Velocity potential

Template:More citations needed 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, $$\mathrm{\nabla }\times 𝐮=0,$$ where Template:Math denotes the flow velocity. As a result, Template:Math can be represented as the gradient of a scalar function Template:Math: $$𝐮=\mathrm{\nabla }\phi =\frac{\partial \phi }{\partial x}𝐢+\frac{\partial \phi }{\partial y}𝐣+\frac{\partial \phi }{\partial z}𝐤.$$

Template:Math is known as a velocity potential for Template:Math.

A velocity potential is not unique. If Template:Math is a velocity potential, then Template:Math is also a velocity potential for Template:Math, where Template:Math 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.

Usage in acoustics

In theoretical acoustics, it is often desirable to work with the acoustic wave equation of the velocity potential Template:Math instead of pressure Template:Mvar and/or particle velocity Template:Math. $${\mathrm{\nabla }}^2\phi -\frac{1}{c^2}\frac{{\partial }^2\phi }{\partial t^2}=0$$ Solving the wave equation for either Template:Mvar field or Template:Math field does not necessarily provide a simple answer for the other field. On the other hand, when Template:Math is solved for, not only is Template:Math found as given above, but Template:Mvar is also easily found—from the (linearised) Bernoulli equation for irrotational and unsteady flow—as $$p=-\rho \frac{\partial \phi }{\partial t}.$$

See also

• Vorticity
• Hamiltonian fluid mechanics
• Potential flow
• Potential flow around a circular cylinder

External links

• Joukowski Transform Interactive WebApp

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