Definition:Navier-Stokes Equations

Definition

The Navier-Stokes equations are a set of partial differential equations that model the flow of fluid in space.

Let $F$ be an ideal incompressible fluid.

Let $\map {\mathbf v} {\mathbf x, t}$ be the velocity of $F$ at point $\mathbf x$ and time $t$.

Let $\map p {\mathbf x, t}$ be the pressure of $F$ at point $\mathbf x$ and time $t$.

The Navier-Stokes equations are the three components of the vector equations:

$\rho \paren {\dfrac {\partial \mathbf v} {\partial t} + \paren {\mathbf v \cdot \nabla} \mathbf v} = -\nabla p + \nu \nabla^2 \mathbf v + \mathbf f$

where:

$\rho$ denotes the density of $F$

$\nu$ denotes the viscosity of $F$

$\mathbf f$ denotes the external force on $F$

$\nabla$ denotes the del operator.

while:

$\nabla \cdot \mathbf v = 0$

represents the incompressibility of $F$.

Also see

• Results about the Navier-Stokes equations can be found here.

Source of Name

This entry was named for Claude-Louis Navier and George Gabriel Stokes.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Navier-Stokes equation