Laminar Friction Factor (f = 64/Re)

This is one of the few exact results in pipe hydraulics: integrate the parabolic Hagen–Poiseuille velocity profile and the Darcy friction factor falls out as precisely 64/Re, with no empirical fitting anywhere. At Re = 1600 that is f = 0.04. Because it is exact, it also anchors the left-hand edge of every Moody diagram ever drawn — the straight line of slope −1 on log-log paper that all the roughness curves eventually peel away from.

The striking implication is that roughness is irrelevant in laminar flow. A rusty pipe and a glass pipe give identical pressure drop, because the fluid nearest the wall is not moving and the flow never sees the bumps. Above Re ≈ 2300 the assumption collapses; in the 2300–4000 transition band no correlation is reliable and prudent engineers simply design out of it, usually by choosing a diameter that puts velocity solidly into turbulent territory.