Fluid · Experiment

Poiseuille Flow

Push a viscous fluid steadily through a pipe and it does not slide as a block — it flows fastest down the middle and clings to the walls, tracing a perfect parabola. The volume delivered scales as the fourth power of the radius, the most consequential number in plumbing and physiology.

flow tracersparabolic velocity profilepipe wall (v = 0, no-slip)

Controls

Centre-line speed vmax—
Mean speed v̄ = vmax/2—
Flow rate Q = πΔPR⁴/8ηL—
Wall shear stress τw—
Reynolds number Re—
Fastest in the middle, zero at the walls — and the flow rate scales as R⁴.
About this experiment ›

What you are looking at

Steady, smooth (laminar) flow of a viscous fluid down a straight pipe, driven by a pressure difference ΔP between the ends. The tracer dots move at the local fluid speed: quick along the axis, frozen against the wall. Their envelope is the gold velocity profile — and it is exactly a parabola.

Why a parabola

Viscosity means each cylindrical shell of fluid drags on its neighbour. Balancing the pressure force on a cylinder of radius r against the viscous shear on its surface, and applying the no-slip condition (fluid touching the wall doesn't move), gives
v(r) = ΔP·(R² − r²) / (4ηL)
Speed peaks on the axis (r = 0) and falls to zero at r = R. The centre-line maximum and the cross-sectional average are
vmax = ΔP·R² / 4ηL   v̄ = vmax / 2

The fourth-power law

Integrate that parabola over the pipe's area and the total volume flow rate is the Hagen–Poiseuille result:
Q = πΔP·R⁴ / 8ηL
The R⁴ is the headline. Widen a pipe by just 19% and you double its throughput; halve a radius and flow collapses to one-sixteenth. This is why a modest narrowing of an artery sharply raises the pressure the heart must supply, and why a slightly wider fire hose delivers vastly more water.

Staying laminar

Poiseuille's parabola only holds while the flow is orderly. The Reynolds number compares inertia to viscosity:
Re = ρ·v̄·(2R) / η
Below roughly 2000 the flow is laminar and the formulas above are exact; push Re higher (wide pipe, thin fluid, big ΔP) and the stream eventually turns turbulent, at which point flow rate grows much more slowly than this ideal law predicts.

Things to try

Nudge the radius up and down and watch Q swing wildly while ΔP barely moves the profile's shape — R⁴ versus linear. Crank the viscosity up to model blood or oil: the profile keeps its parabola but every speed drops in proportion. Then raise ΔP with a thin fluid until Re climbs past 2000 and the readout warns that real flow would break into turbulence.