Hagen-Poiseuille Flow

Hagen-Poiseuille flow describes steady, fully-developed laminar flow of a viscous fluid through a straight circular pipe driven by a pressure difference.

Volumetric Flow Rate

The flow rate $Q$ (m³/s) is:

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

where:

• $r$ — pipe radius (m)
• $\Delta P$ — pressure difference $P_1 - P_2$ (Pa)
• $\mu$ — dynamic viscosity (Pa·s)
• $L$ — pipe length (m)

Notice the strong $r^4$ dependence — halving the radius reduces flow by a factor of 16.

Mean and Maximum Velocity

The mean (average) velocity across the cross-section:

$\bar{v} = \frac{Q}{\pi r^2} = \frac{r^2 \Delta P}{8 \mu L}$

The velocity profile is parabolic, with the maximum velocity at the centreline being exactly twice the mean:

$v_{\max} = 2\bar{v} = \frac{r^2 \Delta P}{4 \mu L}$

Validity

This law applies only to laminar flow, which requires $Re < 2300$. For turbulent flow a different (empirical) approach is needed.

Your Task

Implement:

• poiseuille_flow(r, delta_P, mu, L) — volumetric flow rate $Q$ (m³/s)
• mean_velocity(r, delta_P, mu, L) — mean velocity $\bar{v}$ (m/s)
• max_velocity(r, delta_P, mu, L) — centreline velocity $v_{\max}$ (m/s)