INTERFACIAL TRANSPORT PHENOMENA 2 nd Edition John C. Slattery Department ofaerospace Engineering Texas A&M University Leonard Sagis Department of Agrotechnology & Food Science Wageningen University Eun-Suok Oh LG Chem, Research Park South Korea Spri ringer
Contents 1 Kinematics and Conservation of Mass 1 1.1 Motion 2 1.1.1 Body 2 1.1.2 Stretch and Rotation [19, p. 17] 6 1.2 Motion of Multiphase Bodies 7 1.2.1 What are Phase Interfaces? 7 1.2.2 Three-Dimensional Interfacial Region 7 1.2.3 Dividing surface 8 1.2.4 Dividing Surface as a Model for a Three-Dimensional Interfacial Region 9 1.2.5 Motion of Dividing Surface 9 1.2.6 Stretch and Rotation within Dividing Surfaces 17 1.2.7 More about Surface Velocity 18 1.2.8 Rate of Deformation 21 1.2.9 Moving Common Lines: Qualitative Description 25 1.2.10 Moving Common Lines: Emission of Material Surfaces [16] 37 1.2.11 Moving Common Lines: Velocity is Multivalued on a Rigid Solid 43 1.2.12 Moving Common Lines: Quantitative Description 47 1.3 Mass 52 1.3.1 Conservation of Mass 52 1.3.2 Surface Mass Density 55 1.3.3 Surface Transport Theorem 60 1.3.4 Transport Theorem for Body Containing Dividing Surface 67 1.3.5 Jump Mass Balance 70 1.3.6 Location of Dividing Surface 73 1.3.7 Transport Theorem for Body Containing Intersecting Dividing Surfaces 73 1.3.8 Mass Balance at a Common Line 79
vi Contents 1.3.9 Comment on Velocity Distribution in Neighborhood of Moving Common Line on Rigid Solid 85 1.3.10 More Comments on Velocity Distribution in Neighborhood of Moving Common Line on Rigid Solid 90 1.4 Frame 93 1.4.1 Changes of Frame 93 1.4.2 Frame Indifferent Sealars, Vectors, and Tensors 99 1.4.3 Equivalent Motions 100 1.4.4 Principle of Frame Indifference 105 2 Foundations for Momentum Transfer 107 2.1 Force 107 2.1.1 What are Forces? 107 2.1.2 Momentum and Moment of Momentum Balances 111 2.1.3 Body Forces and Contact Forces 113 2.1.4 Momentum Balance at Dividing Surfaces 115 2.1.5 Surface Stress Tensor 117 2.1.6 Jump Momentum Balance 119 2.1.7 T (ff) is Symmetrie Tangential Tensor 121 2.1.8 Surface Velocity, Surface Stress, and Surface Body Forcel24 2.1.9 Momentum Balance at Common Line 125 2.1.10 Momentum Balance at Common Line on Relatively Rigid Solid 130 2.1.11 Factors Influencing Measured Contact Angles 133 2.1.12 Relationships for Measured Contact Angles 136 2.1.13 More Comments Concerning Moving Common Lines and Contact Angles on Rigid Solids and Their Relation to the Disjoining Pressure 137 2.2 Correcting Material Behavior for Intermolecular Forces from Adjacent Phases [20] 140 2.2.1 The Correction 143 2.2.2 One Unbounded Dividing Surface: View (iv) 146 2.2.3 One Thin Lens or Fracture: View (iv) 150 2.2.4 One Thin Film: View (v) 152 2.2.5 A Discontinuous Thin Film: View (v) 156 2.2.6 One Unbounded Common Line: View (iv) 157 3 Applications of the Differential Balances to Momentum Transfer 159 3.1 Philosophy 159 3.1.1 Structure of Problem 159 3.1.2 Approximations 161 3.2 Only Interfacial Tension 162 3.2.1 Classes of Problems 162 3.2.2 Spinning Drop Interfacial Tensiometer [21] 164
