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 when
If the flow field is irrotational, then it can be also be represented as the gradient of a scalar function :
is known as a velocity potential for . Velocity potentials are unique up to a constant and a function solely of the temporal variable. So if is a velocity potential, then generates the same flow field as .
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.
In theoretical acoustics, it is often desirable to work with the acoustic wave equation of the velocity potential instead of pressure and/or particle velocity .
Solving the wave equation for either field or field does not necessarily provide a simple answer for the other field. On the other hand, when is solved for, not only is found as given above, but is also easily foundâÂÂfrom the (linearised) Bernoulli equation for irrotational and unsteady flowâÂÂas