# Navier-Stokes equation

## Navier-Stokes equation

A partial differential equation which describes the conservation of linear momentum for a linearly viscous (newtonian), incompressible fluid flow. In vector form, this relation is written as Eq. (1),

(1)

where &rgr; is fluid density, **V**is fluid velocity,

*p*is fluid pressure,

**g**is the gravitational acceleration, μ is fluid viscosity, ∇ is the del or grad operator, and ∇

^{2}is the laplacian operator. The equation is named after its two principal developers, French engineer C. L. M. H. Navier (1823) and Irish scientist George G. Stokes (1845). When coupled with the conservation of mass relation, ∇ ·

**V**= 0, Eq. (1) can be solved for the space-time distribution of

**V**and

*p*in a given region of viscous fluid flow. Typical boundary conditions are (1) the knowledge of the velocity and pressure in the far field, and (2) the no-slip condition at solid surfaces (fluid velocity equals solid velocity).

*See*Newtonian fluid, Viscosity

Equation (1) correctly models the continuum behavior of all newtonian fluids, including the disorderly fluctuating motion known as turbulence. However, since the left-hand side is highly nonlinear, only a few score of exact solutions are known, mostly for simple geometries. The primary dimensionless parameter which governs Eq. (1) is the Reynolds number, given by Eq. (2),

(2)

where *L*is a characteristic body dimension. For small Re 1, Eq. (1) can be simplified by neglecting the left-hand side, resulting in a linear approximation called Stokes flow, or creeping flow, for which many solutions are known.

*See*Creeping flow, Reynolds number

For large Re 1, viscous effects are often confined to a thin boundary layer near solid surfaces, with the remaining flow being nearly inviscid. *See* Boundary-layer flow