Contents vii 3.2.3 Meniscal Breakoff Interfacial Tensiometer 171 3.2.4 Pendant Drop 182 3.2.5 Sessile Drop 188 3.3 Applications of Our Extension of Continuum Mechanics to the Nanoscale 194 3.3.1 Supercritical Adsorption [22] 195 3.3.2 Static Contact Angle [20] 202 3.3.3 A Review of Coalescence (with J. D. Chen) 208 3.3.4 Coalescence [23-25] 215 3.3.5 Moving Common Line and Receding Contact Angle... 234 3.3.6 Nanoscale Fracture [26] 248 4 Foundations for Simultaneous Momentum, Energy, and Mass Transfer 261 4.1 Viewpoint 261 4.1.1 Viewpoint in Considering Multicomponent Materials... 261 4.1.2 Body, Motion, and Material Coordinates of Species A.. 262 4.1.3 Motion of Multicomponent Dividing Surface 264 4.1.4 More about Surface Velocity of Species A 267 4.2 Mass Balance 269 4.2.1 Species Mass Balance 269 4.2.2 Concentrations, Velocities, and Mass Fluxes 275 4.2.3 Location of Multicomponent Dividing Surface 277 4.3 Further Comments on Viewpoint 279 4.3.1 Further Comments on Viewpoint of Multicomponent Materials 279 4.4 Mass 281 4.4.1 Conservation of Mass 281 4.5 Force 284 4.5.1 Momentum and Moment of Momentum Balances 284 4.5.2 Jump Momentum Balance 284 4.5.3 T( ff ) is Symmetrie, Tangential Tensor 286 4.6 Energy 287 4.6.1 Rate of Energy Transmission 287 4.6.2 Energy Balance 287 4.6.3 Radiant and Contact Energy Transmission 288 4.6.4 Jump Energy Balance 290 4.7 Entropy 295 4.7.1 Entropy Inequality 295 4.7.2 Radiant and Contact Entropy Transmission 297 4.7.3 Jump Entropy Inequality 299 4.8 Behavior as Restricted by Entropy Inequality 304 4.8.1 Behavior of Multicomponent Materials 304 4.8.2 Bulk Behavior: Implications of Entropy Inequality 304
Vlll Contents 4.8.3 Surface Behavior: Implications of Jump Entropy Inequality 316 4.8.4 Surface Behavior: Adsorption Isotherms and Equations of State 332 4.8.5 Alternative Forms for the Energy Balances and the Entropy Inequalities 349 4.9 Behavior as Restricted by Frame Indifference 352 4.9.1 Other Principles to be Considered 352 4.9.2 Alternative Independent Variables in Constitutive Equations 353 4.9.3 Bulk Behavior: Constitutive Equations for Stress Tensor, Energy Flux Vector and Mass Flux Vector 355 4.9.4 Surface Behavior: Constitutive Equations for Surface Stress Tensor 358 4.9.5 Boussinesq Surface Fluid 358 4.9.6 Simple Surface Material 361 4.9.7 Surface Isotropy Group 366 4.9.8 Isotropie Simple Surface Materials 369 4.9.9 Simple Surface Solid 371 4.9.10 Simple Surface Fluid 373 4.9.11 Fading Memory and Special Cases of Simple Surface Fluid 374 4.9.12 Simple Surface Fluid Crystals 377 4.9.13 Surface Behavior: Constitutive Equations for Surface Energy Flux Vector 377 4.9.14 Surface Behavior: Constitutive Equations for Surface Mass Flux Vector 379 4.10 Intrinsically Stable Equilibrium [27] 382 4.10.1 Stable Equilibrium 382 4.10.2 Constraints on Isolated Systems 383 4.10.3 Implications of (4.10.2-24) for Intrinsically Stable Equilibrium 390 4.10.4 Implications of (4.10.2-25) for Intrinsically Stable Equilibrium 397 4.11 Thermodynamics of Single-Component, Elastic, Crystalline Surface Solids [28] 409 4.11.1 Thermodynamics of Surface Crystals 409 4.11.2 Constraints on Isolated Systems 413 4.11.3 Implications of Equilibrium 416 4.11.4 Stress-Deformation Behavior of Single-Walled Carbon Nanotubes 423 5 Applications of the Differential Balances to Momentum, Energy and Mass Transfer 429 5.1 Philosophy 429
