Navier-Stokes equations

In fluid dynamics, the Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes are a set of nonlinear partial differential equations that describe the flow of fluids such as liquids and gases. For example, they model weather or the movement of air in the atmosphere, ocean currents, water flow in a pipe, as well as many other fluid flow phenomena.

Although the full, unsteady Navier-Stokes equations correctly describe nearly all flows of practical interest, they are too complex for practical solution in many cases and a special "reduced" form of the full equations is often used instead — these are the Reynolds-averaged Navier-Stokes (RANS) equations. The solution of the full steady Navier-Stokes equations is sufficiently accurate alone for cases where the fluid flow is laminar. For turbulent flows the Reynolds-averaged form of the equations are most commonly used. The RANS form of the equations introduce new terms that reflect the additional modeling of the small turbulent motions.

Solution of flow equations by numerical methods is called computational fluid dynamics.

It is a famous open question whether smooth initial conditions always lead to smooth solutions for all times; a $1,000,000 prize was offered in May 2000 by the Clay Mathematics Institute for the answer to this question.