In continuum mechanics the macroscopic velocity, also flow velocity in fluid dynamics or drift velocity in electromagnetism, is a vector field which is used to mathematically describe the motion of a continuum. The length of the flow velocity vector is the flow speed and is a scalar.
The flow velocity u of a fluid is a vector field
u
=
u
(
x
,
t
)
,
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
q
=


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
u
does not vary with time. That is if
∂
u
∂
t
=
0.
If a fluid is incompressible the divergence of
u
is zero:
∇
⋅
u
=
0.
That is, if
u
is a solenoidal vector field.
A flow is irrotational if the curl of
u
is zero:
∇
×
u
=
0.
That is, if
u
is an irrotational vector field.
A flow in a simplyconnected 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.
The vorticity,
ω
, of a flow can be defined in terms of its flow velocity by
ω
=
∇
×
u
.
Thus in irrotational flow the vorticity is zero.
If an irrotational flow occupies a simplyconnected fluid region then there exists a scalar field
ϕ
such that
u
=
∇
ϕ
.
The scalar field
ϕ
is called the velocity potential for the flow. (See Irrotational vector field.)