When is the Hagen-Poiseuille equation valid?

It applies to steady, incompressible, laminar (Re < 2300) flow of a Newtonian fluid in a long, straight, rigid, circular pipe. Turbulent flow, non-Newtonian fluids, or entrance effects require different models.

Why does radius have such a large effect on flow rate?

Flow rate scales with the fourth power of radius (r⁴). Doubling the pipe radius increases flow rate 16×. This strong dependence makes the Hagen-Poiseuille law very sensitive to small changes in tube diameter.

What viscosity value should I use for water?

Water at 20 °C has a dynamic viscosity of about 0.001002 Pa·s (1.002 mPa·s). Viscosity decreases with temperature: ~0.00089 Pa·s at 25 °C and ~0.00055 Pa·s at 50 °C.

What is wall shear stress and why does it matter?

Wall shear stress (τ) is the tangential force per unit area exerted by the fluid on the pipe wall. It determines erosion and fouling rates in industrial pipes, and endothelial cell response in blood vessels.

Can I use this for gas flow?

For gases at low pressure where compressibility and slip effects are negligible, the equation gives a useful approximation. At higher pressures or very small tube diameters, gas-specific corrections (Klinkenberg, Knudsen) are needed.

Poiseuille Flow Calculator

Enter pipe radius, length, fluid viscosity, and pressure drop to compute volumetric flow rate and wall shear stress via the Hagen-Poiseuille equation.

Pipe & Fluid Parameters

Pipe inner radius (r)

Pipe length (L)

Dynamic viscosity (η)

Water at 20 °C ≈ 0.001002 Pa·s

Pressure drop (ΔP)

Fluid density (ρ) — for Reynolds number

kg/m³

Water at 20 °C ≈ 998 kg/m³

Volumetric Flow Rate (Q)

—

Wall Shear Stress (τ)

—

Mean Velocity (v̄)

—

Reynolds Number (Re)

—

Hydraulic Resistance (R_hyd)

—

R = 8ηL / (πr⁴) — pressure drop per unit flow rate

Equations Used

Q = (π r⁴ ΔP) / (8 η L)

τ_w = (r ΔP) / (2 L)

v̄ = Q / (π r²)

Re = (2 ρ v̄ r) / η

Summary

The Hagen-Poiseuille equation describes laminar (viscous) flow of a Newtonian fluid through a straight cylindrical pipe. It relates volumetric flow rate to pipe radius (raised to the fourth power), pipe length, fluid viscosity, and the pressure difference driving the flow. This calculator also computes wall shear stress and the Reynolds number so you can verify the laminar flow assumption holds (Re < 2300).

How it works

Enter the pipe inner radius and length.
Enter the dynamic viscosity of the fluid (e.g. 0.001 Pa·s for water at 20 °C).
Enter the pressure drop across the pipe length.
The calculator applies Q = (π r⁴ ΔP) / (8 η L) to find volumetric flow rate.
Wall shear stress τ = (r ΔP) / (2 L) and Reynolds number are computed automatically.
Click Calculate (or press Enter) to compute results, and adjust parameters to explore sensitivity.

Use cases

Size capillary tubes in microfluidic devices.
Estimate blood flow resistance in physiology and biomedical engineering.
Validate CFD simulations of laminar pipe flow.
Determine pump pressure requirements for a given flow rate in a lab setup.
Teach or learn the Hagen-Poiseuille law interactively.
Check whether a pipe flow regime is truly laminar (Re < 2300).

Frequently Asked Questions

Last updated: 2026-06-14 · Reviewed by Nham Vu