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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).

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.

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 does not vary with time. That is if

If a fluid is incompressible the divergence of is zero:

That is, if is a solenoidal vector field.

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:

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

Thus in irrotational flow the vorticity is zero.

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.)

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.

**^**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.