Poiseuille Flow Calculator (Hagen–Poiseuille Law for Tubes and Blood Vessels)
Flow rate of a thick liquid through a narrow tube from the pressure difference, tube size, length and viscosity, with a check that the flow is smooth enough for the law to hold.
Results
The dashed line marks the value you entered.
What this tool does
The Hagen–Poiseuille law says flow through a tube grows with the fourth power of its radius: halve the radius and you get one sixteenth of the flow, which is why a slightly narrowed artery or a blocked needle matters so much. It holds only for smooth, layered (laminar) flow, so the tool also gives the Reynolds number; above about 2300 the flow turns turbulent and the law overestimates it. Viscosity falls as liquids warm — water is about 1.0 mPa·s at 20 °C and 0.65 at 40 °C.
Formula
Q = π r⁴ ΔP ÷ (8 μ L) (Hagen–Poiseuille) · τ_wall = ΔP r ÷ 2L · Re = ρ v D ÷ μ
Variables
| Symbol | Meaning | Unit |
|---|---|---|
dm | Diameter | mm |
lg | Length | m |
dk | Pressure difference between the two ends | kPa |
mu | Dynamic viscosity | mPa·s |
ro | Density | kg/m³ |
Q | Flow rate | mL/min |
V | Mean speed of the fluid | m/s |
RE | Reynolds number | — |
RG | How the fluid is moving | — |
TW | Shear stress at the wall | Pa |
Worked example
- Diameter1 mm
- Length0.5 m
- Pressure difference between the two ends5 kPa
- Dynamic viscosity1 mPa·s
- Density1000 kg/m³
- Flow rate14.726 mL/min
- Mean speed of the fluid0.3125 m/s
- Reynolds number313
- How the fluid is movingLaminar — smooth, in layers
- Shear stress at the wall2.50 Pa
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- For work that must comply with a standard or be signed off, check the result against the applicable code and have it reviewed by a qualified engineer.