Velocity potential

A velocity potential is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788.

It is used in continuum mechanics, when a continuum occupies a simply-connected region and is irrotational. In such a case, $$\nabla \times \mathbf{u} =0 \,,$$ where $u$ denotes the flow velocity. As a result, $u$ can be represented as the gradient of a scalar function $Φ$: $$ \mathbf{u} = \nabla \Phi\ = \frac{\partial \Phi}{\partial x} \mathbf{i} + \frac{\partial \Phi}{\partial y} \mathbf{j} + \frac{\partial \Phi}{\partial z} \mathbf{k} \,.$$

$Φ$ is known as a velocity potential for $u$.

A velocity potential is not unique. If $Φ$ is a velocity potential, then $Φ + a(t)$ is also a velocity potential for $u$, where $a(t)$ is a scalar function of time and can be constant. In other words, velocity potentials are unique up to a constant, or a function solely of the temporal variable.

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 of the velocity potential $Φ$ instead of pressure $p$ and/or particle velocity $u$. $$ \nabla ^2 \Phi - \frac{1}{c^2} \frac{ \partial^2 \Phi }{ \partial t ^2 } = 0 $$ Solving the wave equation for either $p$ field or $u$ field does not necessarily provide a simple answer for the other field. On the other hand, when $Φ$ is solved for, not only is $u$ found as given above, but $p$ is also easily found—from the (linearised) Bernoulli equation for irrotational and unsteady flow—as $$ p = -\rho \frac{\partial\Phi}{\partial t} \,.$$