We notice that Eq. (5.278) has the same form as that of Eq. (5.180), with vcV defined by Eq. (5.267) instead of vcL as advection velocity. With Eq. (5.226) and Eqs. (5.274), (5.275), and (5.276), the solution of Eq. (5.278) is obtained. Putting subscript V back again to variables, and then, using Eqs. (5.255) and (5.263), the temperature field inside the bubble is obtained: Q(η, s) Tisνc  s [TV w (ς ) − Tis (ς )] vcV (ς ) − TV w,η (ς ) 1 = Tis + √ √ s−ς 2 π 0 ⎧  2 ⎫ * s ⎪ ⎨ η − ς vcV (ζ )dζ ⎪ ⎬ × exp − dς ⎪ ⎪ 4(s − ς ) ⎩ ⎭ *s  s η − ς vcV (ζ )dζ 1 [TV w (ς ) − Tis (ς )] + √ 2(s − ς )3/2 2 π 0 ⎫ ⎧   2 * ⎪ ⎬ ⎨ η − ςs vcV (ζ )dζ ⎪ dς. (5.279) × exp − ⎪ ⎪ 4(s − ς ) ⎭ ⎩

TV (η, s) = Tis +

5.7

Temperature Field of Bubble Interior

195

With the use of Eqs. (5.234), (5.235), (5.274), and (5.277), TV at the bubble wall can be written as  s   TV w,η (ς ) 1 exp −Z (s, ς )2 dς TV w (s) = Tis − √ √ s−ς π 0  s vcV (ς ) − 12 vmV (s, ς ) 1 +√ [TV w (ς ) − Tis (ς )] √ s−ς π 0   × exp −Z V (s, ς )2 dς,

(5.280)

where *s vmV (s, ς ) =

ς

vcV (ζ )dζ s−ς

*s ,

Z V (s, ς ) =

vcV (ζ )dζ . √ 2 s−ς

ς

(5.281)

Similarly, the gradient of vapor temperature inside the bubble can be written as *s  s ( η − ς vcV (ζ )dζ ' 1 ∂ TG (η, s) TV w (ς )vcV (ς ) − TV w,η (ς ) =− √ ∂η 2 π 0 2(s − ς )3/2 ⎧  ⎫  2 * ⎪ ⎨ η − ςs vcV (ζ )dζ ⎪ ⎬ × exp − dς ⎪ ⎪ 4(s − ς ) ⎩ ⎭ ⎧  2 ⎫ *s ⎪ ⎪  s ⎨ ⎬ η − v (ζ )dζ ς cV 1 1 TV w (ς ) − + √ 3/2 ⎪ ⎪ 2 π 0 4(s − ς )5/2 ⎩ 2(s − ς ) ⎭ ⎧  2 ⎫ * ⎪ ⎨ η − ςs vcV (ζ )dζ ⎪ ⎬ × exp − dς. (5.282) ⎪ ⎪ 4(s − ς ) ⎩ ⎭ Then, with the use of Eq. (5.238), we obtain the gradient of the vapor temperature at the bubble wall, TV w,η ,  s ∂ 7[T (ς ) − T (ς )] exp '−Z (s, ς )2 (8 Vw is V 1 ∂ς TV w,η (s) = − √ dς √ s−ς π 0  &   s% 1 1 [TV w (ς ) − Tis (ς )] vcV (ς ) − vmV (ς ) − TV w,η (ς ) + √ 2 2 π 0 5 6 vmV (s, ς ) − vcV (s) × exp −Z V (s, ς )2 dς. (5.283) √ (s − ς )

196

5 Dynamics of Spherical Vapor Bubble

Solving above integral equation with the use of TV w defined by Eq. (5.270), we can obtain TV w,η . Then, we can properly determine T (η, s), by substituting TV w and TV w,η into Eq. (5.279).

5.7.5 Pressure and Velocity In Sect. 5.7.3, we assumed that pressure pV inside the bubble is uniform and a function of time only, i.e., Eq. (5.146). Now, we determine how pV depends on time, and then, determine the explicit functional form of vV (r, t). Adding the conservation equation of mass, Eq. (5.118), multiplied by c pV TV and the conservation equation of energy, Eq. (5.127), leads to      ∂ ∂ TV ∂ d pV (ρV c pV TV ) − (r 2 ρV vV c pV TV ) − r 2 λV = −r 2 . ∂r ∂r ∂t dt (5.284) With the use of Eqs. (5.125), (5.246), and (5.256), we obtain ρV c pV TV = ρV c pV 0 νcp Tisνc TV =

νcp γ0 νc T pV , γ0 − 1 is

(5.285)

which is a function of time only; hence the square bracket term in the right-hand side of Eq. (5.284) is independent of r . Integrating once the both sides of Eq. (5.284) with respect to r leads to 

νcp γ0 ∂ TV vV Tisνc pV − λV γ0 − 1 ∂r



r3 =− 3



d dt





 d pV − r . dt (5.286) An integration constant which should have appeared in Eq. (5.286) has been eliminated with the use of the condition at r = 0. Equation (5.286) leads to 2

νcp γ0 νc T pV γ0 − 1 is

  γ0 − 1 d 3 γ0 − 1 Tis ∂ TV d νc ln(Tis pV ) − ln( pV ) = − vV − , dt r νcp γ0 pV ∂r νcp γ0 Tisνc dt

(5.287)

where Eq. (5.259) is used. Substituting Eq. (5.248) into Eq. (5.287), the left-hand side of Eq. (5.287) becomes d γ0 − 1 d d ln(Tisνc ) + ln( p V ) − ln( p V ) dt dt νcp γ0 Tisνc dt   1 d pV γ0 − 1 = 1− . νcp γ0 (νc + 1)Tisνc pV dt

(5.288)

Since pV is independent of r , we can impose condition at the bubble wall, r = R, on Eq. (5.288), and consequently, we obtain

5.8

Structure of Mathematical Model

197

 ;   ∂ T  γ0 − 1 γ0 − 1 1 − . Tis − p w V V νcp γ0 ∂r r =R − νcp γ0 (νc + 1)Tisνc (5.289) From Eqs. (5.287), (5.288), and (5.289), We obtain d pV 3 = dt R



  r ∂ TV r Tis γ0 − 1 ∂ TV − (r, t) − (r = R , t) . vV (r, t) = wV + R pV νcp γ0 ∂r R ∂r

(5.290)

5.8 Structure of Mathematical Model We have derived a set of equations in Sects. 5.4, 5.5, 5.6, and 5.7, to deal with the dynamics of a spherical vapor bubble accompanied with evaporation and condensation at the bubble wall. We refer to this set of equations as a mathematical model that describes dynamics of a spherical vapor bubbles. Here, we investigate the structure of this model. We have constructed the model from conservation equations (5.110), (5.113), and (5.117) for liquid, and (5.118), (5.119), and (5.127) for vapor, with boundary conditions at the bubble wall, Eqs. (5.129), (5.138), (5.143), and (5.144). We should notice that these equations are partial differential equations. Although t and r are originally taken as a set of the independent variables in these equations, we changed these variables to s and η defined and used in Sects. 5.6.1, 5.6.2, 5.7.2, and 5.7.4 for ease of solving the equations. After these changes of variables, the Laplace transform, and other manipulations, we successfully obtained ordinary differential equations, integral equations, and integro-differential equations with auxiliary algebraic equations, instead of partial differential equations. In addition to the equations we have already obtained, the two equations 

t

R(t) = 

˙ )dτ + R(0), R(τ

(5.291)

w˙ L (τ )dτ,

(5.292)

0 t

w L (t) = 0

are necessary to close the set of equations. All equations that compose the mathematical model describing the dynamics of a spherical vapor bubble with the evaporation and condensation at the bubble wall are listed in Table 5.3 with variables in each equation. The types of equation, ordinary differential equation, integral equation, integro-differential equation, and algebraic equation, are also indicated as O.D.E, I.E., I.D.E., and A.E., respectively in Table 5.3. Here, we verify that the number of variables (24) is the same as the number of equations listed in Table 5.3 (24); hence the set of equations we have derived is successfully closed. It should be also emphasized that we do not have to solve

198

5 Dynamics of Spherical Vapor Bubble

Equation

Table 5.3 Variables and constants in necessary equations Type Contained variables Constants

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

(5.70) (5.114) (5.116) (5.121) (5.122) (5.130) (5.131) (5.134) (5.135) (5.142)

A.E. A.E. A.E. A.E. A.E. A.E. A.E. A.E. A.E. A.E.

