The Navier-Stokes equations (NS equations) describe the motion of fluids, relating viscosity, Newtons laws and pressure.
One general form:
ρ(∂v/∂t+v·∇v) = -∇p + ∇·T + f
No universally-applicable analytic solution is known so solutions are generally found through computation. For incompressible fluids, the constant density simplifies the equations somewhat.
The pluralization ("Navier-Stokes equations") may refer to the inclusion of an associated continuity equation, and possibly a form of the equation based upon energy conservation, in addition to the usual form based upon momentum conservation. Or to the fact that momentum conservation form is often broken down into three equations corresponding to the three dimensions. There are also variants of NS equations for compressible versus incompressible fluids.
They are central to hydrodynamicses, and magnetohydrodynamics uses a form that adds the forces of electromagnetism.