In continuum mechanics the macroscopic velocity, also flow velocity in fluid dynamics 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).
The flow velocity u of a fluid is a vector field
and is a scalar field.
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 does not vary with time. That is if
If a fluid is incompressible the divergence of is zero:
That is, if is a solenoidal vector field.
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:
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
In many engineering applications the flow velocity vector field is not known and the only accessible velocity is the bulk velocity (or average velocity) which can be expressed from the volume flow rate by
where is the cross sectional area.
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