# Flow velocity

Short description: Vector field which is used to mathematically describe the motion of a continuum

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

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

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

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

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

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

### Incompressible flow

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.

### Irrotational flow

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: $\displaystyle{ \Delta\Phi=0. }$

### Vorticity

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.

## The velocity potential

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.)

## Bulk velocity

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) $\displaystyle{ U }$ which is the ratio between the volume flow rate $\displaystyle{ \dot{V} }$ and the cross sectional area $\displaystyle{ A }$, given by

$\displaystyle{ u_{\rm{}av}=\frac{\dot{V}}{A} }$.