# 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

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

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

Main article: 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

Main article: 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

Main article: 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

Main article: Potential flow

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.)