1.3 Navier Stokes Equations

It is assumed that the reader has studied elementary fluid mechanics and has been exposed to the basic notions of fluid flow. These include the role of pressure in momentum transport, conservation of mass in an incompressible, deformable fluid medium, and the origin of viscous, frictional forces. The last have just been discussed. This section provides an informal description of the Navier Stokes momentum equation for constant density, incompressible flow. Because we rely on CFD software for solutions to the equations, we will abbreviate the treatment that can be found more fully in standard texts on viscous flow (White, 1991).

The Navier Stokes equations were named for the French engineer and scientist Claude Louis Marie Henri Navier and the English mathematical physicist George Gabriel Stokes. The essential form of these equations was set forth by Navier in 1822; however, he did not properly treat the origin of viscous stress. The latter was addressed by others, in particular Poisson and Saint-Venant, but independently developed by Stokes in 1845. Stokes constructed a number of solutions to the equations of viscous flow, which confirmed their ability to describe fluid dynamical phenomena. An example is creeping flow, also called Stokes flow, which we discuss in Chapter 3. Navier is properly credited for the seminal formulation of the Navier Stokes equations and Stokes for ushering their entry into theoretical physics.

Essentially, the Navier Stokes momentum equation is an expression of Newton s law m a = F applied to an infinitesimal fluid volume. Here...