Flow velocity


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.

Definition

The flow velocity u of a fluid is a vector field
which gives the velocity of an element of fluid at a position and time
The flow speed q is the length of the flow velocity vector
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:

Steady flow

The flow of a fluid is said to be steady if does not vary with time. That is if

Incompressible flow

If a fluid is incompressible the divergence of is zero:
That is, if is a solenoidal vector field.

Irrotational flow

A flow is irrotational if the curl of is zero:
That is, if 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 with If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero:

Vorticity

The vorticity,, of a flow can be defined in terms of its flow velocity by
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 such that
The scalar field is called the velocity potential for the flow.

Bulk velocity

In many engineering applications the local flow velocity vector field is not known in every point and the only accessible velocity is the bulk velocity which is the ratio between the volume flow rate by
where is the cross sectional area.