Preface xi
About the companion website xiii
1 Introduction 1
1.1 History of the Lattice Boltzmann method 2
1.2 The Lattice Boltzmann method 3
1.3 Multiphase LBM 6
1.3.1 Color-gradient model 7
1.3.2 Shan-Chen model 7
1.3.3 Free-energy model 8
1.3.4 Interface tracking model 9
1.4 Comparison of models 9
1.5 Units in this book and parameter conversion 11
1.6 Appendix: Einstein summation convention 14
1.6.1 Kronecker ? function 15
1.6.2 Lattice tensors 15
1.7 Use of the Fortran code in the book 16
2 Single-component multiphase Shan-Chen-type model 18
2.1 Introduction 18
2.1.1 "Equilibrium" velocity in the SC model 20
2.1.2 Inter-particle forces in the SC SCMP LBM 20
2.2 Typical equations of state 21
2.2.1 Parameters in EOS 27
2.3 Thermodynamic consistency 28
2.3.1 The SCMP LBM EOS 29
2.3.2 Incorporating other EOS into the SC model 31
2.4 Analytical surface tension 32
2.4.1 Inter-particle Force Model A 32
2.4.2 Inter-particle Force Model B 33
2.5 Contact angle 34
2.6 Capillary rise 36
2.7 Parallel flow and relative permeabilities 39
2.8 Forcing term in the SC model 40
2.8.1 Schemes to incorporate the body force 42
2.8.2 Scheme overview 44
2.8.3 Theoretical analysis 44
2.8.4 Numerical results and discussion 46
2.9 Multirange pseudopotential (Inter-particle Force Model B) 55
2.10 Conclusions 58
2.11 Appendix A: Analytical solution for layered multiphase flow in a channel 58
2.12 Appendix B: FORTRAN code to simulate single component multiphase droplet contacting a wall as shown in Figure 2.7(c) 60
3 Shan and Chen-type multi-component multiphase models 71
3.1 Multi-component multiphase SC LBM 71
3.1.1 Fluid-fluid cohesion and fluid-solid adhesion 73
3.2 Derivation of the pressure 73
3.2.1 Pressure in popular papers (2D) 74
3.2.2 Pressure in popular papers (3D) 75
3.3 Determining Gc and the surface tension 76
3.4 Contact angle 78
3.4.1 Application of Young's equation to MCMP LBM 79
3.4.2 Contact angle measurement 79
3.4.3 Verification of proposed equation 80
3.5 Flow through capillary tubes 83
3.6 Layered two-phase flow in a 2D channel 85
3.7 Pressure or velocity boundary conditions 87
3.7.1 Boundary conditions for 2D simulations 87
3.7.2 Boundary conditions for 3D simulations 89
3.8 Displacement in a 3D porous medium 91
4 Rothman-Keller multiphase Lattice Boltzmann model 94
4.1 Introduction 94
4.2 RK color-gradient model 96
4.3 Theoretical analysis (Chapman-Enskog expansion) 99
4.3.1 Discussion of above formulae 103
4.4 Layered two-phase flow in a 2D channel 103
4.4.1 Cases of two fluids with identical densities 104
4.4.2 Cases of two fluids with different densities 106
4.5 Interfacial tension and isotropy of the RK model 110
4.5.1 Interfacial tension 110
4.5.2 Isotropy 110
4.6 Drainage and capillary filling 111
4.7 MRT RK model 113
4.8 Contact angle 114
4.8.1 Spurious currents 115
4.9 Tests of inlet/outlet boundary conditions 117
4.10 Immiscible displacements in porous media 118
4.11 Appendix A 121
4.12 Appendix B 122
5 Free-energy-based multiphase Lattice Boltzmann model 136
5.1 Swift free-energy based single-component multiphase LBM 136
5.1.1 Derivation of the coefficients in the equilibrium distribution function 138
5.2 Chapman-Enskog expansion 143
5.3 Issue of Galilean invariance 146
5.4 Phase separation 149
5.5 Contact angle 154
5.5.1 How to specify a desired contact angle 154
5.5.2 Numerical verification 155
5.6 Swift free-energy-based multi-component multiphase LBM 158
