Velocity potential

A velocity potential is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788.^[1]

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 $Φ$ : $\mathbf {u} =\nabla \Phi \ ={\frac {\partial \Phi }{\partial x}}\mathbf {i} +{\frac {\partial \Phi }{\partial y}}\mathbf {j} +{\frac {\partial \Phi }{\partial z}}\mathbf {k} \,.$

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

A velocity potential is not unique. If $Φ$ is a velocity potential, then $Φ + a (t)$ is also a velocity potential for $u$ , where $a (t)$ is a scalar function of time and can be constant. In other words, 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.

^ Anderson, John (1998). A History of Aerodynamics. Cambridge University Press. ISBN 978-0521669559.^{[page needed]}

[1] Anderson, John (1998). A History of Aerodynamics. Cambridge University Press. ISBN 978-0521669559.^{[page needed]}

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