Flow velocity

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 the fluid.

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Definition

The flow velocity of a fluid is a vector field

which gives the velocity of an element of fluid at a point at a time t.

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

Main article: Steady flow

The flow of a fluid is said to be steady if does not vary with time. That is if

Incompressible flow

Main article: Incompressible flow

A fluid is incompressible if the divergence of is zero:

That is, if is a solenoidal vector field.

Irrotational flow

A flow is irrotational if the curl of is zero:

That is, if is an irrotational vector field.

Vorticity

Main article: 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.

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.)