Velocity potential is used in fluid dynamics, when a fluid occupies a simply-connected region – no sources or sinks - and is irrotational, so the velocity fieldhas zero curl:
As a result,can be represented as the gradient of a scalar function:
- in Cartesian coordinates. Given the velocity field we can find the velocity potential by integration.
The velocity potential is not unique. Ifis a constant thenis also a velocity potential forConversely, ifis a velocity potential forthenfor some constantIn other words, velocity potentials are unique up to a constant.
Example : Supposeshown below:
Example:Find the velocity potential.
These are all equal. Put the first and second equal. Thenare both functions ofonly and both are equal towhich is a function ofandsoandwhere is an arbitrary constant.