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 (as in, e.g., law of the wall).
Definition.
The flow velocity u of a fluid is a vector field
which gives the velocity of an "element of fluid" at a position formula_2 and time formula_3
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 formula_5 does not vary with time. That is if
Incompressible flow.
If a fluid is incompressible the divergence of formula_7 is zero:
That is, if formula_7 is a solenoidal vector field.
Irrotational flow.
A flow is "irrotational" if the curl of formula_7 is zero:
That is, if formula_7 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 formula_13 with formula_14 If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero: formula_15
Vorticity.
The "vorticity", formula_16, 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 formula_18 such that
The scalar field formula_20 is called the velocity potential for the flow. (See Irrotational vector field.)
Bulk velocity.
In many engineering applications the local flow velocity formula_5 vector field is not known in every point and the only accessible velocity is the bulk velocity (or average flow velocity) formula_22 which is the ratio between the volume flow rate formula_23 and the cross sectional area formula_24, given by
where formula_24 is the cross sectional area.