A velocity potential is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788.
It is used in continuum mechanics, when a continuum occupies a simply-connected region and is irrotational. In such a case,
∇
×
u
=
0
,
where
u
denotes the flow velocity. As a result,
u
can be represented as the gradient of a scalar function
Φ
:
u
=
∇
Φ
=
∂
Φ
∂
x
i
+
∂
Φ
∂
y
j
+
∂
Φ
∂
z
k
.
Φ
is known as a velocity potential for
u
.
A velocity potential is not unique. If
a
is a constant, or a function solely of the temporal variable, then
Φ
+
a
(
t
)
is also a velocity potential for
u
. Conversely, if
Ψ
is a velocity potential for
u
then
Ψ
=
Φ
+
b
for some constant, or a function solely of the temporal variable
b
(
t
)
. In other words, velocity potentials are unique up to a constant, or a function solely of the temporal variable.
If a velocity potential satisfies Laplace equation, the flow is incompressible ; one can check this statement by, for instance, developing
∇
×
(
∇
×
u
)
and using, thanks to the Clairaut-Schwarz's theorem, the commutation between the gradient and the laplacian operators.
Unlike a stream function, a velocity potential can exist in three-dimensional flow.
In theoretical acoustics, it is often desirable to work with the acoustic wave equation of the velocity potential
Φ
instead of pressure
p
and/or particle velocity
u
.
∇
2
Φ
−
1
c
2
∂
2
Φ
∂
t
2
=
0
Solving the wave equation for either
p
field or
u
field doesn't necessarily provide a simple answer for the other field. On the other hand, when
Φ
is solved for, not only is
u
found as given above, but
p
is also easily found – from the (linearised) Bernoulli equation for irrotational and unsteady flow – as
p
=
−
ρ
∂
∂
t
Φ
.