(11)

(5.147)

O.D.E.

(12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22)

(5.160) (5.174) (5.234) (5.248) (5.239) (5.256) (5.257) (5.259) (5.267) (5.283) (5.289)

A.E. A.E. I.E. A.E. A.E. A.E. A.E. A.E. A.E. I.D.E. O.D.E.

(23) (24)

(5.291) (5.292)

A.E. A.E.

j, TLw , p ∗ ˙ w L , w˙ L p L , R, R, e L , TL eV w , c pV , TV w ρV w , p V , TV w ˙ j, ρV w wV , R, ˙ wL , j R, ∗ ,T , R p ∗ , p∞ Lw ∗ p∞ , TLw eV w , e Lw , p V w , p L , ρV w , L w˙ L , w L , R, j, pV , ρV w , w L  , j, L, λ TV w , TLw Vw vcL , j, R  ,v TLw , TLw cL Tis , p V ˙ wV TV w , TLw , R, c pV , Tis μV , Tis λV , Tis vcV , j, R, ρV w , pV w TV w , TV w , Tis , vcV pV , R, TV w , TV w , Tis , wV R, R˙ w L , w˙ L

αe , αc , C4∗ , Rc ρL cL γ0 Rc

Conditions p∞ p∞

ρL ρ L , Rc , σ ρL ρ L , σ , μL

p∞

λL ρL , λL , cL νc , νcp , γ0 d4∗ , Rc νc , νcp , c p0 cμV cλV , νc γ0 , cλV , c p0 , νcp

TL0 TV 0 , p V 0

γ0 , νc , νcp R(0)

partial differential equations. This greatly reduces the complexity of mathematical procedures. ˙ wV , In these twenty four variables, the following twenty three variables: R, R,  , T , T , v , v , ∗ , ρ w L , w˙ L , j, pV , p ∗ , p∞ , e , e , L, T , T , T Vw Vw L Vw Lw is cV cL Vw Lw μV w , λV w , and c pV , are either directly or indirectly depend on t only. Only one variable, p L , is a function5 of both t and r ; however, we know that p L can be ˙ w L , and w˙ L are obtained for any given r , as written obtained explicitly once R, R, in Eq. (5.114). Sixteen constant physical properties required are the following: αe , αc 6 , C4∗ , ∗ d4 ,Rc ,γ0 , ρ L , σ , c L , λ L , μ L , c p0 , cλV , cμV , νcp , νc . Then, this set of equations

5

We notice that the independent variable s can be replaced back with t alone, with the use of Eqs. (5.171) and (5.264); hence, variables depending only on s can be rewritten as those depending only on t. We can also replace back independent variable η with both r and t, with the use of Eqs. (5.166), (5.172), (5.253), and (5.265). 6 As we discussed in Chap. 3, α is not a constant; however, we treat α as a constant in this chapter c c for simplicity.

5.9

Bubble Expansion with Uniform Interior

199

can be solved with initial and boundary conditions including p∞ , TL0 , TV 0 , and pV 0 , specified. It should be noted that we use the temperature boundary condition (5.69), listed in the nineteenth row in Table 5.3, with Eqs. (5.234) and (5.280) for TLw and TV w , respectively, and the temperature gradient boundary condition (5.143) which  is listed in the twentieth row in Table 5.3, with Eqs. (5.238) and (5.283) for TLw  and TV w , respectively. It should also be noted that both Eqs. (5.234) and (5.238), which are listed in the twelfth row in Table 5.3, are derived from the same equation (5.226); hence only one of them are used as an independent equation, and that both Eqs. (5.280) and (5.283), which in the eighteenth row in Table 5.3 are derived from the same equation (5.279). Thus, we have derived the set of equations, which can describe dynamics of a spherical vapor bubble by taking phase change at the bubble wall accurately into consideration. However, the derived set of equations is rather complicated; hence can be solved only numerically. Solving this set of equations is out of the scope of this book. Further discussion may be found in, e.g., Beylich [3]. Instead, we further simplify the mathematical model with some crucial assumptions in the next section.

5.9 Bubble Expansion with Uniform Interior We have derived a set of equations for the radial motion of a spherical vapor bubble with taking phase change at the vapor–liquid interface into account correctly in Sects. 5.4, 5.5, 5.6, 5.7, and 5.8. In this section, we show that the derived mathematical model reduces to a conventional mathematical model of bubble expansion with uniform interior, with further crucial assumptions, such that evaporating velocity of liquid is sufficiently small compared with the liquid velocity at the bubble wall. We do not use the boundary conditions at the bubble wall, Eqs. (5.158) and (5.159) which contain C 4∗ and d4∗ . Consequently, the reduced model is less accurate in dealing with mass flux at the bubble wall. Although this model is still hard to solve analytically, we can at least discuss the asymptotic behavior of bubble motion, later in this chapter.

5.9.1 Assumptions We further impose the following assumptions: (i) The quantities ρV , pV , TV , and eV are uniform in the bubble; hence these quantities are functions of time alone, (ii) Evaporating velocity of liquid: j/ρ L is sufficiently small compared with the liquid velocity at the bubble wall w L , defined by Eq. (5.131); hence w L is equated ˙ Then, we replace Eq. (5.111) by to bubble wall velocity R. vL =

R 2 R˙ . r2

(5.293)

200

5 Dynamics of Spherical Vapor Bubble ˙ wL=R TVw = TLw

Vapor

Liquid

Temperature Distribution

Bubble Wall

Fig. 5.7 The velocity of bubble wall and the temperature distribution in the simplified model

The governing equation of bubble dynamics (5.147) becomes  ρL

R R¨ +

3 ˙2 R 2

 = pV (t) − p∞ (t) −

2σ , R

(5.294)

where liquid viscosity μ L is neglected for simplicity, (iii) Temperature discontinuity at the bubble wall written in Eq. (5.144) is negligible. Then, at the bubble wall, we impose TL |r =R = TV .

(5.295)

With the use of above assumptions, the schematic diagram of the model shown in Fig. 5.5 can be greatly simplified as shown in Fig. 5.7

5.9.2 Governing Equations and Conditions Now, we consider how other quantities depend on r by examining the conservation of mass for the spherically symmetric flows, Eq. (5.95), using the fact that ρV is independent of r , ∂ρV 2ρV vV ∂ = − (ρV vV ) − = −ρV ∂t ∂r r



∂vV 2vV + ∂r r

 .

(5.296)

Notice that the left-hand side of Eq. (5.296) is independent of r , so is the right-hand side. Let ∂vV 2vV + = C(t), ∂r r

(5.297)

where C(t) is independent of r , i.e. a function of time only. The dependence of vV on r is easily found that vV = r D(t): under the condition that vV is bounded at r = 0, where D(t) is another function of time only; and then we have

5.9

Bubble Expansion with Uniform Interior

201

∂vV vV = . ∂r r

(5.298)

Substituting Eq. (5.298) into Eq. (5.296) we have, vV = −

r dρV . 3ρV dt

(5.299)

We only have to solve the equations for the outside of the bubble, i.e., those for liquid, due to the assumption of the uniformity inside the bubble. Since vcL becomes zero with the assumption (ii) in Sect. 5.9.1 and Eq. (5.174), the heat equation (5.180) can be further reduced ∂ 2 TL ∂ TL , = DL ∂s ∂z 2

(5.300)

where D L is the coefficient of thermal diffusivity of liquid and defined by DL =

λL . ρL cL

(5.301)

We now consider the boundary conditions at r = R. With the use of Eq. (5.299), the conservation equation of mass at the bubble wall (5.128) becomes ρV R˙ − w L = ρL



R dρV + R˙ 3ρV dt



d 1 = 2 4πρ L R dt



4 π R 3 ρV 3

 .

