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

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

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

 \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} .

\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.

Usage in acoustics[edit]

In theoretical acoustics,[2] it is often desirable to work with the acoustic wave equation of the velocity potential \Phi\; instead of pressure p\; and/or particle velocity \mathbf{u}\;.

 \nabla ^2 \Phi - {1 \over c^2} { \partial^2 \Phi  \over  \partial t ^2 } = 0

Solving the wave equation for either p\; field or \mathbf{u}\; field doesn't necessarily provide a simple answer for the other field. On the other hand, when \Phi\; is solved for, not only is \mathbf{u}\; found as given above, but p\; is also easily found – from the (linearised) Bernoulli equation for irrotational and unsteady flow – as

 p = -\rho {\partial \over \partial t}\Phi .


  1. ^ Anderson John (1998). A History of Aerodynamics. Cambridge University Press. ISBN 0521669553. 
  2. ^ Pierce A.D. (1994). Acoustics: An Introduction to Its Physical Principles and Applications. Acoustical Society of America. ISBN 0883186128. 

See also[edit]