Velocity potential

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, $\nabla \times \mathbf {u} =0\,,$ where $u$ denotes the flow velocity. As a result, $u$ can be represented as the gradient of a scalar function $ϕ$ : $\mathbf {u} =\nabla \varphi \ ={\frac {\partial \varphi }{\partial x}}\mathbf {i} +{\frac {\partial \varphi }{\partial y}}\mathbf {j} +{\frac {\partial \varphi }{\partial z}}\mathbf {k} \,.$

$ϕ$ is known as a velocity potential for $u$ .

A velocity potential is not unique. If $ϕ$ is a velocity potential, then $ϕ + f (t)$ is also a velocity potential for $u$ , where $f (t)$ is a scalar function of time and can be constant. Velocity potentials are unique up to a constant, or a function solely of the temporal variable.

The Laplacian of a velocity potential is equal to the divergence of the corresponding flow. Hence if a velocity potential satisfies Laplace equation, the flow is incompressible.

Unlike a stream function, a velocity potential can exist in three-dimensional flow.