Poiseuille's Law Calculator
Work out the flow rate through a narrow tube from its radius, length, pressure drop and the water temperature.
What it does
Water creeping through a narrow tube is a different kind of flow from water down a drain. It is slow, smooth and steady, with the fluid layers sliding past one another instead of mixing, and it is described exactly by one equation. The OpenStax physics text calls it Hagen-Poiseuille flow: the volume that moves through a tube in a second depends on the pressure difference, the length of the tube, and the fourth power of its radius.
That fourth power is the striking part. Doubling the radius of a tube does not double the flow, it multiplies it by sixteen. It is why a capillary is such a good flow meter, and why medical tubing has such a small bore, and why a slow leak through a pinhole can still empty a tank overnight.
The law also has a hard limit. It describes laminar flow, where the fluid moves in neat layers, and OpenStax puts the boundary at a Reynolds number below 2000, with 2000 to 3000 unstable and above 3000 turbulent. A turbulent flow does not obey the equation, and the calculated value then understates what really happens.
Viscosity enters every part of it. Water at 20 °C has a dynamic viscosity of about 1.002 milli pascal seconds, and it thins as it warms, so the same tube moves more water in summer. The density and viscosity used here come from the NIST reference data for water.
The other assumption worth naming is that the pressure drop is the whole driving force. In a real system the pump, the fittings and the elevation change all have a say, and a horizontal tube with a steady pressure difference at each end is the clean case the equation was derived for.
How to use
- Pick metric or imperial units.
- Enter the inside radius of the tube, not the diameter.
- Enter the length of the tube.
- Enter the pressure drop from end to end.
- Enter the water temperature.
- Check the regime line: if the Reynolds number is above 2000 the law does not apply and the real flow will be higher.
Formula
- Q
- Volume flow rate, in cubic metres per second.
- r
- Inside radius of the tube.
- dP
- Pressure drop along the tube, with the high pressure end first.
- mu
- Dynamic viscosity of the water at its temperature.
- L
- Length of the tube.
- Re
- Reynolds number, which uses the tube diameter.
The equation assumes steady, fully developed, laminar flow of a Newtonian fluid in a straight round tube with no slip at the wall, and a horizontal tube. The Reynolds number here uses the tube diameter, not the radius, and the thresholds are the ones OpenStax gives: laminar below 2000 and turbulent above 3000, with the band in between being neither.
Worked example
Example: a 1 mm capillary, 1 m long, 1,000 Pa drop at 20 °C
Defaults: metric units, a radius of 1 mm, a length of 1 m, a pressure drop of 1,000 Pa and water at 20 °C.
- Water at 20 °C has a dynamic viscosity of 1.0016e-3 Pa s and a density of 998.2 kg/m3.
- Flow: pi x (0.001 to the fourth) x 1,000 / (8 x 1.0016e-3 x 1) = 3.9207e-5 m3/s.
- That is 0.3921 mL/s, since a cubic metre holds a million millilitres.
- Average velocity: 3.9207e-5 / (pi x 0.001 squared) = 0.1248 m/s.
- Reynolds number: 998.2 x 0.1248 x 0.002 / 0.0010016 = 248.8, which is laminar.
- The tool shows 0.3921 mL/s, 0.1248 m/s, 248.8 and laminar.
Reference table
| Pressure drop | Flow rate | Average velocity | Reynolds number |
|---|---|---|---|
| 250 Pa | 0.0980 mL/s | 0.0312 m/s | 62.2 |
| 1,000 Pa | 0.3921 mL/s | 0.1248 m/s | 248.8 |
| 10,000 Pa | 3.9207 mL/s | 1.2480 m/s | 2,487.6 |
| 100,000 Pa | 39.2072 mL/s | 12.4800 m/s | 24,875.5 |
Limits
- Only valid for laminar flow, so a Reynolds number below 2000 is required.
- Assumes a straight, horizontal, round tube running full with a steady pressure drop.
- Uses water only, so another fluid needs its own viscosity and density.
- Leaves out entrance effects, so the first few diameters of the tube behave differently.
Common mistakes
- Using the diameter where the radius belongs, which changes the answer by a factor of sixteen.
- Applying the law to a turbulent flow and believing the small number it gives.
- Forgetting the length in the denominator, which makes a long tube look as fast as a short one.
- Using the kinematic viscosity where the dynamic viscosity belongs.
- Ignoring the fourth power of the radius, which is the whole point of the law.
Try it yourself
Quick checks. Answers sit tucked under each question.
What flow do you get through a 2 mm radius tube, 0.5 m long, with a 250 Pa drop at 20 °C?
Q = pi x 0.002 to the fourth x 250 / (8 x 1.0016e-3 x 0.5) = 3.1366e-6 m3/s, which is 3.1366 mL/s, with a Reynolds number of 995.0 and laminar flow.
Why does the flow rate change so much with temperature?
Water thins as it warms, so the viscosity falls and the same pressure drop moves more water. Between 10 °C and 40 °C the dynamic viscosity drops from 1.3059e-3 to 0.6527e-3 Pa s, nearly half.
A 1 mm tube with a 100,000 Pa drop gives 39.2072 mL/s. Is that valid?
No. The Reynolds number is 24,875.5, which is well into turbulent flow, so the laminar law no longer holds and the real flow would be higher than the calculation.
Questions
What is Poiseuille's law?
The equation that says the flow through a tube is proportional to the pressure difference and to the fourth power of the radius, and inversely proportional to the length and the viscosity.
What is laminar flow?
Flow in which the fluid moves in smooth layers that slide past one another without mixing. It happens at low speeds in narrow channels, and it is the only kind of flow the law describes.
How do I tell whether the law applies?
Compute the Reynolds number. Below 2000 the flow is laminar and the equation is valid. Above 3000 it is turbulent, and the calculated value understates the real flow.
Why does the fourth power of the radius matter so much?
It makes the radius the most powerful variable in the design. A tube with twice the radius passes sixteen times the water, which is why flow metering uses very fine bores.
Can this be used for a garden hose?
No. A hose at any useful flow is turbulent, so the Darcy-Weisbach friction factor is used instead. The law is for narrow tubes such as capillaries, needles and laboratory glassware.
What other fluids can be used?
Any Newtonian fluid, with its own viscosity and density. Oils are far more viscous than water, so the same tube moves much less, while liquid mercury moves more.
Sources
- OpenStax University Physics Volume 1: 14.7, viscosity and turbulence
- NIST Chemistry WebBook: density and viscosity of water at atmospheric pressure
- University of Sao Paulo: viscosity of water against temperature