# Flow velocity

In continuum mechanics the macroscopic velocity, 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

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

{{#invoke:main|main}}

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

{{#invoke:main|main}}

$\nabla \cdot \mathbf {u} =0.$ ### Irrotational flow

{{#invoke:main|main}}

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

{{#invoke:main|main}}

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

{{#invoke:main|main}} If an irrotational flow occupies a simply-connected fluid region then there exists a scalar field $\phi$ such that

$\mathbf {u} =\nabla \mathbf {\phi }$ 