8 - Theoretical Background of Lattice Boltzmann Method

Abstract

The lattice Boltzmann method is a useful simulation technique for numerically solving flow problems and for systems such as a suspension of solid particles or a polymeric liquid. The method is the simplest and provides a solid foundation for application to various flow problems and enables the solution of the motion of suspended particles and the ambient flow field simultaneously. This chapter is designed to review the theoretical background of the lattice Boltzmann method. In a multicomponent system, the motion of the suspended particles or polymers must be solved together with the flow field of the solvent molecules. In a molecular simulation of a suspension composed of solid particles in a liquid, it is very difficult to treat the multibody hydrodynamic interactions among the suspended particles. Hence, it is usual to model the flow field as a simple shear flow, and under this approach only the motion of the suspended particles will be solved during the simulation. An understanding of the theoretical background is essential to employ a new boundary condition or develop a new version of the lattice Boltzmann method that can take into account the random motion of the suspended particles. For a clear, logical development, the fundamental equations for the following derivation are provided. The key equations are indicated for the successive derivation procedure such that the important equations from the key expressions can be easily derived.