How to Use Poiseuille’s Law Calculator
The Poiseuille’s Law Calculator computes the volumetric flow rate through a cylindrical tube under a constant pressure difference, assuming fully developed, steady, laminar flow of a Newtonian fluid.
- Enter pipe radius r — the internal radius (not diameter); because Q scales as r⁴, even a small radius error propagates strongly to the flow estimate.
- Enter pressure drop ΔP — the pressure difference between the tube inlet and outlet ends.
- Enter dynamic viscosity μ — use SI units (Pa·s); water at 20 °C ≈ 1.002 × 10⁻³ Pa·s, blood ≈ 3–4 × 10⁻³ Pa·s, glycerine ≈ 1.5 Pa·s.
- Enter tube length L — the full length over which ΔP acts; exclude entry/exit correction lengths if the tube is short.
- Read Q in m³/s, mL/min and µL/min for compatibility with syringe pumps, flow meters and clinical flow specifications.
Formula & Theory — Poiseuille’s Law Calculator
The Poiseuille’s Law Calculator implements the Hagen–Poiseuille analytical solution to the Navier–Stokes equations for laminar pipe flow:
Q = π · r⁴ · ΔP / (8 · μ · L)
v_max = r² · ΔP / (4 · μ · L) (centreline velocity)
v_avg = v_max / 2 = Q / (π · r²) (mean velocity)
Re = ρ · v_avg · 2r / μ (laminar validity check)| Symbol | Meaning | SI Unit |
|---|---|---|
| Q | Volumetric flow rate | m³/s |
| r | Tube inner radius | m |
| ΔP | Pressure drop (inlet − outlet) | Pa |
| μ | Dynamic viscosity | Pa·s |
| L | Tube length | m |
| v_max | Maximum centreline velocity | m/s |
The r⁴ dependence means doubling the tube radius increases Q by a factor of 16 — the dominant design parameter in both microfluidics and vascular physiology. Hagen–Poiseuille flow is valid for Re < 2300 (laminar) and requires a fully developed parabolic profile, reached after an entry length ≥ 0.06 · Re · D.
Use Cases for Poiseuille’s Law Calculator
- Microfluidic chip design — size channel widths and depths to deliver target flow rates from syringe pump pressures in lab-on-chip assays and point-of-care diagnostics.
- Capillary viscometry — determine unknown fluid viscosity by measuring Q under a known ΔP through a calibrated glass capillary tube.
- Biomedical flow modelling — estimate blood flow in arterioles and venules for pharmacokinetic modelling, oxygen-delivery calculations and drug-infusion rate planning.
- Catheter and dialysis sizing — determine the minimum catheter bore needed to achieve clinically required flow rates at physiological pressure gradients.
- Lubrication engineering — calculate oil flow through journal-bearing supply grooves, nozzle orifices and hydraulic restrictor passages under operating conditions.
- Educational laminar flow — visualise the parabolic velocity profile and demonstrate how viscosity, radius and tube length individually affect hydraulic resistance.