# Flow velocity

In continuum mechanics the macroscopic velocity,[1][2] also flow velocity in fluid dynamics or drift velocity in electromagnetism, is a vector field which is used to mathematically describe the motion of a continuum. The length of the flow velocity vector is the flow speed and is a scalar.

## Definition

The flow velocity u of a fluid is a vector field

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

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

The flow speed q is the length of the flow velocity vector[3]

${\displaystyle q=||\mathbf {u} ||}$

and is a scalar field.

## 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:

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

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

### Incompressible flow

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If a fluid is incompressible the divergence of ${\displaystyle \mathbf {u} }$ is zero:

${\displaystyle \nabla \cdot \mathbf {u} =0.}$

### Irrotational flow

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

${\displaystyle \nabla \times \mathbf {u} =0.}$

A flow in a simply-connected domain which is irrotational can be described as a potential flow, through the use of a velocity potential ${\displaystyle \Phi ,}$ with ${\displaystyle \mathbf {u} =\nabla \Phi .}$ If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero: ${\displaystyle \Delta \Phi =0.}$

### Vorticity

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

${\displaystyle \omega =\nabla \times \mathbf {u} .}$

Thus in irrotational flow the vorticity is zero.

## The velocity potential

{{#invoke:main|main}} If an irrotational flow occupies a simply-connected fluid region then there exists a scalar field ${\displaystyle \phi }$ such that

${\displaystyle \mathbf {u} =\nabla \mathbf {\phi } }$

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

## References

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