Poiseuille's Law Calculator

Poiseuille's law for laminar flow through a straight circular pipe of constant cross-section:

$$Q = \frac{\pi r^4 \Delta P}{8 \mu L}$$

where:

• Q — volumetric flow rate (m³/s)
• r — internal pipe radius (m)
• ΔP — pressure difference between pipe ends (Pa)
• μ — dynamic viscosity (Pa·s)
• L — pipe length (m)

The velocity profile in Poiseuille flow is parabolic:

$$v(r_{pos}) = \frac{\Delta P}{4 \mu L}(r^2 - r_{pos}^2)$$

Key derived quantities:

• Average velocity: $$\bar{v} = \frac{Q}{\pi r^2} = \frac{r^2 \Delta P}{8 \mu L}$$
• Maximum velocity (at centerline): $$v_{max} = 2\bar{v} = \frac{r^2 \Delta P}{4 \mu L}$$
• Flow resistance: $$R = \frac{\Delta P}{Q} = \frac{8 \mu L}{\pi r^4}$$

The flow resistance is analogous to electrical resistance (Ohm's law: V = IR becomes ΔP = QR). This analogy is widely used in hemodynamics to model the vascular system as a circuit of resistances.

The r⁴ dependence has profound consequences: a 10% reduction in artery radius (from atherosclerosis) decreases blood flow by about 34%. This is why even small arterial blockages can significantly impair circulation.

Validity: Poiseuille's law applies only to laminar flow (Re < 2,300), Newtonian fluids, fully developed flow (far from the entrance), rigid walls, and steady (non-pulsatile) conditions.

The calculated flow rate shows how much fluid passes through the pipe per unit time under the given pressure gradient. The average velocity is the flow rate divided by cross-sectional area, while the maximum velocity at the centerline is exactly twice the average for Poiseuille flow. The flow resistance quantifies how much the pipe impedes flow — higher resistance means less flow for the same pressure drop.