Within the applied mathematical study of fluid dynamics and continuum mechanics, a velocity potential is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788.

Suppose a smooth vector field in a simple connected region represents the flow velocity of a fluid at each point. This flow field is said to be irrotational If the flow field is irrotational, then it can be also be represented as the gradient of a scalar

is known as a velocity potential for .

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 instead of pressure and/or particle velocity Solving the wave equation for either field or is solved for, not only is found as given above, but is also easily found—from the (linearised) Bernoulli equation for irrotational and unsteady flow

See also

• Vorticity

• Hamiltonian fluid mechanics

• Potential flow

• Potential flow around a circular cylinder

Notes