Contents ix 5.1.1 Structure of Problems Involving Momentum Transfer..429 5.1.2 Structure of Problems Involving Energy Transfer 429 5.1.3 Structure of Problems Involving Mass Transfer 431 5.2 Problems Involving Momentum Transfer 432 5.2.1 Boussinesq Surface Fluid in a Knife-edge Surface Viscometer 432 5.2.2 Generalized Boussinesq Surface Fluid in a Deep Channel Surface Viscometer 449 5.2.3 Simple Surface Fluid in Curvilineal Surface Flows [29]. 455 5.2.4 Simple Surface Fluid in a Deep Channel Surface Viscometer [29] 460 5.2.5 Simple Surface Fluid in an Oscillating Deep Channel Surface Viscometer [29] 463 5.2.6 Limiting Cases when Effects of Interfacial Viscosities Dominate 470 5.2.7 Displacement in a Capillary [30] 473 5.2.8 Several Interfacial Viscometers Suitable for Measuring Gener alized Boussinesq Surface Fluid Behavior [31]... 480 5.2.9 Stochastic Interfacial Disturbances Created by Thermal Noise and the Importance of the Interfacial Viscosities [32] 491 5.2.10 Capillary Rise [30, 33] 524 5.2.11 Common Line Motion in Systems with Simple Surface Fluid Material Behavior: Implications of the Entropy Inequality [34, 35] 534 5.2.12 More on Common Line Motion in Systems with Simple Surface Fluid Material Behavior: Implications in Polymer Extrusion [36] 563 5.3 Limiting Cases of Energy Transfer 575 5.3.1 Motion of a Drop or Bubble [37; with D. Li] 575 5.4 Limiting Cases of Mass Transfer 580 5.4.1 Motion of a Drop or Bubble [38; with D. Li] 580 5.4.2 Longitudinal and Transverse Waves [32] 587 A Differential Geometry 611 A.l Physical Space 611 A.l.l Euclidean Space 611 A.1.2 Notation in (E 2, V 3 ) 613 A.1.3 Surface in (E 3,V 3 ) 617 A.2 Vector Fields 617 A.2.1 Natural Basis 617 A.2.2 Surface Gradient of Scalar Field 624 A.2.3 Dual Basis 625 A.2.4 Covariant and Contravariant Components 625 A.2.5 Physical Components 626
x Contents A.2.6 Tangential and Normal Components 627 A.3 Second-Order Tensor Fields 629 A.3.1 Tangential Transformations and Surface Tensors 629 A.3.2 Projection Tensor 631 A.3.3 Tangential Cross Tensor 633 A.3.4 Transpose 636 A.3.5 Inverse 637 A.3.6 Orthogonal Tangential Transformation 639 A.3.7 Surface Determinant of Tangential Transformation 641 A.3.8 Polar Decomposition 643 A.4 Third-Order Tensor Fields 646 A.4.1 Surface Tensors 646 A.5 Surface Gradient 647 A.5.1 Spatial Vector Field 647 A.5.2 Vector Field is Explicit Function of Position in Space.. 648 A.5.3 Vector Field is Explicit Function of Position on Surface 649 A.5.4 Second-Order Tensor Field 660 A.5.5 Tensor Field is Explicit Function of Position in Space.. 661 A.5.6 Tensor Field is Explicit Function of Position on Surface 662 A.6 Integration 666 A.6.1 Line Integration 666 A.6.2 Surface Integration 668 A.6.3 Surface Divergence Theorem 669 B Summary of Useful Equations 673 B.l Useful Equations for Single Component Systems 673 B.l.l Bulk Phases 673 B.l.2 Dividing Surfaces 675 B.l.3 Common Lines 693 B.2 Useful Equations for Multicomponent Systems with Simultaneous Momentum, Energy, and Mass Transfer 694 B.2.1 Concentrations, Velocities, and Fluxes 694 B.2.2 Jump Mass, Jump Energy, and Jump Entropy Balance. 700 B.2.3 Specific Forms 704 C Applications of integral averaging to momentum, energy, and mass transfer 735 Ol Integral balances 735 C.l.l Integral overall mass balance 736 C.1.2 The Integral Mass Balance for Species A 738 C.1.3 Integral momentum balance 739 C.1.4 Integral mechanical energy balance 742 C.1.5 The Integral Energy Balance 749 C.1.6 The Integral Entropy Inequality 753
Contents xi Notation 757 References 773 Author Index 809 Index 821