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 scalar, the flow speed. 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

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

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.

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 ${\displaystyle \mathbf {u} }$ does not vary with time. That is if

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

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

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

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

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

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

That is, if ${\displaystyle \mathbf {u} }$ 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 ${\displaystyle \Phi ,}$ with ${\displaystyle \mathbf {u} =\nabla \Phi .}$ If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero: ${\displaystyle \Delta \Phi =0.}$

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

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

If the vorticity is zero, the flow is irrotational.

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.)

In many engineering applications the local flow velocity ${\displaystyle \mathbf {u} }$ vector field is not known in every point and the only accessible velocity is the bulk velocity or average flow velocity ${\displaystyle {\bar {u}}}$ (with the usual dimension of length per time), defined as the quotient between the volume flow rate ${\displaystyle {\dot {V}}}$ (with dimension of cubed length per time) and the cross sectional area ${\displaystyle A}$ (with dimension of square length):

${\displaystyle {\bar {u}}={\frac {\dot {V}}{A}}}$.