Flow velocity
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In fluid dynamics the flow velocity, or velocity field, of a fluid is a vector field which is used to mathematically describe the motion of a fluid.
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[edit] Definition
The flow velocity of a fluid is a vector field
which gives the velocity of an element of fluid at a position and time
.
[edit] 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:
[edit] Steady flow
The flow of a fluid is said to be steady if does not vary with time. That is if
[edit] Incompressible flow
A fluid is incompressible if the divergence of is zero:
That is, if is a solenoidal vector field.
[edit] Irrotational flow
A flow is irrotational if the curl of is zero:
That is, if is an irrotational vector field.
[edit] Vorticity
The vorticity, ω, of a flow can be defined in terms of its flow velocity by
Thus in irrotational flow the vorticity is zero.
[edit] The velocity potential
If an irrotational flow occupies a simply-connected fluid region then there exists a scalar field φ such that
The scalar field φ is called the velocity potential for the flow. (See Irrotational vector field.)