Haibo Huang is an Associate Professor in the University of Science and Technology of China. He was a Courtesy Associate Professor during his stays at Florida International University. Michael C. Sukop is Professor of Hydrogeology at Florida International University in Miami and author of “Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers”. His research emphasis is on flow and transport in porous media. Xiyun Lu is a Professor of Fluid Mechanics in the University of Science and Technology of China. His research interests mainly include computational fluid dynamics, turbulence simulation and biomechanics.

• Preface xiAbout the companion website xiii1 Introduction 11.1 History of the Lattice Boltzmann method 21.2 The Lattice Boltzmann method 31.3 Multiphase LBM 61.3.1 Color-gradient model 71.3.2 Shan–Chen model 71.3.3 Free-energy model 81.3.4 Interface tracking model 91.4 Comparison of models 91.5 Units in this book and parameter conversion 111.6 Appendix: Einstein summation convention 141.6.1 Kronecker δ function 151.6.2 Lattice tensors 151.7 Use of the Fortran code in the book 162 Single-component multiphase Shan–Chen-type model 182.1 Introduction 182.1.1 "Equilibrium" velocity in the SC model 202.1.2 Inter-particle forces in the SC SCMP LBM 202.2 Typical equations of state 212.2.1 Parameters in EOS 272.3 Thermodynamic consistency 282.3.1 The SCMP LBM EOS 292.3.2 Incorporating other EOS into the SC model 312.4 Analytical surface tension 322.4.1 Inter-particle Force Model A 322.4.2 Inter-particle Force Model B 332.5 Contact angle 342.6 Capillary rise 362.7 Parallel flow and relative permeabilities 392.8 Forcing term in the SC model 402.8.1 Schemes to incorporate the body force 422.8.2 Scheme overview 442.8.3 Theoretical analysis 442.8.4 Numerical results and discussion 462.9 Multirange pseudopotential (Inter-particle Force Model B) 552.10 Conclusions 582.11 Appendix A: Analytical solution for layered multiphase flow in a channel 582.12 Appendix B: FORTRAN code to simulate single component multiphase droplet contacting a wall as shown in Figure 2.7(c) 603 Shan and Chen-type multi-component multiphase models 713.1 Multi-component multiphase SC LBM 713.1.1 Fluid–fluid cohesion and fluid–solid adhesion 733.2 Derivation of the pressure 733.2.1 Pressure in popular papers (2D) 743.2.2 Pressure in popular papers (3D) 753.3 Determining Gc and the surface tension 763.4 Contact angle 783.4.1 Application of Young's equation to MCMP LBM 793.4.2 Contact angle measurement 793.4.3 Verification of proposed equation 803.5 Flow through capillary tubes 833.6 Layered two-phase flow in a 2D channel 853.7 Pressure or velocity boundary conditions 873.7.1 Boundary conditions for 2D simulations 873.7.2 Boundary conditions for 3D simulations 893.8 Displacement in a 3D porous medium 914 Rothman–Keller multiphase Lattice Boltzmann model 944.1 Introduction 944.2 RK color-gradient model 964.3 Theoretical analysis (Chapman–Enskog expansion) 994.3.1 Discussion of above formulae 1034.4 Layered two-phase flow in a 2D channel 1034.4.1 Cases of two fluids with identical densities 1044.4.2 Cases of two fluids with different densities 1064.5 Interfacial tension and isotropy of the RK model 1104.5.1 Interfacial tension 1104.5.2 Isotropy 1104.6 Drainage and capillary filling 1114.7 MRT RK model 1134.8 Contact angle 1144.8.1 Spurious currents 1154.9 Tests of inlet/outlet boundary conditions 1174.10 Immiscible displacements in porous media 1184.11 Appendix A 1214.12 Appendix B 1225 Free-energy-based multiphase Lattice Boltzmann model 1365.1 Swift free-energy based single-component multiphase LBM 1365.1.1 Derivation of the coefficients in the equilibrium distribution function 1385.2 Chapman–Enskog expansion 1435.3 Issue of Galilean invariance 1465.4 Phase separation 1495.5 Contact angle 1545.5.1 How to specify a desired contact angle 1545.5.2 Numerical verification 1555.6 Swift free-energy-based multi-component multiphase LBM 1585.7 Appendix 1586 