Flow velocity

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 ${\displaystyle \mathbf{u}}$ does not vary with time. That is if

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

Incompressible flow

If a fluid is incompressible the divergence of ${\displaystyle \mathbf{u}}$ is zero:

${\displaystyle \mathrm{\nabla }\cdot \mathbf{u}=0.}$

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

Irrotational flow

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

${\displaystyle \mathrm{\nabla }\times \mathbf{u}=0.}$

That is, if ${\displaystyle \mathbf{u}}$ 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 ${\displaystyle \mathrm{\Phi },}$ with ${\displaystyle \mathbf{u}=\mathrm{\nabla }\mathrm{\Phi }.}$ If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero: ${\displaystyle \mathrm{\Delta }\mathrm{\Phi }=0.}$

Vorticity

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

${\displaystyle \omega =\mathrm{\nabla }\times \mathbf{u}.}$

If the vorticity is zero, the flow is irrotational.

The velocity potential

If an irrotational flow occupies a simply-connected fluid region then there exists a scalar field ${\displaystyle \varphi }$ such that

${\displaystyle \mathbf{u}=\mathrm{\nabla }\varphi .}$

The scalar field ${\displaystyle \varphi }$ is called the velocity potential for the flow. (See Irrotational vector field.)

Bulk velocity

In many engineering applications the local flow velocity ${\displaystyle \mathbf{u}}$ vector field is not known in every point and the only accessible velocity is the bulk velocity or average flow velocity ${\displaystyle \overline{u}}$ (with the usual dimension of length per time), defined as the quotient between the volume flow rate ${\displaystyle \dot{V}}$ (with dimension of cubed length per time) and the cross sectional area ${\displaystyle A}$ (with dimension of square length):

${\displaystyle \overline{u}=\frac{\dot{V}}{A}}$.