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Generated on: July 20, 2026 at 09:33 PM

Hagen-Poiseuille Flow Calculator

Inputs

Pipe radius5 mm
Pressure difference10 kPa
Dynamic viscosity1 mPa·s
Pipe length1 m

Generated with OneCalc · https://onecalc.app/en/physics/poiseuille-flow

Physics

Hagen-Poiseuille Flow Calculator

Compute the volumetric flow rate of a viscous fluid through a straight circular pipe using the Hagen-Poiseuille equation. Enter the pipe radius, the pressure difference across the pipe, the fluid viscosity, and the pipe length to find the laminar flow rate.

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\begin{aligned} Q &= \frac{\pi\, r^4\, \Delta P}{8\, \mu\, L} \\ &= \frac{\pi\, (5\,\text{mm})^4\, (10\,\text{kPa})}{8\, (1\,\text{mPa·s})\, (1\,\text{m})} \\ &= ?\,\text{L/s} \end{aligned}

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Hagen-Poiseuille Flow

The Hagen-Poiseuille equation describes how a viscous fluid flows through a narrow pipe. It relates the volumetric flow rate to the pressure that drives the fluid, the resistance offered by the fluid's viscosity, and the geometry of the pipe. The relation is central to hydraulics, chemical engineering, microfluidics, and the physiology of blood flow.

The equation

For steady, laminar, incompressible flow of a Newtonian fluid through a long, straight pipe of circular cross-section, the volumetric flow rate is:

Q=π r4 ΔP8 μ LQ = \frac{\pi\, r^4\, \Delta P}{8\, \mu\, L}

where QQ is the flow rate, rr the internal radius, ΔP\Delta P the pressure difference between the ends of the pipe, μ\mu the dynamic viscosity, and LL the pipe length. The flow rate rises with the driving pressure and the fourth power of the radius, and falls as viscosity or length increase.

Symbol table

QuantitySymbolNotes
Volumetric flow rateQQOutput, volume per unit time
Pipe radiusrrInternal radius; enters as r4r^4
Pressure differenceΔP\Delta PDrop between inlet and outlet
Dynamic viscosityμ\muFluid resistance to shear
Pipe lengthLLLength of the straight section

Worked example

Water at 20 °C flows through a straight tube with an internal radius of 5 mm and a length of 1 m. A pressure difference of 10 kPa is applied across the tube. Water has a dynamic viscosity of about 0.001 Pa·s. What is the flow rate?

Q=π (0.005)4 (10,000)8 (0.001) (1)Q = \frac{\pi\, (0.005)^4\, (10{,}000)}{8\, (0.001)\, (1)}

Q=π×6.25×10−10×10,0000.008=2.454×10−3 m3/sQ = \frac{\pi \times 6.25\times10^{-10} \times 10{,}000}{0.008} = 2.454\times10^{-3}\ \text{m}^3/\text{s}

The flow rate is about 2.454 L/s.

Water at 20 °C flows through a straight tube with an internal radius of 5 mm and a length of 1 m under a pressure difference of 10 kPa, with a viscosity of 0.001 Pa·s. Using base SI units for the calculation:

Q=π (0.005)4 (10,000)8 (0.001) (1)=2.454×10−3 m3/sQ = \frac{\pi\, (0.005)^4\, (10{,}000)}{8\, (0.001)\, (1)} = 2.454\times10^{-3}\ \text{m}^3/\text{s}

Converted to US units, the flow rate is about 38.9 gal/min.

The fourth-power dependence on radius

The most striking feature of the equation is that flow rate scales with r4r^4. Because of this, the radius is the single most important dimension of the pipe:

  • Doubling the radius multiplies the flow rate by 24=162^4 = 16.
  • Halving the radius reduces it to 1/161/16 of the original.
  • A 10 % reduction in radius drops the flow rate by about 34 %.

This sensitivity explains why arterial narrowing has such a large effect on blood flow, why a slightly clogged pipe can lose most of its throughput, and why microfluidic channels need precisely controlled dimensions. It also means the radius measurement dominates the uncertainty in any calculation: a small measurement error is amplified fourfold.

Where it applies, and where it does not

The equation assumes laminar flow — smooth, orderly layers of fluid sliding past one another. Whether flow stays laminar is governed by the Reynolds number; for pipe flow the transition to turbulence begins near a Reynolds number of about 2,300. Above that, chaotic mixing raises the resistance and the neat linear relation between pressure and flow breaks down, so an empirical model such as the Darcy-Weisbach equation is used instead.

The other assumptions are a Newtonian fluid (constant viscosity), incompressibility, a constant circular cross-section, no-slip walls, and a fully developed velocity profile. The last point requires the pipe to be long compared with its diameter so that entrance effects near the inlet are negligible. Blood, notably, is non-Newtonian, so the equation gives only a first approximation for physiological flow.

To check whether the laminar assumption is justified, compute the Reynolds number for the flow. If you need to convert viscosity between units such as poise, centipoise, and Pa·s, a viscosity converter handles that. And to relate pressure, speed, and elevation for an ideal (inviscid) flow, Bernoulli's equation is the companion relation.

Frequently Asked Questions (FAQ)

What is the Hagen-Poiseuille equation?

The Hagen-Poiseuille equation gives the volumetric flow rate of a viscous fluid in laminar flow through a straight pipe of circular cross-section: Q = πr⁴ΔP / (8μL). Here Q is the flow rate, r the internal radius, ΔP the pressure difference driving the flow, μ the dynamic viscosity, and L the pipe length.

It is derived from the Navier-Stokes equations for a fully developed parabolic velocity profile and is named after Gotthilf Hagen and Jean Léonard Marie Poiseuille, who studied it in the 1840s.

When does the Hagen-Poiseuille equation apply?

The equation applies to steady, laminar, incompressible flow of a Newtonian fluid in a long, straight pipe of constant circular cross-section, with no-slip walls and a fully developed velocity profile. It ignores entrance effects, so the pipe must be long relative to its diameter. Flow must be laminar — for pipe flow this typically means a Reynolds number below about 2,300. Outside these conditions, a corrected or turbulent flow model is needed.

Why does the pipe radius matter so much?

Flow rate scales with the fourth power of the radius (r⁴), so radius is by far the most influential variable. Doubling the radius increases the flow rate sixteenfold; halving it reduces the flow to one-sixteenth. This strong dependence explains why even slight narrowing of a blood vessel or a pipe can dramatically reduce flow, and why an accurate radius measurement is essential.

What happens when the flow becomes turbulent?

The Hagen-Poiseuille equation is valid only for laminar flow. As the pressure difference or radius grows, the Reynolds number rises; above roughly 2,300 for pipe flow the flow transitions to turbulence, where chaotic mixing increases resistance. In that regime the linear relationship between pressure and flow breaks down, and empirical correlations such as the Darcy-Weisbach equation with a friction factor are used instead.

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