Introduction to Lattice Boltzmann I

In these two lectures, I will introduce the Lattice Boltzmann algorithm, which is used to simulate the Navier-Stokes equations of simple and complex fluids. I will first present the method, then we will see how one can recover the Navier-Stokes equation in the case of a simple Newtonian fluid (with variable density), by means of a Chapman-Enskog expansion. This expansion will impose some constraints on the collision operator which is used in the lattice Boltzmann equation. We will then discuss how to choose in practice a collision operator which satisfies these constraints and with which one can in principle make calculations. If time permits, we will sketch how boundary conditions can be implemented and how it is possible to generalise the basic algorithm to study complex fluids, such as binary fluids or liquid crystals.