velocity potential

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A velocity potential is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788.{{cite book|last=Anderson|first=John|title=A History of Aerodynamics|year=1998|publisher=Cambridge University Press|isbn=978-0521669559}}{{page needed|date=December 2017}}

It is used in continuum mechanics, when a continuum occupies a simply-connected region and is irrotational. In such a case,

$$\backslash nabla\; \backslash times\; \backslash mathbf\{u\}\; =0\; \backslash ,,$$

where {{math|u}} denotes the flow velocity. As a result, {{math|u}} can be represented as the gradient of a scalar function {{math|ϕ}}:

$$\backslash mathbf\{u\}\; =\; \backslash nabla\; \backslash varphi\backslash \; =$$

\frac{\partial \varphi}{\partial x} \mathbf{i} +

\frac{\partial \varphi}{\partial y} \mathbf{j} +

\frac{\partial \varphi}{\partial z} \mathbf{k} \,.

{{math|ϕ}} is known as a velocity potential for {{math|u}}.

A velocity potential is not unique. If {{math|ϕ}} is a velocity potential, then {{math|ϕ + f(t)}} is also a velocity potential for {{math|u}}, where {{math|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.