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# Flow velocity

In fluid dynamics the flow velocity, or velocity field, of a fluid is a vector field which is used to mathematically describe the motion of a fluid.

## Definition

The flow velocity of a fluid is a vector field

$mathbf\left\{u\right\}=mathbf\left\{u\right\}\left(mathbf\left\{x\right\},t\right)$

which gives the velocity of an element of fluid at a position $mathbf\left\{x\right\},$ and time $t,$.

## Uses

The flow velocity of a fluid effectively describes everything about the motion of a fluid. Many physical properties of a fluid can be expressed mathematically in terms of the flow velocity. Some common examples follow:

The flow of a fluid is said to be steady if $mathbf\left\{u\right\}$ does not vary with time. That is if

$frac\left\{partial mathbf\left\{u\right\}\right\}\left\{partial t\right\}=0.$

### Incompressible flow

A fluid is incompressible if the divergence of $mathbf\left\{u\right\}$ is zero:

$nablacdotmathbf\left\{u\right\}=0.$

That is, if $mathbf\left\{u\right\}$ is a solenoidal vector field.

### Irrotational flow

A flow is irrotational if the curl of $mathbf\left\{u\right\}$ is zero:

$nablatimesmathbf\left\{u\right\}=0.$

That is, if $mathbf\left\{u\right\}$ is an irrotational vector field.

### Vorticity

The vorticity, $omega$, of a flow can be defined in terms of its flow velocity by

$omega=nablatimesmathbf\left\{u\right\}.$

Thus in irrotational flow the vorticity is zero.

## The velocity potential

If an irrotational flow occupies a simply-connected fluid region then there exists a scalar field $phi$ such that

$mathbf\left\{u\right\}=nablamathbf\left\{phi\right\}$

The scalar field $phi$ is called the velocity potential for the flow. (See Irrotational vector field.)

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