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Consider water flowing steadily through a long, straight pipe with a uniform cross-section.

In central regions of the pipe, where flow is irrotational, a velocity potential defines the flow, and the velocity is expressed as the gradient of this potential.

Substituting the velocity components directly verifies that the flow satisfies this irrotational condition.

The velocity potential applies to three-dimensional flows, like this pipe flow, while the stream function is specific to two-dimensional flows, resulting from mass conservation.

In incompressible flow, the continuity equation requires the divergence of the velocity vector to be zero.

By substituting the velocity potential, this condition becomes Laplace's equation, a key differential equation governing inviscid, incompressible, and irrotational flows.

In cylindrical coordinates, Laplace's equation adjusts for radial, angular, and axial variations. Solving this equation with boundary conditions at the pipe walls defines the velocity potential for the flow.

From this, the complete velocity field is determined by taking spatial derivatives, revealing the flow pattern within the pipe.