Flow velocity

Definition

The flow velocity u of a fluid is a vector field

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

The flow speed q is the length of the flow velocity vector

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 u does not vary with time. That is if

Incompressible flow

If a fluid is incompressible the divergence of u is zero:

That is, if u is a solenoidal vector field.

Irrotational flow

A flow is irrotational if the curl of u is zero:

That is, if u 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 Φ , with u = ∇ Φ . If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero: Δ Φ = 0.

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.)