(5.302)

The assumption (ii) in Sect. 5.9.1 implies that the order of magnitude of the righthand side of Eq. (5.302) is assumed to be sufficiently small compared with that of ˙ R. As we have discussed in Sect. 5.4.3, the latent heat term usually dominates over the other terms in the right-hand side of the conservation equation of energy at the bubble wall (5.141); hence heat transport to a whole bubble due to the evaporation is written as Ldm = Ld(4π R 3 ρV /3)/dt, where dm is an infinitesimal increment of mass of vapor in the bubble. On the other hand, heat conduction at the bubble wall from the external region dominates in the left-hand side of Eq. (5.141); hence heat transport from the external region to a whole bubble due to heat conduction is written as 4π R 2 λ L (∂ TL /∂r ). Equating these two heat transports, we obtain L d ∂ TL = λL ∂r 4π R 2 dt



4 π R 3 ρV 3

 .

(5.303)

With the use of Eq. (5.264), Eq. (5.303) is also put in Lagrangian coordinates [cf. Eq. (5.164)]:

202

5 Dynamics of Spherical Vapor Bubble

λ

∂T L d = ∂z 4π R 4 dt



4 π R 3 ρV 3

 =

L d 4π ds



4 π R 3 ρV 3

 .

(5.304)

Now we seek for the analytical solution of Eqs. (5.294) and (5.300), following Plesset and Zwick [17, 25] with the assumption that the thermal diffusion length is usually small in comparison with the bubble radius for this class of problems; we solve the thermal problem based on the assumption of thin thermal boundary layer. This expansion is useful since the determination of the motion of the bubble wall depends essentially on the temperature variation in the neighborhood of the wall. In the following, subscript L is dropped for simplicity.

5.9.3 Heat Equation for Liquid We introduce new variables θ and U defined by θ (z, s) =

∂U = T − T∞ , ∂z

(5.305)

where T∞ = T (∞, 0). Introducing U enables us to solve the heat equation even when D in Eq. (5.301) is a function of r , although it is assumed constant in this analysis. Substituting Eq. (5.305) into Eq. (5.300) leads to the heat conduction equation: ∂U ∂U = D 2, ∂s ∂z

(5.306)

with boundary conditions: ∂θ (0, s) ∂ 2 U (0, s) L d =λ λ = 2 ∂z 4π ds ∂z ∂U (∞, s) = θ (∞, s) = 0, ∂z



4 π R 3 ρV 3

 ,

(5.307) (5.308)

and initial condition: θ (z, 0) = g(z).

(5.309)

The temperature at the bubble wall in Eq. (5.295) is defined by ∂U (0, s) = θ (0, s) = TV − T∞ . ∂z

(5.310)

Notice that neither R, ρV in Eq. (5.307) nor TV in Eq. (5.310) are not a priori determined, but they are parts of the solution. Function g(z) in Eq. (5.309) is an initial temperature distribution in the liquid. Suppose g(z) is given by

5.9

Bubble Expansion with Uniform Interior

203

g(z) = [TV (0) − T∞ ] e−z/z 0 ,

(5.311)

where z 0 is so chosen that a variable l, 1/3

l = z0 ,

(5.312)

is the thickness of the initial thermal boundary layer. This statement will be made clear in the following. As the liquid has been assumed incompressible, using Eq. (5.164), we have  z(r, t) =

r

ξ 2 dξ =

R(t)

 1 3 r − R(t)3 . 3

(5.313)

Substituting Eqs. (5.311) and (5.313) into Eq. (5.305), we have 

 (r 3 − R03 ) . T (r, 0) = T∞ + [TV (0) − T∞ ] exp − 3l 3

(5.314)

We integrate Eq. (5.311) once with respect to z and impose that U (∞, s) = 0, to obtain initial condition for U : U (z, 0) = −z 0 [TV (0) − T∞ ] e−z/z 0 .

(5.315)

5.9.4 Solution of Heat Equation Equation (5.306) with conditions (5.307), (5.308), and (5.315) can be solved using the Laplace transform:  U(z, σ ) = L {U (z, s)} =

∞

e−σ τ U (z, τ )dτ.

(5.316)

0

Applying the Laplace transform, Eqs. (5.306), (5.307), and (5.308) become σ U(z, σ ) − U (z, 0) = D

∂2 U(z, σ ), ∂z 2

% & L d ! 3 " ∂ 2 U(0, σ ) { = J (σ ) = L j (s)} = L ρ , R V ∂z 2 3λ ds ∂U(∞, σ ) = 0. ∂z

(5.317)

(5.318)

(5.319)

204

5 Dynamics of Spherical Vapor Bubble

The solution which is bounded at r → ∞ for the ordinary differential equation (5.317) is given by      β D σ J (σ ) − 2 exp −z , U(z, σ ) = β exp (−z/z 0 ) + s D z0

(5.320)

where β is defined as β=−

TV (0) − T∞ σ−

D z 02

.

(5.321)

Here, noticing Eq. (5.305), we should obtain the inverse Laplace transform of not U but ∂U/∂z; β ∂U(z, σ ) = − exp (−z/z 0 ) + ∂z z0

     β D σ J (σ ) − 2 exp −z . σ D z0



(5.322)

With the use of formulas of inverse Laplace transform: ! ) "⎫ ⎧  σ ⎬   ⎨ exp −z D D z2 −1 ) L exp − = , ⎩ ⎭ σ πs 4Ds

(5.323)

D

⎧ ⎫   % & ⎨ TV (0) − T∞ ⎬ Ds β −1 −1 − = [TV (0) − T∞ ] exp L . − =L ⎩ z0 z 02 σ − D2 ⎭

(5.324)

z0

Suppose functions ϕ(s) and ψ(s) are given, then the convolution of these function ϕ(s) ∗ ψ(s) is defined as  ϕ(s) ∗ ψ(s) =

s

ϕ(s − τ )ψ(τ )dτ.

(5.325)

0

Then, we obtain, θ =

% & ∂U ∂U = L−1 ∂z ∂z 



     D z z2 exp − − = [TV (0) − T∞ ] exp exp − z0 πs 4Ds    TV (0) − T∞ Ds L d 3 (R ρV ) + exp . (5.326) ∗ 3λ ds z0 z 02 Ds z 02

5.9

Bubble Expansion with Uniform Interior

205

With the use of Eqs. (5.312), (5.164), and (5.171), we obtain D Ds = 2 z0

*t 0

R(x)4 dx z 02

=

D

*t 0

R(x)4 dx , l6

r 3 − R(t)3 z = . z0 3l 3

(5.327) (5.328)

Substituting Eqs. (5.326), (5.327), and (5.328) into Eq. (5.305), we obtain 

 *t D 0 R(x)4 dx r 3 − R(t)3 T (r, t) = T∞ + [TV (0) − T∞ ] exp − + 3l 3 l6 ⎧  (2 ⎫ ' 3  ⎬ ⎨ r − R(t)3 DT s(t) 1 )* − exp − * s(t) ⎩ 36D s(t) π 0 R(y)4 dy ⎭ R(y)4 dy ξ ξ ⎧ ⎤⎫ ⎡ * s(t) 4 dy ⎬ ⎨ L 4 D R(y) [TV (0) − T∞ ] R(t) d 3 ξ ⎦ dξ, × exp ⎣ (R ρV ) + ⎭ ⎩ 3R(t)4 λ dt l3 l6 (5.329) where Eq. (5.171) is used. Now, we evaluate T at r = R, using Eq. (5.329):   L d 3 D 1 T (R, t) = T∞ − ρ ) ∗ (R V π s(t) 3R(t)4 λ dt 9     : Ds(t) Ds(t) D 1 + [TV (0) − T∞ ] exp ∗ exp − . (5.330) l6 πl 6 s(t) l6 