5.7 Appendix 158
6 Inamuro's multiphase Lattice Boltzmann model 167
6.1 Introduction 167
6.1.1 Inamuro's method 167
6.1.2 Comment on the presentation 169
6.1.3 Chapman-Enskog expansion analysis 170
6.1.4 Cahn-Hilliard equation (equation for order parameter) 173
6.1.5 Poisson equation 174
6.2 Droplet collision 175
6.3 Appendix 178
7 He-Chen-Zhang multiphase Lattice Boltzmann model 196
7.1 Introduction 196
7.2 HCZ model 196
7.3 Chapman-Enskog analysis 199
7.3.1 N-S equations 199
7.3.2 CH equation 202
7.4 Surface tension and phase separation 202
7.5 Layered two-phase flow in a channel 204
7.6 Rayleigh-Taylor instability 205
7.7 Contact angle 210
7.8 Capillary rise 213
7.9 Geometric scheme to specify the contact angle and its hysteresis 215
7.9.1 Examples of droplet slipping in shear flows 218
7.10 Oscillation of an initially ellipsoidal droplet 219
7.11 Appendix A 222
7.12 Appendix B: 2D code 223
7.13 Appendix C: 3D code 238
8 Axisymmetric multiphase HCZ model 253
8.1 Introduction 253
8.2 Methods 253
8.2.1 Macroscopic governing equations 253
8.2.2 Axisymmetric HCZ LBM (Premnath and Abraham 2005a) 255
8.2.3 MRT version of the axisymmetric LBM (McCracken and Abraham 2005) 256
8.2.4 Axisymmetric boundary conditions 258
8.3 The Laplace law 258
8.4 Oscillation of an initially ellipsoidal droplet 259
8.5 Cylindrical liquid column break 263
8.6 Droplet collision 265
8.6.1 Effect of gradient and Laplacian calculation 267
8.6.2 Effect of BGK and MRT 274
8.7 A revised axisymmetric HCZ model (Huang et al. 2014) 276
8.7.1 MRT collision 276
8.7.2 Calculation of the surface tension 277
8.7.3 Mass correction 278
8.8 Bubble rise 279
8.8.1 Numerical validation 281
8.8.2 Surface-tension calculation effect 283
8.8.3 Terminal bubble shape 284
8.8.4 Wake behind the bubble 284
8.9 Conclusion 286
8.10 Appendix A: Chapman-Enskog analysis 288
8.10.1 Preparation for derivation 288
8.10.2 Mass conservation 289
8.10.3 Momentum conservation 289
8.10.4 CH equation 291
9 Extensions of the HCZ model for high-density ratio two-phase flows 292
9.1 Introduction 292
9.2 Model I (Lee and Lin 2005) 293
9.2.1 Stress and potential form of intermolecular forcing terms 293
9.2.2 Model description 294
9.2.3 Implementation 297
9.2.4 Directional derivative 298
9.2.5 Droplet splashing on a thin liquid film 299
9.3 Model II (Amaya-Bower and Lee 2010) 301
9.3.1 Implementation 302
9.4 Model III (Lee and Liu 2010) 304
9.5 Model IV 305
9.6 Numerical tests for different models 306
9.6.1 A drop inside a box with periodic boundary conditions 306
9.6.2 Layered two-phase flows in a channel 311
9.6.3 Galilean invariance 313
9.7 Conclusions 316
9.8 Appendix A: Analytical solutions for layered two-phase flow in a channel 317
9.9 Appendix B: 2D code based on Amaya-Bower and Lee (2010) 319
10 Axisymmetric high-density ratio two-phase LBMs (extension of the HCZ model) 334
10.1 Introduction 334
10.2 The model based on Lee and Lin (2005) 334
10.2.1 The equilibrium distribution functions I 336
10.2.2 The equilibrium distribution functions II 336
10.2.3 Source terms 337
10.2.4 Stress and potential form of intermolecular forcing terms 337
10.2.5 Chapman-Enskog analysis 338
10.2.6 Implementation 340
10.2.7 Droplet splashing on a thin liquid film 342
10.2.8 Head-on droplet collision 342
10.3 Axisymmetric model based on Lee and Liu (2010) 345
10.3.1 Implementation 347
10.3.2 Head-on droplet collision 348
10.3.3 Bubble rise 353
Index 371
