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This book introduces readers to the lattice Boltzmann method (LBM) for solving transport phenomena – flow, heat and mass transfer – in a systematic way. Providing explanatory computer codes throughout the book, the author guides readers through many practical examples, such as:

• flow in isothermal and non-isothermal lid-driven cavities;

• flow over obstacles;

• forced flow through a heated channel;

• conjugate forced convection; and

• natural convection.

Diffusion and advection–diffusion equations are discussed, together with applications and examples, and complete computer codes accompany the sections on single and multi-relaxation-time methods. The codes are written in MatLab. However, the codes are written in a way that can be easily converted to other languages, such as FORTRANm Python, Julia, etc. The codes can also be extended with little effort to multi-phase and multi-physics, provided the physics of the respective problem are known.

The second edition of this book adds new chapters, and includes new theory and applications. It discusses a wealth of practical examples, and explains LBM in connection with various engineering topics, especially the transport of mass, momentum, energy and molecular species.

This book offers a useful and easy-to-follow guide for readers with some prior experience with advanced mathematics and physics, and will be of interest to all researchers and other readers who wish to learn how to apply LBM to engineering and industrial problems. It can also be used as a textbook for advanced undergraduate or graduate courses on computational transport phenomena

### Chapter 1. Introduction and Kinetics of Particles

Abstract
There are two main approaches to solving the transport equations (heat, mass, and momentum) computationally: continuous and discrete. In the continuous approach, ordinary or partial differential equations can be obtained by applying conservation of energy, mass, and momentum with an infinitesimal control volume. Since it is difficult to solve the governing differential equations for many reasons (nonlinearity, complex boundary conditions, complex geometry, etc.), one uses finite difference, finite volume, and finite element methods, among others, to convert the governing differential equations with a given boundary and initial conditions to a system of algebraic equations. Those equations can be solved iteratively until convergence is ensured. Let us discuss the procedure in more detail for a given problem in which the governing equations need to be identified (mainly partial differential equations). This step is called mathematical modeling, which depends on the physics of the problem (and perhaps on the chemistry as well). The next step is to discretize the domain into finite volumes, grids, or elements, depending on the method of the solution. We can consider this step as assigning to each of the finite volumes or nodes or elements a collection of particles (a large number, on the order of $$10 ^{16}$$). The scale is macroscopic. The velocity, pressure, and temperature of all the particles are represented by a nodal value, or averaged over a finite volume, or simply assumed to vary linearly or bilinearly from one node to another. The phenomenological properties such as viscosity, thermal conductivity, and heat capacity are in general known parameters (input parameters, except for inverse problems). For inverse problems, one or more thermophysical properties may be unknown.

### Chapter 2. The Boltzmann Equation

Abstract
Ludwig Eduard Boltzmann (1844–1906), the Austrian physicist whose greatest achievement was in the development of statistical mechanics, explains and predicts how the properties of atoms and molecules (microscopic properties) determine the phenomenological (macroscopic) properties of matter such as viscosity, thermal conductivity, and the diffusion coefficient.

### Chapter 3. Similarities and Scaling

Abstract
In the previous chapters, the theory of LBM applications was discussed. However, before discussing the applications, it is important to create a link between the lattice domain (moment space) and the physical domain. Such a link can be established via similarities (nondimensionalizing the governing equations). It is very common in the thermal sciences (fluid mechanics and heat transfer) to use geometric and dynamic similarities for building a prototype and reducing data. In most engineering practices, the engineers build a model, either small or large depending on the problem, before building a full-sized the prototype. For instance, before building a full-sized airplane or automobile, tests are done on a model, computationally and experimentally.

### Chapter 4. Boundary Conditions

Abstract
Applying boundary conditions to different problems using LBM requires attention because it is different from the application of a boundary condition as in the classical CFD method. However, this chapter may be revisited when one is applying LBM to specific problems in the following chapters.

