In continuum mechanics the **flow velocity** in fluid dynamics, also **macroscopic velocity**^{[1]}^{[2]} in statistical mechanics, or **drift velocity** in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the flow velocity vector is the **flow speed** and is a scalar.
It is also called **velocity field**; when evaluated along a line, it is called a **velocity profile** (as in, e.g., law of the wall).

## 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

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

### Incompressible flow

If a fluid is incompressible the divergence of is zero:

That is, if is a solenoidal vector field.

### Irrotational flow

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

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

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.)

## Bulk velocity

In many engineering applications the local flow velocity vector field is not known in every point and the only accessible velocity is the **bulk velocity** (or average flow velocity) which is the ratio between the volume flow rate and the cross sectional area , given by

where is the cross sectional area.

## See also

## References

**^**Duderstadt, James J.; Martin, William R. (1979). "Chapter 4:The derivation of continuum description from transport equations". In Wiley-Interscience Publications (ed.).*Transport theory*. New York. p. 218. ISBN 978-0471044925.**^**Freidberg, Jeffrey P. (2008). "Chapter 10:A self-consistent two-fluid model". In Cambridge University Press (ed.).*Plasma Physics and Fusion Energy*(1 ed.). Cambridge. p. 225. ISBN 978-0521733175.**^**Courant, R.; Friedrichs, K.O. (1999) [unabridged republication of the original edition of 1948].*Supersonic Flow and Shock Waves*. Applied mathematical sciences (5th ed.). Springer-Verlag New York Inc. pp. 24. ISBN 0387902325. OCLC 44071435.