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.

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:

teady 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:

:$ablacdotmathbf\{u\}=0.$

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

Irrotational flow

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

:$abla\; imesmathbf\{u\}=0.$

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

:$omega=\; abla\; imesmathbf\{u\}.$

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

:$mathbf\{u\}=\; ablamathbf\{phi\}$

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