Poiseuille flow

Poiseuille flow is the steady laminar motion of a Newtonian fluid through a straight cylindrical pipe driven by a pressure gradient. The velocity profile is parabolic: zero at the wall, maximum on the axis, with u(r) = (Δp/(4ηL))·(R² − r²). Integrating gives the volumetric flow rate Q = πR⁴·Δp/(8ηL).

The fourth-power dependence on radius is the headline: a 10% reduction in a blood vessel's radius produces a 34% drop in flow rate. This is why vasoconstriction is such a strong lever for blood pressure, and why calcified arteries are so much more dangerous than they seem.

Jean Poiseuille published the experimental law in 1840; Hagen had reached similar conclusions in Germany in 1839, hence the occasional 'Hagen-Poiseuille equation.' The theoretical derivation from Newton's viscosity ansatz came from Stokes later.