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.[1]

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

\nabla \times \mathbf{u} =0,

where  \mathbf{u} denotes the flow velocity of the fluid. As a result,  \mathbf{u} can be represented as the gradient of a scalar function \Phi\;:

 \mathbf{u} = \nabla \Phi\;,

\Phi\; is known as a velocity potential for \mathbf{u}.

A velocity potential is not unique. If a\; is a constant, or a function solely of the temporal variable, then \Phi+a(t)\; is also a velocity potential for \mathbf{u}\;. Conversely, if \Psi\; is a velocity potential for \mathbf{u}\; then \Psi=\Phi+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 \nabla \times (\nabla \times \mathbf{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.


  1. ^ Anderson John (1998). A History of Aerodynamics. Cambridge University Press. ISBN 0521669553. 

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