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.

Definition

The flow velocity of a fluid is a vector field

$mathbf\{u\}=mathbf\{u\}(mathbf\{x\},t)$

which gives the velocity of an element of fluid at a position $mathbf\{x\},$ and 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

The flow of a fluid is said to be steady if $mathbf\{u\}$ does not vary with time. That is if

$frac\{partial\; mathbf\{u\}\}\{partial\; t\}=0.$

Incompressible flow

A fluid is incompressible if the divergence of $mathbf\{u\}$ is zero:

$nablacdotmathbf\{u\}=0.$

That is, if $mathbf\{u\}$ is a solenoidal vector field.

Irrotational flow

A flow is irrotational if the curl of $mathbf\{u\}$ is zero:

$nablatimesmathbf\{u\}=0.$

That is, if $mathbf\{u\}$ is an irrotational vector field.

Vorticity

The vorticity, $omega$, of a flow can be defined in terms of its flow velocity by

$omega=nablatimesmathbf\{u\}.$

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 $phi$ such that

$mathbf\{u\}=nablamathbf\{phi\}$

The scalar field $phi$ is called the velocity potential for the flow. (See Irrotational vector field.)