The Navier Stokes Equations are partial differential equations that describe the flow of fluids. They govern the flight of airplanes, the air movements in the atmosphere, water flow in a pipe, as well as many other similar phenomena.
The equations are the result of mass and momentum balances to an infinitesimal control volume. The variables to be solved are the velocity components and pressure. The equations can be converted to equations for the secondary variables vorticity and *****. Solution depends on the fluid property viscosity and density and on the boundary conditions of the domain of study.
Solution of flow equations by numerical methods is called computational fluid dynamics. The solution of laminar complex flows is usually made by numerical methods. The solution of turbulent flows usually requires the modelization of the smaller details of the flow. The most used numerical methods are: finite differences, finite element method and finite volumes. There are several commercial software packages to solve the Navier Stokes Equations like Phoenics and Fluent.There is hope, that some problems of this equation can be solved with the help of solution method for flows of any macrostructure.
It is a famous open question whether smooth initial conditions always lead to smooth solutions for all times; a $1,000,000 price has been offered for the answer to this question in 2000.
- Clay Mathematics Institute: Navier Stokes equation prize, http://claymath.org/prizeproblems/navierstokes.htm