Velocity potential

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 $\mathbf {u}$ in a simple connected region represents the flow velocity of a fluid at each point. This flow field is said to be irrotational when $$\nabla \times \mathbf {u} =0\,.$$ If the flow field is irrotational, then it can be also be represented as the gradient of a scalar function $\phi$: $$\mathbf {u} =\nabla \phi \ ={\frac {\partial \phi }{\partial x}}\mathbf {i} +{\frac {\partial \phi }{\partial y}}\mathbf {j} +{\frac {\partial \phi }{\partial z}}\mathbf {k} \,.$$

$\phi$ is known as a velocity potential for u. Velocity potentials are unique up to a constant and a function solely of the temporal variable. So if $\phi (x,y,z)$ is a velocity potential, then $\phi (x,y,z)+f(t)+C$ generates the same flow field as $\phi (x,y,z)$.

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.