Notice that 



  1 Ds(t) ∗ exp s(t) l6    s(t) D 1 D(s(t) − ξ ) dξ exp = ξ πl 6 0 l6     Ds(t) Ds(t) = exp , erf l6 l6 D πl 6 

then, Eq. (5.330) becomes

(5.331)

206

5 Dynamics of Spherical Vapor Bubble

 T (R, t) = T∞ − c

u 0

   d (yρ )dv u u dv√ V + [TV (0) − T∞ ] exp erfc , u u u−v 0 0 (5.332)

where we introduce new variables and constants:  y=

R R0

3

α and u = 4 R0



t

R(y)4 dy,

(5.333)

αl 6 . D R04

(5.334)

0

and L R0 c= 3λ



αD π

and u 0 =

Thus the temperature field T (r, t) is completely determined from Eq. (5.329), if R(t) and ρV (t) are known. On the other hand, if we turn our attention to Eq. (5.332), the role of temperature field in analysis of solution is slightly different. Notice that the left-hand side of Eq. (5.332) is equated with the uniform temperature inside bubble, TV (t), by using temperature boundary condition (5.295); hence Eq. (5.332) is rewritten as  TV (t) = T∞ − c

u 0

   d (yρ )dv u u dv√ V erfc . (5.335) + [TV (0) − T∞ ] exp u0 u0 u−v

Then, R(t) may be determined from Eq. (5.335), since y and u in Eq. (5.333) are functions of R(t), if ρV (t), pV (t), and TV (t) are known. We know that ρV (t), pV (t), and TV (t) are not given a priori, in general, but given as the solution of Eq. (5.335) in conjunction with (5.329), the equation of state [cf. Eq. (5.116)], and the equation for the phase equilibrium that typically determines the equilibrium vapor pressure at given temperature.

5.9.5 Asymptotic Growth of Vapor Bubble Assuming that pV = pe (TV ), and hence ρe = ρV (TV ),

(5.336)

where pe and ρe are the equilibrium vapor pressure and density, respectively, we study the asymptotic growth of a vapor bubble in a superheated liquid as t → ∞, by simplifying Eq. (5.335), noticing that the last term in the right-hand side of Eq. (5.335) vanishes as t → ∞, since exp(y)erfc(y) → 0 as y → ∞. Then, Eq. (5.335) can be approximated to become,

5.9

Bubble Expansion with Uniform Interior



207 u

T0 − Tb ∼ cρe (Tb ) 0

d y dv , as t → ∞. √dv u−v

(5.337)

Here, we assumed that, for sufficiently large t, the temperature inside the bubble is regarded as constant, TV (t) = Tb ; hence density of bubble is also regarded as constant, ρV (t; TV ) = ρe (t; TV ) = ρe (Tb ). We consider the following equation: 

u

0

d y dv √dv = G, u−v

(5.338)

where G=

3λ R0 Lρe (Tb )



π (T0 − Tb ) . α DT

(5.339)

Equation (5.338) belongs to the class of Abel’s integral equations [18] (see Appendix C at the end of this book). We simply substitute Eq. (5.338) into Eq. (C.28), noticing that G is constant, and we obtain sin 12 π d dy = du π du



u 0

G 2G d √ dv = u. 1/2 (u − v) π du

(5.340)

Then solution of Eq. (5.338) is given by y=

2G √ u. π

(5.341)

By substituting Eq. (5.341) into Eq. (5.337), we obtain, 

1 6λ (T0 − Tb ) π D Lρe (Tb )

R3 ∼



t

R 4 (y)dy.

(5.342)

0

Here we are only interested in the asymptotic behavior of bubble radius R as t → ∞; hence we assume that R has the form of a power-low function of t, R ∼ t k . Then Eq. (5.342) becomes  t

3k

∼

 4k+1 1 1 6λ (T0 − Tb ) t 2 . π D Lρe (Tb ) 4k + 1

(5.343)

Equating exponents of t in both sides of Eq. (5.343), we obtain k = 1/2. By dividing both sides by t 2 , we have [25]  R∼2

λ 3 (T0 − Tb )t 1/2 , π Lρe (Tb )

as t → ∞.

(5.344)

208

5 Dynamics of Spherical Vapor Bubble

5.9.6 Bubble Motion Coupled with Heat Conduction We now try to obtain the complete picture of the bubble motion. We need to couple Eq. (5.332) with Eq. (5.294). Notice that R = R0 y 1/3 , du/dt = α(R/r0 )4 = αy 4/3 and α R0 y 2/3 dy , R˙ = 3 du

(5.345)

the left-hand side of Eq. (5.294) becomes d 1 3 R R¨ + R˙ 2 = (R0 α)2 2 6 dy



% y

7/3

dy du

& ,

(5.346)

and the right-hand side of Eq. (5.294) becomes   ( 2σ 2σ 1' 1 . pV (R(t), t) − p∞ (t) − = pV (R(t), t) − p∞ (t) − ρ R ρ p R0 y 1/3 (5.347) Equating Eqs. (5.346) and (5.347), we have 1 d 6 dy



 y

1 d = 6 dy

7/3

dy du



2 

 y

7/3

+φ+ dy du

1 1 2σ 2 1/3 ρ R0 (α R0 ) y

2  +φ+

1 y 1/3

= 0,

(5.348)

where φ=

( R0 ' p∞ − pe (TV ) , 2σ

(5.349)

) and α = (2σ/ρ R02 ). Notice that Eqs. (5.332) and (5.348) are connected when the equilibrium vapor pressure is specified as a function of temperature, this fact which is pointed out by Plesset and Zwick [17]. They assume that, for superheats not too far above the boiling temperature Tb of the liquid at the external pressure p∞ , the vapor pressure may be approximated by a linear function of the temperature: pe (TV ) − p∞ = A(T − Tb ). ρ

(5.350)

References

209

Thus ⎧  u d (yρ )dv R0 ρ A ⎨ dv√ V φ(TV ) = − T∞ − Tb − c 2σ ⎩ u−v 0    & u u erfc . + [TV (0) − T∞ ] exp u0 u0

(5.351)

With the use of the following expressions; r0 ρ A 1 ρe (TV ) cρe (Tb ) = , , ξ= , b= TΔ 2σ ρV (Tb ) TΔ

(5.352)

Eq. (5.348) with Eq. (5.332) becomes 1 d 6 dy



 y

7/3

dy du

2 

 u d (yρ )dv 1 T∞ − Tb dv√ V − 1/3 − b = TΔ y u−v 0    u u (TV (0) − T∞ ) exp . + erfc TΔ u0 u0

(5.353)

Boundary conditions for Eq. (5.353) are obtained from Eqs. (5.333) and (5.345) as y = 1,

dy =0 du

at u = 0.

(5.354)

Solving Eq. (5.353) with (5.354) is out of the scope of this book. Further discussion may be found in, e.g. Plesset and Zwick [17, 25].

