Hagen-Poiseuille Pipe Flow Calculator

The Hagen-Poiseuille equation gives the volumetric flow rate of a Newtonian fluid undergoing steady, laminar flow through a round pipe of constant cross-section, based on the pipe's radius, length, the fluid's viscosity, and the pressure difference driving the flow.

Q = πr⁴ΔP/(8μL), where Q is volumetric flow rate, r is the pipe's inner radius, ΔP is the pressure difference across the pipe, μ is the fluid's dynamic viscosity, and L is the pipe's length.

This strong dependence arises because increasing the radius both enlarges the cross-sectional area available for flow (radius squared) and reduces the velocity gradient near the walls that creates viscous drag (another factor of radius squared), compounding to a fourth-power relationship overall.

It applies specifically to steady, laminar, incompressible flow of a Newtonian fluid through a straight round pipe with a constant cross-section, generally valid when the Reynolds number stays below about 2300. It does not apply to turbulent flow.

It is the basis for understanding blood flow through blood vessels, IV drip line sizing, and dialysis catheter design. The radius-to-the-fourth-power dependence explains why even modest arterial narrowing (as in atherosclerosis) can dramatically reduce blood flow, a key concept in cardiovascular physiology.

The flow forms a parabolic velocity profile: zero velocity at the pipe wall (no-slip condition), increasing smoothly to a maximum at the centerline. The average velocity across the whole cross-section works out to exactly half the centerline maximum velocity.

Flow rate is inversely proportional to viscosity, so a more viscous fluid (like honey compared to water) flows dramatically slower through the same pipe at the same pressure difference. This is why highly viscous fluids like heavy oils often need to be heated (which lowers viscosity) to pump efficiently.

Flow rate is inversely proportional to pipe length because a longer pipe means the fluid experiences more total viscous drag along its path for the same driving pressure difference, exactly analogous to how a longer wire has more electrical resistance for the same voltage.

Yes, extensively. Lab-on-a-chip and microfluidic devices typically operate at very small scales and low velocities where flow is reliably laminar, making the Hagen-Poiseuille equation a standard and highly accurate design tool for these systems.

No, once flow becomes turbulent, the simple fourth-power radius relationship breaks down and flow rate no longer scales the same way with pressure difference; turbulent pipe flow instead requires empirical friction-factor correlations like the Darcy-Weisbach equation.