Inamuro's multiphase Lattice Boltzmann model 1676.1 Introduction 1676.1.1 Inamuro's method 1676.1.2 Comment on the presentation 1696.1.3 Chapman–Enskog expansion analysis 1706.1.4 Cahn–Hilliard equation (equation for order parameter) 1736.1.5 Poisson equation 1746.2 Droplet collision 1756.3 Appendix 1787 He–Chen–Zhang multiphase Lattice Boltzmann model 1967.1 Introduction 1967.2 HCZ model 1967.3 Chapman–Enskog analysis 1997.3.1 N–S equations 1997.3.2 CH equation 2027.4 Surface tension and phase separation 2027.5 Layered two-phase flow in a channel 2047.6 Rayleigh–Taylor instability 2057.7 Contact angle 2107.8 Capillary rise 2137.9 Geometric scheme to specify the contact angle and its hysteresis 2157.9.1 Examples of droplet slipping in shear flows 2187.10 Oscillation of an initially ellipsoidal droplet 2197.11 Appendix A 2227.12 Appendix B: 2D code 2237.13 Appendix C: 3D code 2388 Axisymmetric multiphase HCZ model 2538.1 Introduction 2538.2 Methods 2538.2.1 Macroscopic governing equations 2538.2.2 Axisymmetric HCZ LBM (Premnath and Abraham 2005a) 2558.2.3 MRT version of the axisymmetric LBM (McCracken and Abraham 2005) 2568.2.4 Axisymmetric boundary conditions 2588.3 The Laplace law 2588.4 Oscillation of an initially ellipsoidal droplet 2598.5 Cylindrical liquid column break 2638.6 Droplet collision 2658.6.1 Effect of gradient and Laplacian calculation 2678.6.2 Effect of BGK and MRT 2748.7 A revised axisymmetric HCZ model (Huang et al. 2014) 2768.7.1 MRT collision 2768.7.2 Calculation of the surface tension 2778.7.3 Mass correction 2788.8 Bubble rise 2798.8.1 Numerical validation 2818.8.2 Surface-tension calculation effect 2838.8.3 Terminal bubble shape 2848.8.4 Wake behind the bubble 2848.9 Conclusion 2868.10 Appendix A: Chapman–Enskog analysis 2888.10.1 Preparation for derivation 2888.10.2 Mass conservation 2898.10.3 Momentum conservation 2898.10.4 CH equation 2919 Extensions of the HCZ model for high-density ratio two-phase flows 2929.1 Introduction 2929.2 Model I (Lee and Lin 2005) 2939.2.1 Stress and potential form of intermolecular forcing terms 2939.2.2 Model description 2949.2.3 Implementation 2979.2.4 Directional derivative 2989.2.5 Droplet splashing on a thin liquid film 2999.3 Model II (Amaya-Bower and Lee 2010) 3019.3.1 Implementation 3029.4 Model III (Lee and Liu 2010) 3049.5 Model IV 3059.6 Numerical tests for different models 3069.6.1 A drop inside a box with periodic boundary conditions 3069.6.2 Layered two-phase flows in a channel 3119.6.3 Galilean invariance 3139.7 Conclusions 3169.8 Appendix A: Analytical solutions for layered two-phase flow in a channel 3179.9 Appendix B: 2D code based on Amaya-Bower and Lee (2010) 31910 Axisymmetric high-density ratio two-phase LBMs (extension of the HCZ model) 33410.1 Introduction 33410.2 The model based on Lee and Lin (2005) 33410.2.1 The equilibrium distribution functions I 33610.2.2 The equilibrium distribution functions II 33610.2.3 Source terms 33710.2.4 Stress and potential form of intermolecular forcing terms 33710.2.5 Chapman–Enskog analysis 33810.2.6 Implementation 34010.2.7 Droplet splashing on a thin liquid film 34210.2.8 Head-on droplet collision 34210.3 Axisymmetric model based on Lee and Liu (2010) 34510.3.1 Implementation 34710.3.2 Head-on droplet collision 34810.3.3 Bubble rise 353Index 371

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Haibo Huang is an Associate Professor in the University of Science and Technology of China. He was a Courtesy Associate Professor during his stays at Florida International University. Michael C. Sukop is Professor of Hydrogeology at Florida International University in Miami and author of “Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers”. His research emphasis is on flow and transport in porous media. Xiyun Lu is a Professor of Fluid Mechanics in the University of Science and Technology of China. His research interests mainly include computational fluid dynamics, turbulence simulation and biomechanics.

