Velocity potential
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A velocity potential is used in fluid dynamics, when a fluid occupies a simply-connected region and is irrotational. In such a case, \[\nabla \times \mathbf{u} =0,\] where \( \mathbf{u} \) denotes the flow velocity of the fluid. As a result, \( \mathbf{u} \) can be represented as the gradient of a scalar function \(\Phi\;\): \[ \mathbf{u} = \nabla \Phi\;\], \(\Phi\;\) is known as a velocity potential for \(\mathbf{u}\).
A velocity potential is not unique. If \(a\;\) is a constant then \(\Phi+a\;\) is also a velocity potential for \(\mathbf{u}\;\). Conversely, if \(\Psi\;\) is a velocity potential for \(\mathbf{u}\;\) then \(\Psi=\Phi+b\;\) for some constant \(b\;\). In other words, velocity potentials are unique up to a constant.If value of Φ satisfies Laplace equation,it indicates case of fluid flow.
Unlike a stream function, a velocity potential can exist in three-dimensional flow.
See also
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