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

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

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.

Usage in acoustics

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

See also

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

Notes

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