Navier-Stokes equations are partial differential equations that describe motions of fluids (liquids and gases). In addition to airflow analysis for airplanes, they are useful for modeling weather, ocean currents and many other systems. A simplified form of these equations within a Cartesian frame of reference is as follows:
These equations establish the relations among the fluxes of the variables of interest. By solving these equations, you can describe the motion of fluid substances such as liquids and gases.
FEM: Because of the complexity of the Navier-Stokes equations, it is impossible to solve them algebraically. The techniques of Finite Element Methods (FEM) have been developed to compute approximations of solutions. FEM divides the area of interest into small related regions, makes approximations for each region, then combines the results to produce a single overall approximation to the fluid behavior. FEM techniques are also used to analyze deformation of metals and electro-magnetic fields.