What equation does the velocity potential of an incompressible fluid satisfy?

The velocity potential (\(\phi \)) of an incompressible, irrotational fluid satisfies Laplace's equation, which is written as \(\nabla ^{2}\phi =0\). This equation arises because, for such flows, the continuity equation (\(\nabla \cdot \mathbf{v}=0\)) becomes a second-order partial differential equation for the velocity potential, \(\phi \), when the velocity field (\(\mathbf{v}\)) is expressed as the gradient of the velocity potential (\(\mathbf{v}=\nabla \phi \)).