### Navier-Stokes equations

**(NS equations)**
(equations that describe fluid dynamics)

The **Navier-Stokes equations** (**NS equations**)
describe the motion of fluids, relating viscosity, Newton's laws
and pressure.
One general form:

ρ(∂**v**/∂*t*+**v**·∇**v**) = -∇*p* + ∇·**T** + **f**

**v** - flow velocity.
`ρ` - fluid density.
*p* - pressure.
**T** - deviatoric component of the stress tensor.
**f** - body of forces acting on the fluid.

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.

These are central to hydrodynamicses, and magnetohydrodynamics uses a form that
adds the forces of electromagnetism including those consequent to the fluid's
flow.

(*fluid mechanics,hydrodynamics,fluid dynamics*)
**Further reading:**

https://en.wikipedia.org/wiki/Navier-Stokes_equations

**Referenced by pages:**

Darcy velocity field

general circulation model (GCM)

hydrodynamic equations

MagIC

magnetohydrodynamics (MHD)

Reynolds decomposition

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