One of the greatest virtues of lattice-gas models and lattice-Boltzmann methods is the ease with which they allow one to include microscopic complexity in a model of a fluid. Throughout much of this book we have already considered models of immiscible two-fluid mixtures, the collision rules of which include interactions between particles at neighboring sites. In this chapter, we consider some of the ways in which these ideas may be generalized to create fluids of even greater complexity.

We will cover a fair bit of ground. The models range from toys for the study of pattern formation to methods for the simulation of multi-phase flow. Aside from their intrinsic interest as models of complex fluids, these models also serve to indicate how collision rules may be designed to introduce other kinds of microscopic physics into momentum-conserving hydrodynamic models.

We proceed roughly in the order of increasing complexity.

Stripes and bubbles

We first describe a model that produces some fascinating patterns. Although it conserves momentum, our discussion here is motivated less by fluid mechanics than by pattern formation itself.

The model is a nearly trivial extension of the 2D immiscible lattice gas (ILG) of Chapter 9. In the ILG, red particles and blue particles interact in a way that results in a kind of short-range attraction between particles of the same color. We now introduce a competing long-range repulsion.

We employ the same notation as we used in Section 9.1.