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# Navier-Stokes equations

In fluid mechanics, the Navier-Stokes equations are nonlinear partial differential equations that describe the flow of fluids. For example: they govern the movement of air in the atmosphere, ocean currents[?], water flow in a pipe, as well as many other fluid flow phenomena.

$-\nabla p + \mu \left( \nabla^2 \mathbf{u} + {1 \over 3} \nabla (\nabla \cdot \mathbf{u} ) \right) + \rho \mathbf{u} = \rho \left( { \partial\mathbf{u} \over \partial t } + \mathbf{u} \cdot \nabla \mathbf{u} \right)$

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 stream function. Solution depends on the fluid properties viscosity and density and on the boundary conditions of the domain of study. For a derivation of the Navier-Stokes equation, see Further Reading below.

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[?], CFX[?] 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 prize was offered in May 2000 by the Clay Mathematics Institute for the answer to this question.

1. Clay Mathematics Institute: Navier Stokes equation prize, http://claymath.org/prizeproblems/navierstokes.htm
2. Derivation of the Navier-Stokes equation: http://www.allstar.fiu.edu/aero/Flow2.htm

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