• Preface xiAbout the companion website xiii1 Introduction 11.1 History of the Lattice Boltzmann method 21.2 The Lattice Boltzmann method 31.3 Multiphase LBM 61.3.1 Color-gradient model 71.3.2 Shan–Chen model 71.3.3 Free-energy model 81.3.4 Interface tracking model 91.4 Comparison of models 91.5 Units in this book and parameter conversion 111.6 Appendix: Einstein summation convention 141.6.1 Kronecker δ function 151.6.2 Lattice tensors 151.7 Use of the Fortran code in the book 162 Single-component multiphase Shan–Chen-type model 182.1 Introduction 182.1.1 "Equilibrium" velocity in the SC model 202.1.2 Inter-particle forces in the SC SCMP LBM 202.2 Typical equations of state 212.2.1 Parameters in EOS 272.3 Thermodynamic consistency 282.3.1 The SCMP LBM EOS 292.3.2 Incorporating other EOS into the SC model 312.4 Analytical surface tension 322.4.1 Inter-particle Force Model A 322.4.2 Inter-particle Force Model B 332.5 Contact angle 342.6 Capillary rise 362.7 Parallel flow and relative permeabilities 392.8 Forcing term in the SC model 402.8.1 Schemes to incorporate the body force 422.8.2 Scheme overview 442.8.3 Theoretical analysis 442.8.4 Numerical results and discussion 462.9 Multirange pseudopotential (Inter-particle Force Model B) 552.10 Conclusions 582.11 Appendix A: Analytical solution for layered multiphase flow in a channel 582.12 Appendix B: FORTRAN code to simulate single component multiphase droplet contacting a wall as shown in Figure 2.7(c) 603 Shan and Chen-type multi-component multiphase models 713.1 Multi-component multiphase SC LBM 713.1.1 Fluid–fluid cohesion and fluid–solid adhesion 733.2 Derivation of the pressure 733.2.1 Pressure in popular papers (2D) 743.2.2 Pressure in popular papers (3D) 753.3 Determining Gc and the surface tension 763.4 Contact angle 783.4.1 Application of Young's equation to MCMP LBM 793.4.2 Contact angle measurement 793.4.3 Verification of proposed equation 803.5 Flow through capillary tubes 833.6 Layered two-phase flow in a 2D channel 853.7 Pressure or velocity boundary conditions 873.7.1 Boundary conditions for 2D simulations 873.7.2 Boundary conditions for 3D simulations 893.8 Displacement in a 3D porous medium 914 Rothman–Keller multiphase Lattice Boltzmann model 944.1 Introduction 944.2 RK color-gradient model 964.3 Theoretical analysis (Chapman–Enskog expansion) 994.3.1 Discussion of above formulae 1034.4 Layered two-phase flow in a 2D channel 1034.4.1 Cases of two fluids with identical densities 1044.4.2 Cases of two fluids with different densities 1064.5 Interfacial tension and isotropy of the RK model 1104.5.1 Interfacial tension 1104.5.2 Isotropy 1104.6 Drainage and capillary filling 1114.7 MRT RK model 1134.8 Contact angle 1144.8.1 Spurious currents 1154.9 Tests of inlet/outlet boundary conditions 1174.10 Immiscible displacements in porous media 1184.11 Appendix A 1214.12 Appendix B 1225 Free-energy-based multiphase Lattice Boltzmann model 1365.1 Swift free-energy based single-component multiphase LBM 1365.1.1 Derivation of the coefficients in the equilibrium distribution function 1385.2 Chapman–Enskog expansion 1435.3 Issue of Galilean invariance 1465.4 Phase separation 1495.5 Contact angle 1545.5.1 How to specify a desired contact angle 1545.5.2 Numerical verification 1555.6 Swift free-energy-based multi-component multiphase LBM 1585.7 Appendix 1586 Inamuro's multiphase Lattice Boltzmann model 1676.1 Introduction 1676.1.1 Inamuro's method 1676.1.2 Comment on the presentation 1696.1.3 Chapman–Enskog expansion analysis 1706.1.4 Cahn–Hilliard equation (equation for order parameter) 1736.1.5 Poisson equation 1746.2 Droplet collision 1756.3 Appendix 1787 He–Chen–Zhang