Wiktionary
n. (context fluid mechanics of an inviscid irrotational flow English) A scalar potential function relating to the the flow's velocity distribution, defined as
Wikipedia
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 :
$$\mathbf{u} = \nabla \Phi\ = \frac{\partial \Phi}{\partial x} \mathbf{i} +
\frac{\partial \Phi}{\partial y} \mathbf{j} +
\frac{\partial \Phi}{\partial z} \mathbf{k} .$$
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.