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 .
