The main goal of this article is to give a detailed and self-contained description of solution methods for the Navier-Stokes equations modeling the unsteady flow of incompressible Newtonian viscous fluids. We shall privilegiate here the velocity-pressure formulation of the Navier-Stokes equations since it involves "simple" physical quantities {velocity and pressure are often called primitive variables) instead of more
subtle ones such as vorticity and stream functions; another advantage of the primitive variables approach is that it facilitates, in general, the formulation and treatment of the boundary conditions.