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.
The flow velocity u of a fluid is a vector field
The flow speed q is the length of the flow velocity vector
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
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