Hagen–Poiseuille Flow Calculator
The Hagen–Poiseuille law gives the volumetric flow rate for fully developed laminar flow in a circular pipe. This calculator computes Q, converts it to practical units, and checks its own validity with a Reynolds number.
Frequently asked questions
Why does flow rate scale with the fourth power of radius?
Doubling the radius multiplies Q by 16. Two effects compound: the cross-sectional area grows as r², and the parabolic velocity profile lets the core move faster as r² again. This is why small arterial narrowing drastically reduces blood flow.
When does Hagen–Poiseuille fail?
It requires laminar (Re < 2300), fully developed, steady, incompressible Newtonian flow in a straight circular pipe. Entrance regions, bends, pulsating flow and non-Newtonian fluids like blood in small vessels all break the assumptions.
How is this derived?
It follows from the Navier–Stokes equations reduced to one dimension: balancing the pressure force on a fluid cylinder against viscous shear on its surface gives a parabolic velocity profile, and integrating that profile over the cross-section yields Q = πΔPr⁴/8μL.