Viscosity and Poiseuille flow

Real fluids resist being sheared. Slide one layer of liquid across the layer beneath it and there is internal friction opposing the motion; the measure of that resistance is viscosity. It is a distinct property from density and the two do not track each other — honey and water differ in density by a few tens of per cent but in viscosity by a factor of thousands, and mercury is thirteen times denser than water while flowing more readily than oil.

Viscosity has a consequence at the pipe wall: the fluid in contact with the wall does not move at all. Speed rises from zero there to a maximum at the centre, giving laminar flow a parabolic velocity profile rather than a uniform one. Working out the flow that results gives Poiseuille's law, and its most striking feature is the radius dependence — flow rate goes as r⁴. Halving a pipe's radius cuts the flow to one sixteenth at the same pressure. This is why a modest narrowing of an artery has such disproportionate effects, and why a slightly wider hose empties a pool so much faster.

All of that describes orderly, layered flow. Push the fluid faster and at some point the layers break down into eddies and mixing, and the flow becomes turbulent. The transition is governed by the Reynolds number, which weighs inertia against viscosity, and once past it Poiseuille's law no longer applies: resistance rises much more steeply, and the neat parabolic profile is replaced by chaotic motion that has to be handled statistically.