Our objectives for this chapter are twofold. First, we review some elementary aspects of fluid mechanics. We include in that discussion a classical derivation of the Navier-Stokes equations from the conservation of mass and momentum in a continuum fluid. We then discuss the analogous conservation relations in a lattice gas. Finally, we briefly describe the derivation of hydrodynamic equations for the lattice gas, but defer our first detailed discussion of this subject to the following chapter.

Molecular dynamics versus continuum mechanics

The study of fluids typically proceeds in either of two ways. Either one begins at the microscopic scale of molecular interactions, or one assumes that at a particular macroscopic scale a fluid may be described as a smoothly varying continuum. The latter approach allows us to write conservation equations in the form of partial-differential equations. Before we do so, however, it is worthwhile to recall the basis of such a point of view.

The macroscopic description of fluids corresponds to our everyday experience of flows. Figure 2.1 shows that a flow may have several characteristic length scales li. These lengths scales may be related either to geometric properties of the flow such as channel width or the diameter of obstacles or to intrinsic properties such as the size of vortical structures. The smallest of these length scales will be called Lhydro.