Flow velocity

The flow velocity u of a fluid is a vector field

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

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

The flow speed q is the length of the flow velocity vector^[3]

${\displaystyle q=\|\mathbf {u} \|}$ 

and is a scalar field.

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:

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

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

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

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

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

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

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

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

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

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

If the vorticity is zero, the flow is irrotational.

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

${\displaystyle \mathbf {u} =\nabla \mathbf {\phi } .}$ 

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

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 ${\displaystyle {\bar {u}}}$  (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):

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