### Chapter 5. The Diffusion Equation

Abstract
The one-dimensional diffusion equation can be written as
\begin{aligned} \frac{\partial \phi }{\partial t}=\alpha \frac{\partial ^{2}\phi }{\partial x^{2}}\,. \end{aligned}
The dependent variable $$\phi$$ (such as temperature, species, momentum) diffuses in an infinite medium in both directions (to the left and right, $$x^{+}$$ and $$x^{-}$$) without any preference due to molecular activity. On the macroscopic scale, the rate of diffusion depends on the parameter $$\alpha$$, where $$\alpha$$ stands for the thermal diffusion coefficient, mass diffusion coefficient, or kinematics viscosity, for energy, species, and momentum diffusion, respectively. The diffusion process becomes faster as the parameter $$\alpha$$ increases. An order of magnitude analysis of the above equation yields
\begin{aligned} \frac{1}{\tau }\approx \alpha \frac{1}{\ell ^{2}}\,, \end{aligned}
where $$\tau$$ and l are time and length scales, respectively.

### Chapter 6. The Laplace, Poisson, and Biharmonic Equations

Abstract
The Poisson and Laplace equations arise in many engineering applications, such as the potential theory of hydrodynamics and electromagnetism. Also, in solving problems in incompressible flow, the pressure equation (Poisson equation) needs to be solved in updating the Navier–Stokes solver. In solid mechanics, the Poisson equation arises in the analysis of small transverse deflections of a tensed membrane subjected to a transverse load. If the membrane is replaced by a linearly elastic plate with bending stiffness, the transverse deflections must obey the inhomogeneous biharmonic equation. The biharmonic equation arises also in the analysis of two-dimensional elasticity problems formulated in terms of the Airy stress function. In the following sections, the solution of the above-mentioned equations by the lattice Boltzmann method (LBM) will be undertaken, and the results will be compared with available analytic solutions or with conventional numerical methods predictions.

Abstract
In this chapter, the physics of advection and advection–diffusion will be explained. The lattice Boltzmann method will be discussed for solving different advection–diffusion problems for one- and two-dimensional cases. Extending the method to three-dimensional problems is straightforward.

### Chapter 8. Isothermal Incompressible Fluid Flow

Abstract
In this chapter, fluid flow problems will be explained. The lattice Boltzmann method for solving various isothermal two-dimensional fluid flow problems will be discussed. The implementation of different boundary and flow conditions will be detailed. Extending the method to three-dimensional problems is straightforward.

### Chapter 9. Nonisothermal Incompressible Fluid Flow

Abstract
In Chap. 8, isothermal fluid flow problems were discussed. In many applications, heat and/or mass transfer is associated with flows and convection. In this chapter, two kinds of nonisothermal flows will be discussed, namely forced and natural convection. In forced convection, the energy equation can be solved after one has obtained the flow field; i.e., the momentum equation is not coupled with the energy equation. In natural convection, the momentum equation is coupled with the energy equation. Hence both equations need to be solved simultaneously.

### Chapter 10. Multi-Relaxation Schemes

Abstract
The single-relaxation scheme has been discussed extensively, and many problems were solved in the pervious chapters. There is a claim that multi-relaxation schemes offer greater stability and accuracy than the single-relaxation scheme. This chapter is devoted to explaining the multi-relaxation-time scheme (MRT).

### Chapter 11. Complex Flows

Abstract
Many real-life engineering problems are more complex than what has been described in the previous chapters. However, the materials covered in those chapters are building blocks for dealing with more complex flows. For instance, boiling and condensation take place in many industrial and power plant systems. Flows of oil–water–solid are very common in oil extraction and oil sand processes. And water treatment by macro- and microbubbles, combustion in furnaces, incineration of waste materials, are a few examples worthy of mention. The question is how to model and solve such problems. The natural way is to incorporate complex physics in the source term. In fact, LBM is more powerful for dealing with multiphase and multicomponent flows than the Navier–Stokes equation of continuum mechanics.