A velocity potential is used in fluid dynamics

Fluid dynamics

In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, when a fluid occupies a simply-connected region and is irrotational. In such a case,

where denotes the flow velocity

Flow velocity

In fluid dynamics the flow velocity, or velocity field, of a fluid is a vector field which is used to mathematically describe the motion of a fluid...

 of the fluid. As a result, can be represented as the gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

 of a scalar

Scalar field

In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...

 function :,
is known as a velocity potential for .

A velocity potential is not unique. If is a constant then is also a velocity potential for . Conversely, if is a velocity potential for then for some constant . In other words, velocity potentials are unique up to a constant.If value of Φ satisfies Laplace equation,it indicates case of fluid flow.

Unlike a stream function

Stream function

The stream function is defined for two-dimensional flows of various kinds. The stream function can be used to plot streamlines, which represent the trajectories of particles in a steady flow. Streamlines are perpendicular to equipotential lines...

, a velocity potential can exist in three-dimensional flow.