The Navier-Stokes equations are nonlinear partial differential equations governing the velocity and pressure of a viscous fluid. For an incompressible Newtonian fluid with constant density, they can be written
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is the fluid velocity, 
is the pressure, 
is the constant fluid density, 
is the kinematic viscosity, and 
is the body force per unit mass. The first equation expresses
conservation of momentum, while the second is the incompressibility condition.
The nonlinear term 
is the convective part of the convective
derivative. Setting 
and 
gives Euler's
equations of inviscid motion. Taking the divergence of the momentum equation
and using incompressibility gives Poisson's equation
 |
(3)
|
so the pressure is coupled to the velocity field. A particular flow problem also requires initial data and boundary conditions.
In three spatial dimensions, it remains unknown whether every smooth divergence-free initial velocity with suitable decay gives a globally smooth solution. This existence and smoothness question is one of the Millennium Prize Problems (Fefferman 2000).
The Navier-Stokes equations appear in Big Weld's office in the 2005 animated film Robots.
