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
Key Notes
- Formula: Q = πr⁴ΔP / (8ηL): Q is volumetric flow rate, r is pipe radius, ΔP is pressure difference, η is dynamic viscosity, and L is pipe length. Applicable to laminar, incompressible flow in a straight cylindrical pipe.
- Fourth-power radius dependence: Flow rate scales as r⁴ — halving the pipe radius reduces flow by a factor of 16. This is critical in medicine: a 50% reduction in arterial radius (plaque buildup) reduces blood flow to 1/16 of normal, not 1/2.
- Validity requires laminar flow: Poiseuille's law only holds when the Reynolds number Re = ρvD/η < 2,300. At higher Re, turbulence develops and flow resistance increases beyond what this formula predicts.
- Viscosity is temperature-dependent: Water viscosity drops ~40% between 20°C and 60°C. Engine oil viscosity changes dramatically with temperature, which is why multi-grade oils (e.g., 5W-30) are used in vehicles.
- Applications: Poiseuille's law is used in IV drip rate calculation, arterial blood flow modeling, pipe sizing for HVAC and plumbing, microfluidic chip design, and estimating resistance in hydraulic systems.