How this instrument works
Poiseuille's law describes how a viscous fluid moves through a straight, narrow, circular pipe when a pressure difference pushes it from one end to the other. Because the fluid sticks to the pipe wall (the no-slip condition) and moves fastest along the centerline, the flow forms a parabolic velocity profile — a shape the French physician Jean Léonard Marie Poiseuille worked out experimentally in the 1840s while studying blood flow, and which Gotthilf Hagen derived independently around the same time. Integrating that parabola across the whole cross-section produces the formula: volumetric flow rate equals pressure difference times radius to the fourth power, divided by eight times viscosity times length.
The radius term is raised to the fourth power because two separate effects compound: a wider pipe has more cross-sectional area to carry fluid, and the parabolic profile inside a wider pipe also runs faster on average, since the slow region dragging near the wall matters less relative to the whole radius. Multiply those two effects together and a doubled radius carries sixteen times the flow, not twice — the same relationship that makes a partly blocked artery far more restrictive than its percentage of narrowing suggests, and why plumbers size drain pipes generously rather than by simple proportion.
The formula only holds for laminar flow — smooth, layered motion without eddies — which requires the Reynolds number to stay below roughly 2300; push the same fluid faster, or through a rougher and wider pipe, and the flow transitions to turbulence, where resistance climbs far faster than this equation predicts. It also assumes a Newtonian fluid with constant viscosity, rigid pipe walls, and flow that has fully developed away from the pipe's entrance. Blood is famously non-Newtonian at the low shear rates found in small vessels, which is why physiologists treat Poiseuille's law as a first approximation for vascular resistance rather than an exact law.
- Enter the Pipe (or vessel) radius. This value is raised to the fourth power in the formula, so a small mistake here shifts the result far more than an equal mistake anywhere else.
- Set the Pressure difference driving flow from one end of the pipe to the other.
- Enter the fluid's Dynamic viscosity, Pa·s — water at room temperature is about 0.001 Pa·s; glycerin runs closer to 1 Pa·s.
- Enter the Pipe length the fluid travels along in a straight run.
- Read the Volumetric flow rate result, computed directly from the four inputs above.
Worked example — water through a 2 mm tube
Take a tube with a 2 mm radius (0.002 m), 1 meter long, carrying water at a dynamic viscosity of 0.001 Pa·s, pushed by a pressure difference of 5 kPa (5000 Pa) between its two ends — numbers realistic for a short lab line or a large peripheral vein. Plugging in: Q = π × (0.002)⁴ × 5000 ⁄ (8 × 0.001 × 1) = π × 1.6 × 10⁻¹¹ × 5000 ⁄ 0.008 = 3.14159265359 × 10⁻⁵ cubic meters per second.
That reads as a small number because a cubic meter is a large unit for a thin stream; converted to more familiar terms it is about 0.0314 liters per second — roughly 31.4 milliliters per second, or 1.88 liters per minute, a believable trickle from a narrow tube. Widen the same tube to a 4 mm radius and, with every other input unchanged, the flow rate multiplies by exactly sixteen to 0.000503 cubic meters per second — not by four, which is the entire point of the fourth-power term and the reason a partly narrowed vessel or pipe restricts flow so severely.
Questions
What flow regime does Poiseuille's law require?
Laminar flow only — smooth, layered movement with no swirling eddies, which needs the Reynolds number to stay below roughly 2300. Above that threshold the fluid transitions to turbulence, where friction and resistance climb faster than the formula predicts, so results here stop applying to fast flows in wide, rough, or low-viscosity pipes.
Why does doubling the radius multiply flow by sixteen instead of two?
Because radius appears to the fourth power in the formula, compounding two effects: a wider pipe has more cross-sectional area, and its parabolic velocity profile also runs faster on average since the slow region near the wall matters less. Area's square relationship times the profile's own square relationship gives sixteen — verified here by doubling the radius input and watching the result change by exactly that factor.
Does pipe length affect flow the same way radius does?
No — length sits in the denominator on its own, so flow scales inversely and linearly with it. Double the pipe length and flow rate exactly halves; there is no exponent involved, which is why engineers worry far more about a pipe's diameter than its run length when trying to move more fluid.
Can this formula be used for blood flow in a real vessel?
As a working approximation, yes — physiologists use it routinely to estimate vascular resistance, and it explains why a narrowed artery restricts flow so drastically. It is not exact: blood is a non-Newtonian fluid whose effective viscosity drops at the high shear rates near a vessel wall, and real vessels are elastic rather than rigid, so the formula tends to overstate resistance in the smallest vessels.
What if the pipe's cross-section is not circular?
This exact formula only holds for a circular cross-section; a rectangular duct, an annulus, or an elliptical channel each follows its own flow relation with a different geometric constant, generally producing more resistance for the same cross-sectional area than a circular pipe would.
Why is viscosity entered in Pa·s instead of the more familiar centipoise?
Pa·s is the SI coherent unit and keeps the rest of the formula consistent without extra conversion factors. One centipoise equals 0.001 Pa·s, so water at room temperature — about 1 cP — enters here as 0.001, the same figure used in this page's worked example.