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## Summary

In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity 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

$\mathbf {u} =\mathbf {u} (\mathbf {x} ,t),$ which gives the velocity of an element of fluid at a position $\mathbf {x} \,$ and time $t.\,$ The flow speed q is the length of the flow velocity vector

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

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

${\frac {\partial \mathbf {u} }{\partial t}}=0.$ ### Incompressible flow

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

$\nabla \cdot \mathbf {u} =0.$ That is, if $\mathbf {u}$ is a solenoidal vector field.

### Irrotational flow

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

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

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

$\omega =\nabla \times \mathbf {u} .$ 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 $\phi$ such that

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

## Bulk velocity

In many engineering applications the local flow velocity $\mathbf {u}$ vector field is not known in every point and the only accessible velocity is the bulk velocity (or average flow velocity) $U$ which is the ratio between the volume flow rate ${\dot {V}}$ and the cross sectional area $A$ , given by

$u_{\rm {{}av}}={\frac {\dot {V}}{A}}$ where $A$ is the cross sectional area.