References 1. G.K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge, 1967) 2. C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, Singapore, 1978) 3. A.E. Beylich, Dynamics and thermodynamics of spherical vapour bubbles. VDIForshungsheft 630, 1–27 (1985) 4. R.B. Bird, W.E. Stewart, E.N. Lightfood, Transport Phenomena (Wiley, New York, NY, 1960) 5. R.S. Borden, A Course in Advanced Calculus (Dover Press, New York, NY, 1997) 6. E.B. Dusssan, On the difference between a bounding surface and a material surface. J. Fluid. Mech. 75, 609–623 (1976) 7. H. Goldstein, Classical Mechanics, 2nd edn. (Addison Wesley Publishing, London, 1980) 8. M. Gurin, An Introduction to Continuum Mechanics (Academic, Orlando, FL, 1981) 9. D.Y. Hsieh, Some analytical aspects of bubble dynamics. J. Basic Eng. 87, 991–1005 (1965) 10. JSME Data Book: Thermophysical Properties of Fluids (Japan Society of Mechanical Engineers, Tokyo, 1983)

210

5 Dynamics of Spherical Vapor Bubble

11. D. Kondepudi, I. Prigogine, Modern Thermodynamics: From Heat Engines to Dissipative Structures, 2nd edn. (Wiley, London, 1999) 12. L. Landau, E. Lifshitz, Fluid Mechanics, 2nd edn. (Pergamon Press, Oxford, 1987) 13. L. Landau, E. Lifshitz, Statistical Physics, 3rd edn. (Pergamon Press, Oxford, 1980) 14. M.J. Miksis, L. Ting, Nonlinear radial oscillations of a gas bubble including thermal effects. J. Acoust. Soc. Am. 79, 997–905 (1984) 15. M. Minnarert, Musicalair-bubbles and sounds of running water. Philos. Mag. 16, 235–248 (1933) 16. S. Osher, R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, New York, NY, 2003) 17. M.S. Plesset, S.A. Zwick, The growth of vapor bubbles in superheated liquids. J. Appl. Phys. 25, 493–500 (1954) 18. D. Porter, D.S.G. Stirling, Integral Equations (Cambridge University Press, Cambridge, 1990) 19. A. Prosperetti, Boundary conditions at a liquid-vapor interface. Meccanica 14, 34–47 (1979) 20. J.S. Rowlinson, B. Widom, Molecular Theory of Capillarity (Clarendon Press, Oxford,1982) 21. I. Stackgold, Green’s Functions and Boundary Value Problems (Wiley, New York, NY, 1979) 22. T. Tanahashi, Continuum Mechanics, vol. 6, (Rikoh Tosho, Tokyo, 1988) 23. L. Trilling, The collapse and rebound of a gas bubble. J. Appl. Phys. 23, 14–17 (1952) 24. A.H. Zemanian, Generalized Integral Transformations (Dover, New York, NY, 1987) 25. S.A. Zwick, M.S. Plesset, On the dynamics of small vapor bubbles in liquids. J. Math. Phys. 33, 308–330 (1955)

Appendix A

Vectors, Tensors, and Their Notations

A.1 Scalar, Vector, and Tensor A physical quantity appears in this book is a scalar, vector, or tensor. A quantity expressed by a number, such as mass, volume, temperature, is a scalar. If a quantity a is expressed as a one-dimensional array of numbers, a = (a1 , a2 , . . . , a N ), where N is the dimension of the physical space on which a is considered, it is a vector, and ai (i = 1, 2, . . . , N ) is called the ith component of vector a. The number N is three in this book. Displacement, velocity, and force are vectors. If a and b are vectors, a linear combination αa + β b,

(α and β are scalars),

(A.1)

gives a vector. Sometimes, ai is used as a representative of vector a. A tensor is a multi-dimensional array of numbers. If P is an n-dimensional array of numbers, it is called an nth-order tensor. A scalar and a vector may be called a zeroth-order tensor and a first-order tensor, respectively. An example of a secondorder tensor is a stress tensor in continuum mechanics. A second-order tensor can be expressed by a matrix. It is important to notice that a tensor is a multilinear functional of tensors.1 For example, a second-order tensor P(a, b), which is a functional of two first-order tensors (vectors) a and b, satisfies the bilinearity relations P(a 1 + a2 , b) = P(a1 , b) + P(a2 , b),

(A.2)

P(a, b1 + b2 ) = P(a, b1 ) + P(a, b2 ),

(A.3)

P(αa, b) = P(a, αb) = α P(a, b),

(A.4)

where α is a scalar. Remember that a stress tensor gives a stress if two unit vectors are specified, one determines the normal direction of a surface and another does the component of the stress acting on the surface. 1A

functional is a mapping of some functions to a number.

S. Fujikawa et al., Vapor-Liquid Interfaces, Bubbles and Droplets, Heat and Mass C Springer-Verlag Berlin Heidelberg 2011 Transfer, DOI 10.1007/978-3-642-18038-5, 

211

212

Appendix A

If a tensor (scalar or vector) is a function of time t and position x, it is called a tensor field (scalar field or vector field). Usually, they are assumed to be continuously differentiable with respect to t and x. In fluid dynamics, a pressure field, velocity field, and stress field are, respectively, a scalar field, vector field, and tensor field, and all of them are continuously differentiable. Thus, the basic conservation laws of physics can be expressed in the forms of partial differential equations consisted of scalars, vectors, and tensors. It is essential for physics and its applications to engineering that the basic conservation laws expressed by scalars, vectors, and tensors are unchanged under the rotation of coordinate system with a fixed origin and the Galilean transformation.

A.2 Einstein Summation Convention After a Cartesian coordinate system x = (x1 , x2 , x3 ) is specified, the components ai ’s (i = 1, 2, 3) of vector field a and the components Pi j ’s (i, j = 1, 2, 3) of a second-order tensor field P are determined.2 The expressions of mathematically complicated equations can often be made compact by using symbols like a and P instead of ai ’s and Pi j ’s. However, the numerical evaluations of vectors and tensors require the handling of their components. In the following, we summarize notations of some binary-product operations of vectors and tensors presented in component forms. The inner product (or scalar product) of a second-order tensor P and a vector a gives a vector b, i.e., b = P · a,

(A.5)

and this can be written as bi =

3 

Pi j a j ,

(i = 1, 2, 3).

(A.6)

j=1

According to the Einstein summation convention, we can eliminate the summation symbol to yield bi = Pi j a j ,

(i = 1, 2, 3).

(A.7)

The Einstein summation convention is a rule of notation of binary-product operations of vectors and tensors in a single term, which states that if a same index (subscript) appears twice in a single term, then the summation is taken from one to

2 The

representations of vectors and tensors in component forms are possible in arbitrary curvilinear coordinate systems.

Appendix A

213

three for the index in the term. The index is called a dummy index. Hereafter, we use the Einstein summation convention. The scalar product of two vectors, a and b, gives a scalar α, α = a · b = ai bi .

(A.8)

The dyadic of two vectors gives a second-order tensor P, P = ab = ai b j = Pi j ,

(i, j = 1, 2, 3),

(A.9)

where, as usual, we do not distinguish a tensor P from its representative expression Pi j , although the notation Pi j is often used as the (i, j)th component of tensor P. The scalar product (or contraction) of two tensors T and U, denoted by T : U, gives a scalar α, α = T : U = Ti j Ui j .

(A.10)

The gradient of a scalar field f is a vector, and can be expressed as grad f = ∇ f =

∂f , ∂ xi

(i = 1, 2, 3).

(A.11)

The divergence of a vector field v is a scalar expressed as div v = ∇ · v =

∂vi . ∂ xi

(A.12)

The strain rate tensor3 in fluid dynamics, ε, can be constructed by the dyadic of vectors ∇ and v as4   1  ∂v ∂v j 1 i T = εi j , (i, j = 1, 2, 3), = ε= + (A.13) ∇v + (∇v) 2 2 ∂x j ∂ xi where the superscript T denotes the transpose of matrix. Here, to simplify the notation further, we can indicate the differentiation with respect to xi by index i after an index denoting a component of vector or tensor with a comma separating the two indices. That is, grad f = f ,i ,

3 It

div v = vi,i ,

ε=

$ 1# vi, j + v j,i , 2

(A.14)

is sometimes called the rate-of-strain tensor or rate-of-deformation tensor.

∇ is not a vector because it is not an array of numbers but an array of differential operators, ∇ = (∂/∂ x 1 , ∂/∂ x2 , ∂/∂ x3 ). 4 Precisely,

214

Appendix A

where since f is a scalar, no index appears before the comma before index i indicating the differentiation with respect to xi . The Einstein summation convention is also applied to this type of simplified notation as shown in the second equation in Eq. (A.14). The Kronecker delta δi j is a representation of the second-order identity tensor I, given by 9 δi j =

1 if i = j, 0 otherwise.

(A.15)

The identity transformation from a vector a to a vector b can be written with the Kronecker delta as b = I · a = δi j a j = ai ,

(i = 1, 2, 3).

