## Lattice-Gas Cellular Automata: Simple Models of Complex HydrodynamicsThe text is a self-contained, comprehensive introduction to the theory of hydrodynamic lattice gases. Lattice-gas cellular automata are discrete models of fluids. Identical particles hop from site to site on a regular lattice, obeying simple conservative scattering rules when they collide. Remarkably, at a scale larger than the lattice spacing, these discrete models simulate the Navier-Stokes equations of fluid mechanics. This book addresses three important aspects of lattice gases. First, it shows how such simple idealised microscopic dynamics give rise to isotropic macroscopic hydrodynamics. Second, it details how the simplicity of the lattice gas provides for equally simple models of fluid phase separation, hydrodynamic interfaces, and multiphase flow. Lastly, it illustrates how lattice-gas models and related lattice-Boltzmann methods have been used to solve problems in applications as diverse as flow through porous media, phase separation, and interface dynamics. Many exercises and references are included. |

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### Contents

1 A simple model of fluid mechanics | 1 |

2 Two routes to hydrodynamics | 12 |

3 Inviscid twodimensional latticegas hydrodynamics | 29 |

4 Viscous twodimensional hydrodynamics | 46 |

5 Some simple threedimensional models | 61 |

6 The latticeBoltzmann method | 73 |

7 Using the Boltzmann method | 82 |

8 Miscible fluids | 91 |

15 Hydrodynamics in the Boltzmann approximation | 184 |

16 Phase separation | 203 |

17 Interfaces | 220 |

18 Complex fluids and patterns | 239 |

Tensor symmetry | 253 |

Poly topes and their symmetry group | 265 |

Classical compressible flow modeling | 271 |

Incompressible limit | 276 |

9 Immiscible lattice gases | 106 |

10 LatticeBoltzmann method for immiscible fluids | 119 |

11 Immiscible lattice gases in three dimensions | 128 |

12 Liquidgas models | 141 |

13 Flow through porous media | 151 |

14 Equilibrium statistical mechanics | 168 |

Derivation of the Gibbs distribution | 281 |

Hydrodynamic responce to force at fluid interfaces | 284 |

Answers to exercises | 288 |

Author Index | 290 |

293 | |

### Common terms and phrases

amphiphile Appendix average Boltzmann approximation Boltzmann equation Boltzmann method Boolean boundary conditions Bravais lattices bubbles calculation cellular automata Chapter coefficient collision operator collision rules computed conﬁguration conservation consider deﬁned deﬁnition density derivation described diffusion discrete discussion dynamics equa ergodic components Euler equation expression FCHC lattice Fermi-Dirac ﬁeld ﬁnd ﬁnite ﬁrst ﬁxed ﬂat ﬂow ﬂuctuations ﬂuid force function Gibbs distribution given by equation H-theorem hexagonal lattice hydrodynamic immiscible immiscible ﬂuids immiscible lattice gas incompressible interaction interface invariant isometries isotropic lattice gases lattice-Boltzmann method lattice-gas models length scale linear liquid-gas model macroscopic mass and momentum microdynamics mixture momentum ﬂux Navier-Stokes equation obtain parameter phase separation Phys physics polytope populations porous media pressure properties reﬂections rest particles Rothman Section shown in Figure signiﬁcant simulation space speciﬁc square lattice surface tension symmetry group tensor theoretical theory tion tracer line vectors velocity viscosity wetting ﬂuid