In laminar flow through a pipe the flow rate rises with the fourth power of the radius. Narrowing an artery by 20% cuts blood flow by almost 60% at the same pressure – which is why atherosclerosis is so dangerous.
How to use the Poiseuille’s Law Calculator
- Dynamic viscosity μ
- Tube radius r
- Tube length L
- Pressure difference ΔP
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Enter the viscosity, tube radius, length and pressure difference.
Inputs explained
| Field | What to enter |
|---|---|
| Dynamic viscosity μ | Units: mPa·s (cP), Pa·s |
| Tube radius r | Units: mm, cm, m, in, µm |
| Tube length L | Units: cm, mm, m, in, ft |
| Pressure difference ΔP | Units: Pa, kPa, mmHg, bar |
What each result means
- Flow rate Q = pi×[Tube radius r]^4×[Pressure difference ΔP]÷(8×[Dynamic viscosity μ]×[Tube length L])
- Flow resistance = 8×[Dynamic viscosity μ]×[Tube length L]÷(pi×[Tube radius r]^4)
- Mean velocity = [Tube radius r]^2×[Pressure difference ΔP]÷(8×[Dynamic viscosity μ]×[Tube length L])
Example values
How the main result changes when Dynamic viscosity μ changes and everything else stays at the example values above:
| Dynamic viscosity μ (mPa·s (cP)) | Flow rate Q (mL/s) |
|---|---|
| 1.75 | 3.5904 |
| 2.63 | 2.3936 |
| 3.5 | 1.7952 |
| 4.38 | 1.4362 |
| 5.25 | 1.1968 |
| 7 | 0.8976 |
Formula
Q = π·r⁴·ΔP / (8·μ·L), R = 8μL/(πr⁴)
Worked example
Inputs — Dynamic viscosity μ: 3.5 mPa·s (cP); Tube radius r: 2 mm; Tube length L: 10 cm; Pressure difference ΔP: 100 Pa.
Results — Flow rate Q = 1.7952 mL/s; Flow resistance = 55,704,230.08216 Pa·s/m³; Mean velocity = 14.28571 cm/s.
Key terms
- Laminar flow
- Smooth, layered flow without mixing.
Frequently asked questions
Why the fourth power of r?
Both area and velocity grow with r².
When is the law valid?
Laminar, steady flow of a Newtonian fluid in a long rigid tube.
Does it apply to blood?
Approximately – blood is non-Newtonian and vessels are elastic.
What is the maximum velocity?
Twice the mean, at the centre.