Hong Life CalculatorsUnderstanding the Hagen-Poiseuille laminar flow calculator — concept and principle
The Hagen-Poiseuille law gives the volume flow of a viscous fluid through a long, straight, round tube when the flow is laminar. Enter the pressure difference between the ends, the tube radius, its length and the fluid viscosity, and the calculator returns the flow in mL/s. It is used for capillaries, microfluidics, needles, drip lines and viscometers.
How it is calculated
Q = πr⁴Δp ÷ (8μL)
- Δp: pressure difference between inlet and outlet, in Pa.
- r: inner radius, entered in mm and converted to m.
- L: tube length in m; μ: dynamic viscosity, entered in mPa·s and converted to Pa·s.
- Q comes out in m³/s and is multiplied by 10⁶ to give mL/s.
The principle
In steady laminar flow the fluid moves in concentric cylindrical layers that slide over each other. The pressure force pushing a central cylinder forward balances the viscous shear on its surface. Solving this balance with zero velocity at the wall gives a parabolic profile: fastest in the centre, zero at the wall, with the mean velocity exactly half the maximum. Integrating that profile over the cross-section yields the r⁴ law. Hagen (1839) and Poiseuille (1840s) found it independently by experiment. The same solution gives the laminar friction factor f = 64/Re used in pipe design.
Worked example
- Inputs: Δp = 1000 Pa, r = 1 mm = 0.001 m, L = 1 m, μ = 1 mPa·s = 0.001 Pa·s.
- r⁴ = (0.001)⁴ = 1 × 10⁻¹² m⁴.
- Numerator: π × 10⁻¹² × 1000 = 3.1416 × 10⁻⁹.
- Denominator: 8 × 0.001 × 1 = 0.008.
- Q = 3.1416 × 10⁻⁹ ÷ 0.008 = 3.927 × 10⁻⁷ m³/s = 0.3927 mL/s (shown as 0.3926990817).
| Tube radius (mm) | Flow (mL/s) |
|---|---|
| 0.25 | 0.0015339808 |
| 0.5 | 0.0245436926 |
| 0.75 | 0.1242524438 |
| 1 | 0.3926990817 |
| 1.25 | 0.9587379924 |
| 1.5 | 1.988039101 |
| 2 | 6.283185307 |
How to read the result
The striking feature is the fourth power of radius: doubling the radius gives 16 times the flow at the same pressure, and narrowing it by 10% cuts flow by about 34%. Always check that the flow really is laminar: here the mean velocity is 0.125 m/s and Re = ρvD/μ ≈ 250 for water, well below 2300, so the law applies. Very short tubes also need an entrance length (roughly 0.05·Re·D in laminar flow) before the parabolic profile develops.
Common mistakes
- Entering the diameter instead of the radius, which overstates flow 16 times.
- Applying it to turbulent flow, where flow grows much more slowly with Δp.
- Using it for gases with large pressure drops or for non-Newtonian fluids such as blood in large vessels.
Frequently asked questions
How do I get the pressure needed for a given flow?
Rearrange to Δp = 8μLQ ÷ (πr⁴). This is also written as Δp = Q·R, with hydraulic resistance R = 8μL ÷ (πr⁴).
Why is flow proportional to r⁴ and not r²?
Area grows with r², and the mean velocity at a fixed pressure gradient also grows with r², because the wall is farther from the centre. Together they give r⁴.
Can I combine tubes in series or parallel?
Yes. Resistances add in series; in parallel, their reciprocals add, just like electrical resistors.
Sources: S. P. Sutera and R. Skalak, The history of Poiseuille's law, Annual Review of Fluid Mechanics 25 (1993) 1–19 · F. M. White, Viscous Fluid Flow, McGraw-Hill