(A.16)

The Eddington epsilon i jk defined by

i jk

⎧ ⎪ 1 (i, j, k) = (1, 2, 3), (2, 3, 1), (3, 1, 2), ⎪ ⎨ = −1 (i, j, k) = (3, 2, 1), (2, 1, 3), (1, 3, 2), ⎪ ⎪ ⎩ 0 i = j or j = k or k = i,

(A.17)

is the third-order alternating unit tensor. A vector product of two vectors and curl operation to vector field v can be expressed as c = a × b = i jk a j bk = ci , (i = 1, 2, 3), ∂vk = i jk vk, j , (i = 1, 2, 3). curl v = ∇ × v = i jk ∂x j

(A.18) (A.19)

Several relations involving δi j and i jk are useful in manipulations of vectors and tensors: δi j δi j = 3, i jk i jk = 6, i jk h jk = 2δi h , i jk mnk = δim δ jn − δin δ jm .

(A.20) (A.21) (A.22)

A second-order tensor P is called symmetric, if P = P T , or Pi j = P ji ,

(i, j = 1, 2, 3).

(A.23)

Clearly, the strain rate tensor ε and the Kronecker delta are the symmetric secondorder tensors.

Appendix B

Equations in Fluid Dynamics

B.1 Conservation Equations Let the macroscopic variables be defined everywhere in a space filled with a fluid, and let them be continuously differentiable functions of time t and position x. The macroscopic variables that should be defined at this stage are the density ρ, velocity v, internal energy per unit mass e, stress tensor P, and heat flux q. Then, the conservation equations of mass, momentum and energy of the fluid, in general, are respectively written as ∂ρ + ∇ · (ρv) = 0, ∂t

(B.1)

∂ρv + ∇ · (ρvv + P) = ρb, (B.2) ∂t     1  2 ∂ 1  2 ρ v + ρe + ∇ · ρ v + ρe v + v · P + q ∂t 2 2 = ρb · v + ρ S, (B.3) where b is a body force exerted on the fluid per unit mass, 1  2 ρ v + ρe 2

(B.4)

is the total energy of the fluid per unit volume, and S is a heat generated in the fluid per unit mass and per unit time. The body force b and heat generation S are independent of the motion of fluid and prescribed by some other rules. Equations (B.1), (B.2), and (B.3) are the most fundamental equations in fluid dynamics, and can be derived, for example, by considering the conservation of mass, momentum, and energy in a volume element in the physical space without specifying the explicit forms of P and q. Furthermore, the relation between the density ρ and the internal energy e is not necessary for the derivation of Eqs. (B.1), 215

216

Appendix B

(B.2), and (B.3).1 Clearly, the number of unknown variables in Eqs. (B.1), (B.2), and (B.3) exceeds the number of Eqs. (B.1), (B.2), and (B.3), and therefore we have to add some equations. Usually, fluid dynamics assumes that (1) The fluid is a Newtonian fluid in the sense that the stress tensor is given by the sum of the pressure p and the viscous stress tensor τ ,2 P = pI − τ,   2μ τ = 2με + μb − (ε : I)I, 3

(B.5) (B.6)

where μ is the viscosity coefficient, μb is the bulk viscosity coefficient,3 ε is the strain rate tensor defined by Eq. (A.13) in Appendix A, and the operator : means the contraction of two second-order tensors defined by Eq. (A.10) in Appendix A. Since the strain rate tensor ε and the identity tensor are symmetric, the viscous stress tensor τ is also symmetric. The viscosity coefficients are usually assumed as functions of temperature and pressure.4 (2) The heat flux obeys the Fourier law, q = −λ∇T,

(B.7)

where λ is the thermal conductivity coefficient and T is the temperature of fluid. The thermal conductivity coefficient is usually assumed as a function of temperature and pressure. (3) The thermodynamic relations hold among the pressure p, temperature T , internal energy e, and density ρ. This is the assumption of local equilibrium state. For the above four thermodynamic variables, there exist two independent thermodynamic relations. For example, if the fluid is an ideal gas, we have p = ρ RT,

e = cv T,

(B.8)

where the first one is the (thermal) equation of state of ideal gas (R = k/m is the gas constant, k is the Boltzmann constant, and m is a mass of a molecule) and the second is the (caloric) equation of state of ideal gas (cv is the specific heat for constant volume per unit mass). If the gas is treated as an incompressible fluid, the density ρ is not a thermodynamic variable. Then, the first equation in Eq. (B.8) should be discarded and the definition of incompressible flows

1 In the incompressible fluid flows, we cannot assume any relation between ρ

and other thermodynamics variables. Nevertheless, the conservation laws (B.1), (B.2), and (B.3) should be satisfied.

2 In

many textbooks of fluid dynamics, the sign of stress tensor P is opposite to Eq. (B.5).

3 The 4 The

bulk viscosity coefficient is sometimes called the second viscosity coefficient.

viscosity coefficients and thermal conductivity coefficient of an ideal gas are functions of temperature.

Appendix B

217

∂ρ + v · ∇ρ = 0, ∂t

(B.9)

should be used instead. At least for ideal gases, the above three statements are theoretically validated by the kinetic theory of gases in the limit that the Knudsen number goes to zero, if the nonlinearity is sufficiently weak.5 For liquids, although there are no theoretical validations for Eqs. (B.5), (B.6), and (B.7), they are as a whole admitted and significant objections have never been raised against them.6 Thus, the system of equations in fluid dynamics is closed. In principle, we can solve it under appropriate boundary conditions and initial condition. The set of equations, Eqs. (B.1), (B.2), and (B.3) with Eqs. (B.5), (B.6), and (B.7) may be called the set of Navier–Stokes equations.7 Equations (B.1), (B.2), and (B.3) are written in the so-called conservation law form, ∂ (ρ f ) = −∇ · (ρ f v + φ) + ρϑ, ∂t

(B.10)

where f and ϑ are vectors or scalars and φ is a vector or a tensor. In fact, Eqs. (B.1), (B.2), and (B.3) are recovered as follows: Eq. (B.1) :

f = 1,

φ = 0,

Eq. (B.2) :

f = v,

φ = P,

Eq. (B.3) :

f =

1  2 v + e, 2

ϑ = 0,

(B.11)

ϑ = b, φ = v · P + q,

(B.12) ϑ = b · v + S.

(B.13)

In the above three equations, ρ f v+φ is very important for understanding the physics related to the interface: ρv in Eq. (B.1) is called the mass flux density vector, ρvv + P in Eq. (B.2) is called the momentum flux density tensor, and ρ( 12 |v|2 + e)v + v · P + q in Eq. (B.3) is called the energy flux density vector. In fluid dynamics, in addition to Eq. (B.3), there are several variations in the equation associated with the energy. For example, the equation of the internal energy 5 See

Footnotes 20 and 21 in Chap. 2. fluids are excluded, of course.

6 Non-Newtonian

7 The name “Navier–Stokes equations” is often used to indicate the momentum conservation equa-

tions Eq. (B.2) with the stress tensor of Newtonian fluid (B.5) and (B.6) or its variations, ρ 

∂v = −ρ(v · ∇)v − ∇ p + ∇ · τ + ρb, ∂t

   ∂v 1 + (v · ∇)v = −∇ p + μ∇ 2 v + μb + μ ∇ (∇ · v) + ρb, ∂t 3 for constant μ and μb .

and

ρ

218

Appendix B

per unit volume can be written as ∂ρe = −∇ · (ρev) − p∇ · v + τ : ε − ∇ · q + ρ S. ∂t

(B.14)

B.2 Conservation Equations in Component Forms As mentioned in Appendix A, actual numerical evaluations of vectors and tensors require the handling of their components. We therefore write down Eqs. (B.1), (B.2), and (B.3) and Eqs. (B.5), (B.6), and (B.7) in component forms with indices using the Einstein summation convention explained in Appendix A. The mass conservation equation (B.1): ∂ρvi ∂ρ = 0. + ∂t ∂ xi

(B.15)

The momentum conservation equation (B.2): ∂ρvi v j + Pi j ∂ρvi = ρbi , + ∂t ∂x j

(i = 1, 2, 3).

