Introduction

The Navier-Stokes equations of fluid dynamics are a formulation of Newton's laws of motion for a continuous distribution of matter in the fluid state, characterized by an inability to support shear stresses. We will restrict our attention to the incompressible Navier-Stokes equations for a single component Newtonian fluid. Although they may be derived systematically from the microscopic description in terms of a Boltzmann equation, albeit with some additional fundamental assumptions, in this chapter we present a heuristic derivation designed to illustrate the elements of the physics contained in the equations.

Euler's equations for an incompressible fluid

First we consider an ideal inviscid fluid. The dependent variables in the so-called Eulerian description of fluid mechanics are the fluid density ρ(x, t), the velocity vector field u(x, t), and the pressure field ρ(x, t). Here x ∈ Rd is the spatial coordinate in a d-dimensional region of space (d typically takes values 2 or 3, with a default value of 3 in this chapter). An infinitesimal element of the fluid of volume δ V located at position x at time t has mass δm = ρ(x,t)δV and is moving with velocity u(x,t) and momentum δmu(x,t). The normal force directed into the infinitesimal volume across a face of area nda centered at x, where n is the outward directed unit vector normal to the face, is —np(x, t)δa. The pressure is the magnitude of the force per unit area, or normal stress, imposed on elements of the fluid from neighboring elements. These definitions are illustrated in Figure 1.1.