The flow velocity of a fluid is a vector field
which gives the velocity of an element of fluid at a position and time .
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
A fluid is incompressible if 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.
The vorticity, , of a flow can be defined in terms of its flow velocity by
Thus in irrotational flow the vorticity is zero.
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