(B.16)

The stress tensor of Newtonian fluid (B.5) and (B.6): Pi j = pδi j − τi j ,    ∂v j ∂vi 2μ ∂vk τi j = μ + μb − + δi j , ∂x j ∂ xi 3 ∂ xk 

(B.17) (B.18)

where i, j = 1, 2, 3. The energy conservation equation (B.3):     1 2 ∂ 1 2 ρvi + ρe + ρvi + ρe v j + vi Pi j + q j = ρb j v j + ρ S. 2 ∂x j 2 (B.19) The heat flux based on the Fourier law (B.7): ∂ ∂t



q j = −λ

∂T , ∂x j

( j = 1, 2, 3).

(B.20)

Appendix C

Supplements to Chapter 5

C.1 Generalized Stokes Theorem We here prove the generalized Stokes theorem by using the Gauss theorem: .





∇ · WdV = V

W · nd S,

.

or

S

V

∂n W..n d V =

S

W..i n i d S,

(C.1)

where W (W..n ) is a tensorial quantity of any order. The Gauss theorem turns the surface integral of W over a closed surface S which is enclosing a volume V into the volume integral of a derivative of W (the divergence) over the interior of S, i.e., over the volume of V . We assume here that the surface S where the integration is evaluated is the plane surface as shown in Fig. C.1, for simplicity. Although this choice of the integral surface is rather special, the following discussion is also valid for general integral surfaces by considering the integration over an infinitesimal area element on the tangential surface at a point of contact. nT = n ST SS

n

nS = nC

S

nC nB = – n

tC

h

C SB

Fig. C.1 A volume considered in the proof of the generalized Stokes theorem

We draw a smooth closed line C on the plane surface S, and construct a column perpendicular to its base whose peripheral edge is C, as shown in Fig. C.1. The 219

220

Appendix C

height of this column is h. The unit normal vector to the plane surface S is denoted as n. The column is enclosed by the lateral closed surface S S and the top and bottom base surfaces S T and S B . The unit normal vectors to these three surfaces are n S , nT , and n B , respectively. The unit normal and tangential vectors to the closed line C are defined as nC and t C , respectively. Now we substitute T × n (nml T..m nl ) into W of Eq. (C.1) to obtain  V

. ∇ · (T × n)d V = (T × n) · n S d S S S S   + (T × n) · nT d S T + (T × n) · n B d S B , ST

(C.2)

SB

where T is a tensor of any order. The integrand of the left-hand side of Eq. (C.2) can be rewritten as 



h

∇ · (T × n)d V =

. (∇ × T ) · nd Sdh,

0

V

(C.3)

S

by using the fact that n is constant since the surface S is plane. Now we rewrite the right-hand side of Eq. (C.2). Notice that the following holds: (T × n) · n = i jk T.. j n k n i = [n 1 (T..2 n 3 − T..3 n 2 ) + n 2 (T..3 n 1 − T..1 n 3 ) + n 3 (T..1 n 2 − T..2 n 1 )] = [T..1 (n 2 n 3 − n 3 n 2 ) + T..2 (n 3 n 1 − n 1 n 3 ) + T..3 (n 1 n 2 − n 2 n 1 )] = 0, (C.4) and that n T is equal to n and n B is equal to −n. Therefore only the integration over the lateral surface of the column contributes to Eq. (C.2). Since n S is written as nC on the lateral surface, the left-hand side of Eq. (C.2) is rewritten as 

. SS

(T × n) · n S d S, = 0

h

. (T × n) · nC dldh.

(C.5)

C

With the use of nC = t C × n and n · t C = 0, the integrand of the left-hand side of Eq. (C.5) can be rewritten as ! " (T × n) · nC = (T × n) · t C × n = i jk T.. j n k imn tmC n n # $ = δ jm δkn − δ jn δkm T.. j n k tmC n n = T.. j n k t Cj n k − T.. j n k tkC n j " ! ! " = T · t C (n · n) − (T · n) t C · n = T · t C . (C.6) So we have the right-hand side of Eq. (C.2) as

Appendix C

221

.

 SS

.

h

S

(T × n) · n d S, =

T · t C dldh.

0

(C.7)

C

Equating Eqs. (C.3) and (C.7), the Gauss theorem is rewritten as 

h 0

.



h

(∇ × T ) · nd Sdh =

.

0

S

T · t C dldh.

(C.8)

C

Equation (C.8) holds for arbitrary choice of h, and therefore we finally obtain  . (∇ × T ) · nd S = T · t C dl. (C.9) S

C

C.2 Characteristic Time of Heat Conduction We discuss the characteristic time of heat conduction by considering the simplest case, i.e., heat conduction in a uniform rod. Temperature u at position x in a uniform rod is governed by one-dimensional heat conduction equation: ∂ 2u ∂u = D 2, ∂t ∂x

(D > 0),

(C.10)

where D is the coefficient of thermal diffusivity. We first investigate the case that the rod is infinitely long; hence the domain of definition is −∞ < x < ∞. Suppose that temperature distribution at t = 0 is given as u|t=0 = ϕ(x),

(−∞ < x < ∞).

(C.11)

It is easily verified that solution of Eq. (C.10) is written as:  u(x, t) =

∞

−∞

! " exp −Dλ2 t [A(λ) cos λx + B(λ) sin λx] dλ,

(C.12)

with coefficients A and B to be determined using initial condition (C.11). The substitution of the formal solution (C.12) into the initial condition (C.11) provides the Fourier Integral representation of ϕ(x):  ϕ(x) =

∞ −∞

[A(λ) cos λx + B(λ) sin λx] dλ.

(C.13)

Coefficients A(λ) and B(λ) in Eq. (C.13) are obtained as A(λ) =

1 2π



∞

−∞

ϕ(ξ ) cos λξ dξ, B(λ) =

1 2π



∞

−∞

ϕ(ξ ) sin λξ dξ.

(C.14)

222

Appendix C

Substitution of relation (C.14) into Eq. (C.13) provides the following representation of u(x, t): u(x, t) =

1 2π





∞

−∞ ∞

dλ



∞ −∞

! " exp −Dλ2 t ϕ(ξ ) cos λ(x − ξ )dξ

 ∞ ! " 1 = ϕ(ξ )dξ exp −Dλ2 t cos λ(x − ξ )dλ 2π −∞ −∞    ∞ 1 (ξ − x)2 ϕ(ξ ) √ dξ. exp − = 4Dt 2 π Dt −∞

(C.15)

Now we solve the heat conduction problem in a semi-infinite rod. The governing equation is Eq. (C.10). We consider that the initial temperature of the rod is uniform and the temperature of an end of a rod is set as 0. Then, the boundary and initial conditions are written as u(0, t) = 0,

(C.16)

lim u(x, t) = 1,

(C.17)

⎧ ⎨ 1 if x > 0, 0 if x = 0, u(x, 0) = ϕ(x) = ⎩ −1 if x < 0,

(C.18)

x→∞

where all variables have been nondimensionalized. We require that ϕ(x) should be an odd function for Eq. (C.17) to be satisfied. This can be easily shown as follows. First we divide the solution (C.15) into two parts and rewrite as   (ξ − x)2 ϕ(ξ ) exp − u(x, t) = √ dξ 4Dt 2 π Dt 0   &  ∞ (ξ + x)2 + ϕ(−ξ ) exp − dξ . 4Dt 0 1

%

∞

(C.19)

Now we seek for the condition on which the solution satisfies Eq. (C.17) to be imposed on ϕ(x). We have from Eq. (C.19), 1 u(0, t) = √ 2 π Dt



∞ 0

  ξ2 [ϕ(ξ ) + ϕ(−ξ )] exp − dξ = 0. 4Dt

(C.20)

We find that Eq. (C.20) holds if and only if the relation ϕ(−ξ ) = −ϕ(ξ ),

(C.21)

Appendix C

223

is satisfied; hence ϕ(ξ ) should be an odd function. √ Substituting √ Eq. (C.18) in (C.19), and Setting σ1 = (ξ − x)/(2 Dt) and σ2 = (ξ + x)/(2 Dt), the solution is given by 1 u(x, t) = √ 2 π Dt 1 = √ π 2 = √ π



∞

χ

   & (ξ − x)2 (ξ + x)2 exp − − exp − dξ 4Dt 4Dt

0

−χ



∞%



 " ! 2 exp −σ1 dσ1 −

∞ χ

 " ! 2 exp −σ2 dσ2

" ! exp −σ 2 dσ = erf(χ ),

(C.22)

0

where x χ= √ . 2 Dt

(C.23)

Therefore the solution of the one-dimensional heat conduction equation (C.10) can be written by using only χ defined by (C.23). In Sect. 5.5, we use this χ to discuss the characteristic time of heat conduction.

