Poiseuille Law Calculator
Poiseuille law describes laminar viscous flow through a cylinder. Flow rate scales with r to the fourth power — halving the radius reduces flow to just 6.25% of original.
💧 Fluid Dynamics📐 Q = πr⁴ΔP/8ηL🔧 Engineering
Q = πΔPr⁴8ηL
Qvolumetric flow ratem³/s
ΔPpressure differencePa
rtube inner radiusm
ηdynamic viscosityPa·s
Ltube lengthm
Pipe Radius (m)
Pipe Length (m)
Pressure Diff ΔP (Pa)
Viscosity (Pa·s)
Please enter valid values.
Formula & Reference
| Variable | Symbol | Formula | Units |
|---|---|---|---|
| Flow Rate | Q | Q = πr⁴ΔP/(8ηL) | m³/s |
| Radius | r | r = (8ηLQ/πΔP)¹⁄⁴ | m |
| Pressure | ΔP | ΔP = 8ηLQ/(πr⁴) | Pa |
| Viscosity | η | Water: 0.001 Pa·s | Pa·s |
| Length | L | Pipe length | m |
Step-by-Step Examples
Example 1
Blood in Artery
Artery r=2 mm, L=10 cm, dP=2660 Pa, eta=0.003 Pa·s.
- Q = π(0.002)⁴×2660/(8×0.003×0.1)
- Q = 5.57×10⁻⁵ m³/s
✓ 55.7 mL/s for this idealized rigid-tube model
Example 2
Water Pipe
r=5 mm, L=2 m, dP=5000 Pa, water eta=0.001.
- Q = π(0.005)⁴×5000/(8×0.001×2)
- Q = 6.14×10⁻⁴ m³/s = 614 mL/s
✓ 614 mL/s, provided the resulting flow remains laminar
Example 3
Radius Doubles
How does doubling pipe radius change flow?
- Q ∝ r⁴
- New Q = 2⁴ × original = 16 × original
✓ 16x more flow — the fourth-power law
Real-World Applications
Idealized Vessel Flow
The fourth-power radius sensitivity is instructive, but real blood flow is pulsatile and vessels are compliant; this calculator is not a clinical model.
Laminar Tube Design
The equation supports capillary and small-tube design when flow is fully developed, Newtonian and laminar.
Microfluidics
Lab-on-chip devices rely on precise Poiseuille flow at microscale.
Viscometry
Capillary viscometers infer viscosity from pressure-driven laminar flow through a tube of known dimensions.
Common Mistakes to Avoid
⚠️
Forgetting r to the fourth
Q is proportional to r⁴, not r. A 50% radius reduction gives only (0.5)⁴ = 6.25% of original flow.
⚠️
Applying to turbulent flow
Valid only for laminar flow (Re < 2300). Turbulent flow needs different equations.
⚠️
Wrong viscosity units
Must use Pa·s. Water = 0.001; blood = 0.003; honey = ~10 Pa·s.
Connected Formulas
validity gate: laminar flow →Reynolds Numbercomputes the same output Q →Flow Rategeneral pipe-loss relation →Darcy–Weisbach
Frequently Asked Questions
What is the Reynolds number limit for this law? ▾
Valid for Re = rho*v*D/eta < 2300 (laminar). Above ~4000 flow is turbulent.
Why does r to the fourth appear? ▾
Area scales as r² and average velocity as r² (parabolic profile). Together Q scales as r⁴.
What is hydraulic resistance? ▾
R_hyd = 8*eta*L/(pi*r⁴), analogous to electrical resistance. dP = R_hyd * Q (like Ohm law).
How does this model arterial disease? ▾
Plaque reduces r. A 50% blockage (r halved) reduces flow to (0.5)⁴ = 6.25% of normal.
What is the velocity profile? ▾
Parabolic: v(r) = (dP/4*eta*L)(R² - r²). Max velocity at center = 2 * average velocity.
Does it apply to gases? ▾
Yes for slow gas flows where compressibility is negligible.
What is the Hagen-Poiseuille equation? ▾
Same equation, named after Hagen and Poiseuille who derived it experimentally in 1839-1840.
How does temperature affect results? ▾
Temperature affects viscosity strongly. Water viscosity drops 3.5x from 20 to 100 deg C.