This monograph elaborates a fundamental topic of the theory of fluid dynamics which is introduced in most textbooks on the theory of flow of a viscous fluid. A knowledge of this introductory background, for which reference may be made to Batchelor (1967), will be assumed here. However, it will be helpful to summarise a little of the background wherever we need it. In particular, we begin by introducing the scope of the book by loosely defining the terms of the title.

The Navier–Stokes equations are the system of non-linear partial differential equations governing the motion of a Newtonian fluid, which may be liquid or gas. In essence, they represent the balance between the rate of change of momentum of an element of fluid and the forces on it, as does Newton's second law of motion for a particle, where the stress is linearly related to the rate of strain of the fluid. Newton himself did not understand well the nature of the forces between elemental particles in a continuum, but he did (Newton 1687, Vol. II, Section IX, Hypothesis, Proposition LI) initiate the theory of the dynamics of a uniform viscous fluid in an intuitive and imaginative way. It was many years later that the Navier–Stokes equations, as we now know them, were deduced from various physical hypotheses, and in various forms, by Navier (1827), Poisson (1831), Saint-Venant (1843) and Stokes (1845).