Push a fluid down a thin pipe and it moves fastest at the axis, dead-still at the wall — a perfect parabola. And the throughput obeys a merciless law: the flow scales with the fourth power of the radius, so a pipe just twice as wide carries sixteen times as much. Down the center, the terms go in, the engine solves the laminar profile, the volume comes out. The blue team builds and defends it; the red team tries to break it.
source G. Hagen (1839), Ann. Phys. Chem.; J. L. M. Poiseuille (1840), "Recherches expérimentales sur le mouvement des liquides dans les tubes de très-petits diamètres," Comptes Rendus Acad. Sci. 11, 961–967, 1041–1048 — Comptes Rendus, t.11 (Gallica). No stable single-article link; cited by author/title/year. Rendered, not quoted.
Steady, laminar, Newtonian flow down a straight circular pipe of radius R and length L, driven by pressure drop Δp, has an exact closed-form solution:
profile v(r) = (Δp / 4μL)·(R² − r²) — a parabola, max on the axis, zero at the wall (no-slip).
throughput Q = π R⁴ Δp / (8 μ L) — the fourth-power law.
Live values for the current pipe (computed, not stored):
| quantity | value |
|---|
Hagen–Poiseuille is not a fit; it is the exact solution of the-navier-stokes equations when the flow is fully developed and laminar — the nonlinear inertial term vanishes and momentum balance collapses to μ∇²v = ∂p/∂x.
It holds only while the-reynolds-number stays low (Re ≲ 2300). Above that, turbulence shreds the parabola and Q no longer follows r⁴. Each sphere is the next one's premise.
The blue team's live check: recompute the radius-doubling ratio, the mean-to-max ratio, and the no-slip value straight from the engine and compare to the known truth (16, ½, 0). If red tampers, this badge is where it shows.
Four physical inputs, each with real units. Feed them to the engine below:
| symbol | meaning | enters as |
|---|---|---|
| R | pipe radius | R⁴ (quartic) |
| Δp/L | pressure gradient | linear |
| μ | dynamic viscosity | 1/μ (inverse) |
| r | radial position 0…R | R²−r² (parabola) |
The single fact that makes the pipe special: R enters to the fourth power. Everything else is linear or inverse. That is what you feed the panel below.
Arrow length = local speed v(r); walls are no-slip (zero). Pipe half-height is drawn to scale in R.
Move any slider — Q, v_max and v_mean are computed from the closed-form laws on the spot, never looked up.
What the machine produces, proven: Q = π R⁴ Δp / (8 μ L). Because of the exponent, doubling R multiplies Q by exactly 16 (2⁴), the mean velocity is exactly half the axial maximum, and the wall speed is zero. Those three numbers — 16, ½, 0 — are the output.
The blue team's witness (left) confirms these live; the red team (right) tries to make them wrong.
Above Re ≈ 2300 the flow turns turbulent and Q scales roughly as Δp~0.57, not linearly; the parabola flattens to a blunt plug. Blood is non-Newtonian (shear-thinning) and pulsatile. In micro/nano channels the no-slip boundary itself can slip. Poiseuille is the answer to one clean question, not to pipes in general.
"Double the radius, double the flow." Cut. Flow goes as r⁴ — doubling R gives 16×, not 2×. This is the single most-missed fact about pipes.
"Poiseuille derived the law from theory." Cut, corrected. Hagen (1839) and Poiseuille (1840) found it experimentally; the r⁴ derivation from viscosity came later — Wiedemann, Hagenbach (1860), Stokes. The name honours the measurement.
"It describes blood flow in arteries." Kept, corrected. Poiseuille (a physician) studied it for capillaries, where it nearly holds; large arteries are pulsatile and non-Newtonian, outside its clean scope.
The red team's move: swap the exponent — make Q scale as r² instead of r⁴, so doubling R only quadruples the flow. The blue team's witness (window 7) is watching.
Force the r² law and the doubling ratio collapses from 16 to 4 — the witness recomputes, disagrees with the known value, and turns red. Nothing is faked; the attack is real and it is caught.