multiphase Lattice Boltzmann model 1967.1 Introduction 1967.2 HCZ model 1967.3 Chapman–Enskog analysis 1997.3.1 N–S equations 1997.3.2 CH equation 2027.4 Surface tension and phase separation 2027.5 Layered two-phase flow in a channel 2047.6 Rayleigh–Taylor instability 2057.7 Contact angle 2107.8 Capillary rise 2137.9 Geometric scheme to specify the contact angle and its hysteresis 2157.9.1 Examples of droplet slipping in shear flows 2187.10 Oscillation of an initially ellipsoidal droplet 2197.11 Appendix A 2227.12 Appendix B: 2D code 2237.13 Appendix C: 3D code 2388 Axisymmetric multiphase HCZ model 2538.1 Introduction 2538.2 Methods 2538.2.1 Macroscopic governing equations 2538.2.2 Axisymmetric HCZ LBM (Premnath and Abraham 2005a) 2558.2.3 MRT version of the axisymmetric LBM (McCracken and Abraham 2005) 2568.2.4 Axisymmetric boundary conditions 2588.3 The Laplace law 2588.4 Oscillation of an initially ellipsoidal droplet 2598.5 Cylindrical liquid column break 2638.6 Droplet collision 2658.6.1 Effect of gradient and Laplacian calculation 2678.6.2 Effect of BGK and MRT 2748.7 A revised axisymmetric HCZ model (Huang et al. 2014) 2768.7.1 MRT collision 2768.7.2 Calculation of the surface tension 2778.7.3 Mass correction 2788.8 Bubble rise 2798.8.1 Numerical validation 2818.8.2 Surface-tension calculation effect 2838.8.3 Terminal bubble shape 2848.8.4 Wake behind the bubble 2848.9 Conclusion 2868.10 Appendix A: Chapman–Enskog analysis 2888.10.1 Preparation for derivation 2888.10.2 Mass conservation 2898.10.3 Momentum conservation 2898.10.4 CH equation 2919 Extensions of the HCZ model for high-density ratio two-phase flows 2929.1 Introduction 2929.2 Model I (Lee and Lin 2005) 2939.2.1 Stress and potential form of intermolecular forcing terms 2939.2.2 Model description 2949.2.3 Implementation 2979.2.4 Directional derivative 2989.2.5 Droplet splashing on a thin liquid film 2999.3 Model II (Amaya-Bower and Lee 2010) 3019.3.1 Implementation 3029.4 Model III (Lee and Liu 2010) 3049.5 Model IV 3059.6 Numerical tests for different models 3069.6.1 A drop inside a box with periodic boundary conditions 3069.6.2 Layered two-phase flows in a channel 3119.6.3 Galilean invariance 3139.7 Conclusions 3169.8 Appendix A: Analytical solutions for layered two-phase flow in a channel 3179.9 Appendix B: 2D code based on Amaya-Bower and Lee (2010) 31910 Axisymmetric high-density ratio two-phase LBMs (extension of the HCZ model) 33410.1 Introduction 33410.2 The model based on Lee and Lin (2005) 33410.2.1 The equilibrium distribution functions I 33610.2.2 The equilibrium distribution functions II 33610.2.3 Source terms 33710.2.4 Stress and potential form of intermolecular forcing terms 33710.2.5 Chapman–Enskog analysis 33810.2.6 Implementation 34010.2.7 Droplet splashing on a thin liquid film 34210.2.8 Head-on droplet collision 34210.3 Axisymmetric model based on Lee and Liu (2010) 34510.3.1 Implementation 34710.3.2 Head-on droplet collision 34810.3.3 Bubble rise 353Index 371

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Haibo Huang is an Associate Professor in the University of Science and Technology of China. He was a Courtesy Associate Professor during his stays at Florida International University. Michael C. Sukop is Professor of Hydrogeology at Florida International University in Miami and author of “Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers”. His research emphasis is on flow and transport in porous media. Xiyun Lu is a Professor of Fluid Mechanics in the University of Science and Technology of China. His research interests mainly include computational fluid dynamics, turbulence simulation and biomechanics.

