Fully developed flow ($\mathbf{u}=u(z)\hat{\mathbf{x}}$)

A fully developed parallel flow has a velocity profile independent of the streamwise coordinate. In a planar channel a representative profile is $\mathbf{u}=u(z)\hat{\mathbf{x}}$; wall conditions and translational symmetry remove cross-stream motion. The streamwise pressure gradient may remain nonzero, while the convective acceleration vanishes. Fully developed flow is a modeling branch, not a uniqueness assertion for all flows in that geometry.