C.3 Abel’s Integral Equation We shall seek the solution x of Abel’s integral equation, 

t

f (t) = 0

x(η) dη (t − η)ν

(0 < ν < 1),

(C.24)

where f (t) is a given continuously differentiable function. Introducing a function φ(t) defined by  φ(t) =

t

x(η)dη,

(C.25)

0

and using a formula π = sin νπ



t η

dξ (t

− ξ )1−ν (ξ

− η)ν

,

(C.26)

224

Appendix C

we can easily carry out the following integrations: π φ(t) = sin νπ





t

dη 

t

=

x(η) dξ (t − ξ )1−ν (ξ − η)ν

η

0



t

ξ

x(η) dη (t − ξ )1−ν (ξ − η)ν

dξ 0

 = 0

0 t

f (ξ ) dξ, (t − ξ )1−ν

(C.27)

where Eq. (C.24) has been used in the last equation of Eq. (C.27). Differentiating Eq. (C.27) with respect to t gives the solution of Eq. (C.24) as x(t) =

sin νπ d π dt

 0

t

f (ξ ) dξ. (t − ξ )1−ν

(C.28)

Index

A Abel’s integral equation, 207, 223 Adsorbed liquid film, 78 Antoine’s equation, 8, 80 Association, 88 degree of, 88 B Berthelot equation, 138 BKW equation, 42 Boltzmann constant, 14, 20, 32, 113 Boltzmann equation, 1, 39 Bubble wall, 168, 200 Bulk viscosity, 57, 216 C Clausius–Clapeyron equation, 127 Coefficient of thermal diffusivity, 174, 201, 221 Collision frequency, 40 Collision term, 39, 41 Complete-condensation condition, 44 Compression factor, 2, 116 Condensation coefficient, 4, 44, 53, 64 Condensation mass flux, 72 Conservation equation of energy at bubble wall, 171, 175, 201 for spherical bubble, 164 on interface, 161 Conservation equation of mass at bubble wall, 168, 201 for spherical bubble, 163 on interface, 158 Conservation equation of momentum at bubble wall, 170 for spherical bubble, 164 on interface, 160 Conservation equations, 25, 153, 215 Convective derivative, 149, 151

Convolution, 183, 204 Critical temperature, 112 Cut-off radius, 34, 113 D Density distribution function, 23 Diffuse-reflection condition, 43 Dirac delta function, 25, 182 E Einstein summation convention, 39, 85, 160, 212 Energy flux density vector, 217 Energy reflectance, 89 Equipartition theorem, 20 Error function, 10, 184, 205 Euler equations, 54, 55 Evaporation coefficient, 4, 44, 50, 64, 131 Evaporation into vacuum, 46, 132 F Flux balance on interface, 157 Fourier law, 165, 170, 216, 218 G Gauss divergence theorem, 26, 145, 147, 150, 151 Gaussian–BGK Boltzmann equation, 5, 55, 77 Generalized Stokes theorem, 160, 221 Ghost effect, 54 Gibbs dividing surface (equimolor dividing surface), 129 Grad–Boltzmann limit, 41 H Half-space problem, 55, 62 Hamilton’s canonical equations of motion, 22 Hamiltonian, 22, 34, 37 Heat conduction, 174

225

226 Heat equation, 178, 194, 202, 221 Heat flux, 29 Heaviside function, 179 Helmholtz free energy, 153 Hertz–Knudsen–Langmuir formula, 66, 132 H-theorem, 42, 56 I Ideal gas, 20, 41 Incident shock wave, 79 Incompressible fluid, 216 Interface, 3 Interface velocity, 145 Interferometer, 89 Intermolecular force, 21 Internal degrees of freedom, 51, 56, 74 Intramolecular force, 21 Inverse Laplace transform, 181 K Kelvin equation, 111, 127, 169 Kinetic boundary condition, 3, 43, 50, 53, 57 Kinetic theory, 38 Kinetic theory of gases, 25, 38 Knudsen layer, 3, 55, 61 Knudsen layer analysis, 61 Knudsen number, 2, 55 L Laplace equation, 126 Laplace transform, 179, 203 Latent heat, 171, 201 Leap-frog scheme, 33, 113 Lennard-Jones potential, 31, 113 Liouville equation, 24 Liquid film, 79 Liquid temperature at bubble wall, 188 gradient at bubble wall, 189 of bubble interior, 186 Liquid velocity at bubble wall, 165, 169 Local equilibrium, 20, 28, 38, 55, 59 Local Maxwellian, 41 Loschmidt constant, 20 M Mach number, 82 Mass flux across the interface, 45, 47, 53, 65, 158 of molecules spontaneously evaporating, 47, 72, 131 Mass flux density vector, 217 Mass fraction, 87

Index Maxwell distribution function, 40 Mean collision frequency, 40–41 Mean free path, 3, 40–42, 57, 116 Mean free time, 14, 40 Molecular dynamics, 31, 46 Molecular gas dynamics, 25 Momentum flux density tensor, 217 Monatomic molecule, 21, 39, 72 Moving boundary problem, 176 N Nanodroplet, 112 Navier–Stokes equations, 1, 54–55, 217 Net mass flux of condensation, 78 Net mass flux of evaporation, 76 Newton’s equation of motion, 21 Newtonian fluid, 216, 218 Noncondensable gas, 87 Nonequilibrium state, 2, 46, 78, 131 N V E simulation, 31, 113 P Partition function, 74 Periodic boundary condition, 35, 113 Permanently absorbed liquid film, 93 Phase space, 23, 25 Polyatomic molecule, 50, 55, 73 Prandtl number, 42, 57, 192 Pressure of bubble interior, 196 R Ratio of specific heats, 56, 75 Rayleigh–Plesset equation, 172 Reflected shock wave, 79 Reflection mass flux, 72 S S expansion, 58 Saturated vapor density, 8, 44, 72 Saturated vapor pressure, 8, 116 Schrage formula, 66 Second harmonics, 108 Shock tube, 6, 78 Shock wave, 6, 13, 78 Slip coefficient, 62, 63 Solvability condition, 59 Sound resonance, 107 Sound resonance method, 106 Standing wave, 108 State equation of real gas, 138 Stress tensor, 28, 29 Surface entropy, 152 Surface of tension, 129 Surface tension, 116, 124, 159

Index T Temperature discontinuity at bubble wall, 158, 171 Temporal transition phenomenon, 12 Temporarily adsorbed liquid film, 94 Thermal diffusion-controlled condensation, 13 Thickness of interface, 2, 46, 129 Tolman equation, 111, 130 Tolman length, 129 Total curvature, 147 Transition layer, 2, 35, 116 Transition time, 11 Triple point temperature, 36, 50, 112, 138 U Unit normal of interface, 145

227 V Vapor pressure, 116 Vapor temperature at bubble wall, 195 gradient at bubble wall, 195 of bubble exterior, 194 outside thermal boundary layer, 190 with uniform interior, 206 Vapor velocity at bubble wall, 165, 169 Vapor–liquid interface, 143 Velocity distribution function, 39, 48, 73, 77 Velocity field of bubble interior, 197 Velocity scaling, 47, 114, 133 Volterra integral equation of the second kind, 9, 80