Hagen-Poiseuille Flow: Why Pipe Radius Rules Everything

Steady laminar flow in a straight pipe

Push a viscous fluid through a straight cylindrical pipe slowly enough and the flow settles into Hagen-Poiseuille flow: smooth concentric layers sliding past one another with no mixing between them, named for Gotthilf Hagen and Jean Léonard Marie Poiseuille, who independently worked out the relationship between flow rate and pressure drop in the 1840s. Balancing the pressure force pushing fluid forward against the viscous shear force resisting it at every radius gives a differential equation whose solution is a clean parabola: velocity is maximum on the centreline and falls smoothly to zero at the wall, where the no-slip condition pins the fluid in direct contact with the pipe.

v(r) = (ΔP / 4μL) · (R² - r²)

ΔP = pressure drop along the pipe    μ = dynamic viscosity
L  = pipe length                     R = pipe radius, r = distance from centre
v(0) = ΔP·R² / 4μL   (centreline, maximum)
v(R) = 0              (wall, no-slip)

Averaged over the whole cross-section, the mean velocity is exactly half the centreline value — a direct consequence of the parabola's shape — and integrating that average over the circular area gives the total volumetric flow rate, the Hagen-Poiseuille equation:

The fourth-power law

Q = (π·ΔP·R⁴) / (8μL)

double the radius  → flow rate increases 16×  (2⁴)
halve the radius   → flow rate drops to 1/16

This R⁴ dependence is the single most consequential number in the whole result. It is why a partially clogged artery or a slightly undersized pipe restricts flow so dramatically more than its percentage narrowing would suggest, why blood vessels that dilate by even 20% carry roughly twice the blood, and why plumbing and HVAC engineers oversize pipes rather than trust a narrow margin — a pipe 20% too small carries only about half the flow, not 80% of it.

When the parabola breaks down: turbulence

Hagen-Poiseuille flow is only valid while the flow stays laminar — layered and orderly. Whether it does is governed by the dimensionless Reynolds number, the ratio of inertial to viscous forces:

Re = ρ·v·D / μ      (ρ = density, v = mean velocity, D = pipe diameter, μ = viscosity)

Re < ~2300   laminar    → clean parabolic profile, Poiseuille law holds
Re > ~4000   turbulent  → chaotic eddies, flatter time-averaged profile
2300-4000     transitional, unstable, hard to predict

Below roughly Re ≈ 2300 viscosity damps out any disturbance before it can grow, and the flow is laminar. Above roughly Re ≈ 4000, inertial forces dominate, small perturbations amplify into turbulent eddies, and the smooth parabola is replaced by a much flatter, chaotic time-averaged velocity profile with strong mixing across the whole cross-section. In the turbulent regime pressure drop no longer scales linearly with flow rate as Poiseuille predicts — it scales closer to the square of velocity, and pumping costs rise sharply.

Where this shows up

The same fourth-power relationship governs three very different systems built on the same physics: blood flow in arterioles and capillaries (where the body actively controls vessel diameter via smooth muscle to modulate blood flow far more efficiently than it could by changing pressure), microfluidics chips in lab-on-a-chip devices where channel widths of tens of micrometres make viscous forces dominate completely (Reynolds numbers well under 1), and industrial piping for water, oil and gas, where engineers use the Hagen-Poiseuille and, for turbulent flow, the empirical Darcy-Weisbach equation to size pumps.

Viscosity itself is temperature-dependent — most liquids thin out as they warm, roughly following an exponential law — so a pipe network's flow rate and pumping power both shift with temperature even at constant pressure, which is why the simulation on this page lets you vary viscosity independently of pressure gradient and radius: it is the one parameter operators can change without touching the hardware.

Frequently asked questions

Try it live

Everything above runs in your browser — open Pipe Flow Profiles and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.