Poiseuille's Law
Calculate fluid flow rate through a pipe based on pressure, viscosity, and pipe dimensions.
Used in medical and engineering applications.
The Formula
Q = (π × r⁴ × ΔP) / (8 × μ × L)
Poiseuille's law describes laminar flow of a viscous fluid through a cylindrical pipe. Flow rate is extremely sensitive to pipe radius — doubling the radius increases flow 16 times.
Variables
| Symbol | Meaning |
|---|---|
| Q | Volumetric flow rate (m³/s) |
| r | Inner radius of the pipe (meters) |
| ΔP | Pressure difference between the two ends (Pascals) |
| μ | Dynamic viscosity of the fluid (Pa·s) |
| L | Length of the pipe (meters) |
Example 1
Water (μ = 0.001 Pa·s) flows through a 1 cm radius pipe, 2 m long, with ΔP = 500 Pa
Q = (π × (0.01)⁴ × 500) / (8 × 0.001 × 2)
Q = (π × 10⁻⁸ × 500) / 0.016
Q ≈ 9.82 × 10⁻⁴ m³/s ≈ 0.98 liters/s
Example 2
Blood (μ = 0.003 Pa·s) flows through an artery (r = 2 mm, L = 10 cm, ΔP = 400 Pa)
Q = (π × (0.002)⁴ × 400) / (8 × 0.003 × 0.1)
Q = (π × 1.6 × 10⁻¹¹ × 400) / 0.0024
Q ≈ 8.38 × 10⁻⁶ m³/s ≈ 8.38 mL/s
When to Use It
Use Poiseuille's law when:
- Modeling blood flow through arteries and veins
- Designing piping systems for viscous fluids
- Understanding how artery narrowing (stenosis) reduces blood flow
- Calculating flow in microfluidic and lab-on-a-chip devices