Blood Flow & Vessel Mechanics: Poiseuille's Law, Shear Stress and Stenosis

Poiseuille's law: flow from a pressure drop

Blood in a straight, unbranched vessel behaves, to a good first approximation, like a Newtonian fluid in laminar flow through a rigid pipe. Jean Léonard Marie Poiseuille worked out the relationship experimentally in the 1840s (he was a physician studying blood pressure, not an engineer): the volumetric flow rate Q through a cylindrical tube depends on the fourth power of the radius.

Q = (π · ΔP · r⁴) / (8 · η · L)

  Q  = volumetric flow rate
  ΔP = pressure drop along the vessel
  r  = vessel radius
  η  = dynamic viscosity of blood
  L  = vessel length

The r⁴ term is the whole story of vascular disease. Halve a vessel's radius and, all else equal, flow drops to 1/16th unless the pressure gradient rises to compensate — which is exactly what a stenosed artery forces the heart to do. The velocity profile across the vessel is a parabola: zero at the wall (the no-slip condition) and maximum at the centreline, at twice the mean velocity.

Wall shear stress and why the endothelium cares

The quantity that matters biologically is not the flow itself but the wall shear stress (WSS) — the tangential drag the moving blood exerts on the vessel lining. For steady Poiseuille flow it is τ_w = 4ηQ / (πr³), so it scales with r⁻³: a mild narrowing raises local shear sharply even before flow itself drops. The endothelial cells lining every vessel sense this shear through mechanoreceptors and respond by releasing nitric oxide, which relaxes the smooth muscle and dilates the vessel — a feedback loop called flow-mediated dilation that keeps WSS within a narrow physiological band (roughly 1–2 Pa in large arteries).

Chronically low or oscillatory WSS, which occurs at bends and branch points where flow separates and recirculates, is pro-atherogenic: it downregulates the same nitric-oxide pathway and promotes the inflammatory changes that seed atherosclerotic plaque. High, uniform shear (as in a straight arterial segment) is broadly protective. This is why plaques cluster at bifurcations — the carotid sinus, coronary branch points — rather than uniformly along a vessel.

Stenosis: why turbulence appears where the vessel narrows

Poiseuille's law assumes laminar flow, which holds while the Reynolds number Re = ρvD/η stays below roughly 2000–2300 for pipe flow. Blood in most vessels sits comfortably under that threshold. But a stenosis locally shrinks the diameter while conservation of mass forces the velocity up in direct proportion, so Re can jump past the transition point exactly inside and just downstream of the narrowing, where the jet decelerates and separates from the wall. The result is an audible bruit and a measurable pressure loss beyond what Poiseuille's law alone predicts — clinically captured by the extended Young formula, which adds a viscous (linear in velocity) and a separation-loss (quadratic in velocity) term on top of the Poiseuille term.

ΔP_stenosis ≈ Kᵥ · (η / D) · v         [viscous, Poiseuille-like]
            + K_t · ρ · (A0/As - 1)² · v²   [turbulent, expansion loss]

  A0 = normal cross-section, As = stenosed cross-section

That quadratic term is why a stenosis feels almost harmless until it crosses roughly 50-70% area reduction and then decompensates quickly — the pressure drop needed to maintain resting flow grows nonlinearly, and by the time the vessel is critically narrowed, even modest increases in demand (exercise) cannot be met without symptoms.

Why the pulse matters: Womersley flow

Real blood flow is not steady — it is driven by a pulsatile pressure wave from the heart, and Poiseuille's parabola only forms after the flow has had time to diffuse viscously across the vessel. John Womersley's 1955 analysis introduces a dimensionless number α = r√(ωρ/η) comparing the vessel radius to the thickness of the oscillatory viscous boundary layer. In small arterioles α is small and the flow is quasi-Poiseuille at every instant; in the aorta α is large (around 15-20) and the velocity profile becomes flat and even reverses near the wall while the centre is still moving forward — a signature seen on real Doppler ultrasound traces that a naive steady-flow model completely misses.

What this simulation shows

The model on this page combines these pieces: a pulsatile pressure gradient drives Poiseuille flow through an adjustable vessel radius, a stenosis constriction is applied as a local area reduction with the extended Young pressure-loss term, and wall shear stress is recomputed every frame from the local velocity profile. Push the stenosis far enough and the model flags the local Reynolds number crossing into the turbulent regime, exactly where real diagnostic ultrasound looks for spectral broadening.

Frequently asked questions

Try it live

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