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

Contents 1 Definition 2 Uses 2.1 Steady flow 2.2 Incompressible flow 2.3 Irrotational flow 2.4 Vorticity 3 The velocity potential 4 Notes and references

Definition

The flow velocity u of a fluid is a vector field

which gives the velocity of an element of fluid at a position and time .

The flow speed q is the length of the flow velocity vector^[1]

and is a scalar field.

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

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

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

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.)

Notes and references