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 1788.¹

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

${\displaystyle \phi }$ 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 ${\displaystyle \phi (x,y,z)}$ is a velocity potential, then ${\displaystyle \phi (x,y,z)+f(t)+C}$ generates the same flow field as ${\displaystyle \phi (x,y,z)}$.

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.

In theoretical acoustics,² it is often desirable to work with the acoustic wave equation of the velocity potential ${\displaystyle \phi }$ instead of pressure p and/or particle velocity u. $${\displaystyle \nabla ^{2}\phi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\phi }{\partial t^{2}}}=0}$$ 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 ${\displaystyle \phi }$ 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 $${\displaystyle p=-\rho {\frac {\partial \phi }{\partial t}}\,.}$$