• Preface xiAbout the companion website xiii1 Introduction 11.1 History of the Lattice Boltzmann method 21.2 The Lattice Boltzmann method 31.3 Multiphase LBM 61.3.1 Color-gradient model 71.3.2 Shan–Chen model 71.3.3 Free-energy model 81.3.4 Interface tracking model 91.4 Comparison of models 91.5 Units in this book and parameter conversion 111.6 Appendix: Einstein summation convention 141.6.1 Kronecker δ function 151.6.2 Lattice tensors 151.7 Use of the Fortran code in the book 162 Single-component multiphase Shan–Chen-type model 182.1 Introduction 182.1.1 "Equilibrium" velocity in the SC model 202.1.2 Inter-particle forces in the SC SCMP LBM 202.2 Typical equations of state 212.2.1 Parameters in EOS 272.3 Thermodynamic consistency 282.3.1 The SCMP LBM EOS 292.3.2 Incorporating other EOS into the SC model 312.4 Analytical surface tension 322.4.1 Inter-particle Force Model A 322.4.2 Inter-particle Force Model B 332.5 Contact angle 342.6 Capillary rise 362.7 Parallel flow and relative permeabilities 392.8 Forcing term in the SC model 402.8.1 Schemes to incorporate the body force 422.8.2 Scheme overview 442.8.3 Theoretical analysis 442.8.4 Numerical results and discussion 462.9 Multirange pseudopotential (Inter-particle Force Model B) 552.10 Conclusions 582.11 Appendix A: Analytical solution for layered multiphase flow in a channel 582.12 Appendix B: FORTRAN code to simulate single component multiphase droplet contacting a wall as shown in Figure 2.7(c) 603 Shan and Chen-type multi-component multiphase models 713.1 Multi-component multiphase SC LBM 713.1.1 Fluid–fluid cohesion and fluid–solid adhesion 733.2 Derivation of the pressure 733.2.1 Pressure in popular papers (2D) 743.2.2 Pressure in popular papers (3D) 753.3 Determining Gc and the surface tension 763.4 Contact angle 783.4.1 Application of Young's equation to MCMP LBM 793.4.2 Contact angle measurement 793.4.3 Verification of proposed equation 803.5 Flow through capillary tubes 833.6 Layered two-phase flow in a 2D channel 853.7 Pressure or velocity boundary conditions 873.7.1 Boundary conditions for 2D simulations 873.7.2 Boundary conditions for 3D simulations 893.8 Displacement in a 3D porous medium 914 Rothman–Keller multiphase Lattice Boltzmann model 944.1 Introduction 944.2 RK color-gradient model 964.3 Theoretical analysis (Chapman–Enskog expansion) 994.3.1 Discussion of above formulae 1034.4 Layered two-phase flow in a 2D channel 1034.4.1 Cases of two fluids with identical densities 1044.4.2 Cases of two fluids with different densities 1064.5 Interfacial tension and isotropy of the RK model 1104.5.1 Interfacial tension 1104.5.2 Isotropy 1104.6 Drainage and capillary filling 1114.7 MRT RK model 1134.8 Contact angle 1144.8.1 Spurious currents 1154.9 Tests of inlet/outlet boundary conditions 1174.10 Immiscible displacements in porous media 1184.11 Appendix A 1214.12 Appendix B 1225 Free-energy-based multiphase Lattice Boltzmann model 1365.1 Swift free-energy based single-component multiphase LBM 1365.1.1 Derivation of the coefficients in the equilibrium distribution function 1385.2 Chapman–Enskog expansion 1435.3 Issue of Galilean invariance 1465.4 Phase separation 1495.5 Contact angle 1545.5.1 How to specify a desired contact angle 1545.5.2 Numerical verification 1555.6 Swift free-energy-based multi-component multiphase LBM 1585.7 Appendix 1586 