Velocity potential

A velocity potential is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788.^[1]

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

ϕ is known as a velocity potential for u.

A velocity potential is not unique. If ϕ is a velocity potential, then ϕ + f(t) is also a velocity potential for u, where f(t) is a scalar function of time and can be constant. 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,^[2] 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. $${\displaystyle \nabla ^{2}\varphi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}=0}$$ 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 $${\displaystyle p=-\rho {\frac {\partial \varphi }{\partial t}}\,.}$$

See also

• Vorticity
• Hamiltonian fluid mechanics
• Potential flow
• Potential flow around a circular cylinder