Velocity potential

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

ϕ is known as a velocity potential for u.

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

Usage in acoustics

In theoretical acoustics, it is often desirable to work with the acoustic wave equation of the velocity potential ϕ instead of pressure Template:Mvar and/or particle velocity u. $${\mathrm{\nabla }}^2\phi -\frac{1}{c^2}\frac{{\partial }^2\phi }{\partial t^2}=0$$ Solving the wave equation for either Template:Mvar field or u field does not necessarily provide a simple answer for the other field. On the other hand, when ϕ is solved for, not only is u 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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