Inamuro's multiphase Lattice Boltzmann model 1676.1 Introduction 1676.1.1 Inamuro's method 1676.1.2 Comment on the presentation 1696.1.3 Chapman–Enskog expansion analysis 1706.1.4 Cahn–Hilliard equation (equation for order parameter) 1736.1.5 Poisson equation 1746.2 Droplet collision 1756.3 Appendix 1787 He–Chen–Zhang multiphase Lattice Boltzmann model 1967.1 Introduction 1967.2 HCZ model 1967.3 Chapman–Enskog analysis 1997.3.1 N–S equations 1997.3.2 CH equation 2027.4 Surface tension and phase separation 2027.5 Layered two-phase flow in a channel 2047.6 Rayleigh–Taylor instability 2057.7 Contact angle 2107.8 Capillary rise 2137.9 Geometric scheme to specify the contact angle and its hysteresis 2157.9.1 Examples of droplet slipping in shear flows 2187.10 Oscillation of an initially ellipsoidal droplet 2197.11 Appendix A 2227.12 Appendix B: 2D code 2237.13 Appendix C: 3D code 2388 Axisymmetric multiphase HCZ model 2538.1 Introduction 2538.2 Methods 2538.2.1 Macroscopic governing equations 2538.2.2 Axisymmetric HCZ LBM (Premnath and Abraham 2005a) 2558.2.3 MRT version of the axisymmetric LBM (McCracken and Abraham 2005) 2568.2.4 Axisymmetric boundary conditions 2588.3 The Laplace law 2588.4 Oscillation of an initially ellipsoidal droplet 2598.5 Cylindrical liquid column break 2638.6 Droplet collision 2658.6.1 Effect of gradient and Laplacian calculation 2678.6.2 Effect of BGK and MRT 2748.7 A revised axisymmetric HCZ model (Huang et al. 2014) 2768.7.1 MRT collision 2768.7.2 Calculation of the surface tension 2778.7.3 Mass correction 2788.8 Bubble rise 2798.8.1 Numerical validation 2818.8.2 Surface-tension calculation effect 2838.8.3 Terminal bubble shape 2848.8.4 Wake behind the bubble 2848.9 Conclusion 2868.10 Appendix A: Chapman–Enskog analysis 2888.10.1 Preparation for derivation 2888.10.2 Mass conservation 2898.10.3 Momentum conservation 2898.10.4 CH equation 2919 Extensions of the HCZ model for high-density ratio two-phase flows 2929.1 Introduction 2929.2 Model I (Lee and Lin 2005) 2939.2.1 Stress and potential form of intermolecular forcing terms 2939.2.2 Model description 2949.2.3 Implementation 2979.2.4 Directional derivative 2989.2.5 Droplet splashing on a thin liquid film 2999.3 Model II (Amaya-Bower and Lee 2010) 3019.3.1 Implementation 3029.4 Model III (Lee and Liu 2010) 3049.5 Model IV 3059.6 Numerical tests for different models 3069.6.1 A drop inside a box with periodic boundary conditions 3069.6.2 Layered two-phase flows in a channel 3119.6.3 Galilean invariance 3139.7 Conclusions 3169.8 Appendix A: Analytical solutions for layered two-phase flow in a channel 3179.9 Appendix B: 2D code based on Amaya-Bower and Lee (2010) 31910 Axisymmetric high-density ratio two-phase LBMs (extension of the HCZ model) 33410.1 Introduction 33410.2 The model based on Lee and Lin (2005) 33410.2.1 The equilibrium distribution functions I 33610.2.2 The equilibrium distribution functions II 33610.2.3 Source terms 33710.2.4 Stress and potential form of intermolecular forcing terms 33710.2.5 Chapman–Enskog analysis 33810.2.6 Implementation 34010.2.7 Droplet splashing on a thin liquid film 34210.2.8 Head-on droplet collision 34210.3 Axisymmetric model based on Lee and Liu (2010) 34510.3.1 Implementation 34710.3.2 Head-on droplet collision 34810.3.3 Bubble rise 353Index 371

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