In steady, incompressible flow through a long, straight pipe with a uniform cross-section, the flow in the central region (far from the pipe walls) is irrotational. This irrotational nature means that fluid particles do not rotate around their axes, and a scalar function called the velocity potential, represented by ϕ, can be used to describe their movement. In irrotational flows, the velocity field V is defined as the gradient of the velocity potential:

Since taking the curl of a gradient is always zero, defining the velocity in terms of ϕ ensures the flow has no vorticity, making it irrotational.

For incompressible flows, the continuity equation states that the divergence of the velocity field must be zero:

Substituting V into this equation, the equation becomes:

This is known as Laplace's equation. In regions where the flow is irrotational and incompressible, the velocity potential ϕ must satisfy Laplace's equation.

In the case of pipe flow, cylindrical coordinates (r, θ, z) are a natural choice for describing the flow, where r is the radial distance from the pipe centreline, θ is the angular position, and z is the axial distance along the pipe. Laplace's equation in cylindrical coordinates, accounting for radial, angular, and axial variations, is expressed as:

Finding the exact form of ϕ in the pipe requires boundary conditions. For instance, at the pipe walls, the no-slip condition would imply zero tangential velocity, while in the central irrotational region, the potential function would follow the symmetry of the pipe. Solving Laplace's equation with these conditions defines the velocity potential across the flow field.

Once ϕ is determined, the velocity components in cylindrical coordinates can be derived by taking partial derivatives of ϕ. These components are:

Radial velocity:

Angular velocity

Axial velocity

These expressions for velocity components confirm that the flow satisfies the irrotational condition, as they stem from a gradient.