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Velocity potential

Scalar potential used in fluid dynamics

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Source updated February 4, 2026

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.

01Usage in acoustics

In theoretical acoustics, it is often desirable to work with the acoustic wave equation of the velocity potential $\phi$ 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 $\phi$ 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 $p=-\rho {\frac {\partial \phi }{\partial t}}\,.$

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Velocity potential

01Usage in acoustics

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Related topics

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