# 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

which gives the velocity of an *element of fluid* at a position and time .

The flow speed *q* is the length of the flow velocity vector^{[3]}

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:

### Steady flow

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The flow of a fluid is said to be *steady* if does not vary with time. That is if

### Incompressible flow

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If a fluid is incompressible the divergence of is zero:

That is, if is a solenoidal vector field.

### Irrotational flow

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A flow is *irrotational* if the curl of is zero:

That is, if is an irrotational vector field.

A flow in a simply-connected domain which is irrotational can be described as a potential flow, through the use of a velocity potential with If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero:

### Vorticity

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

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 such that

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