Poiseuille Pipe Flow

Worked Example

Let's calculate the flow rate and speeds for the baseline pipe configuration:

• Radius r = 1 mm = 0.001 m
• Length L = 20 cm = 0.2 m
• Viscosity μ = 1 mPa·s = 0.001 Pa·s
• Pressure drop ΔP = 10 Pa

First, calculate the maximum (centerline) speed using the formula v_max = ΔP · r² / (4 · μ · L):

vmax = 10 · (0.001)2 / (4 · 0.001 · 0.2)\n vmax = 10 · 10-6 / 0.0008 = 0.0125 m/s

For a parabolic profile, the mean speed v_mean is exactly half the maximum speed:

vmean = 0.0125 / 2 = 0.00625 m/s

Now, calculate the volumetric flow rate Q using the cross-sectional area A = πr²:

Q = vmean · A = 0.00625 · π · (0.001)2\n Q = 0.00625 · π · 10-6 ≈ 0.019634954... × 10-6 m3/s

Converting to mL/s (where 1 m³ = 10⁶ mL):

Q = 0.019634954... mL/s

Finally, we find the Reynolds number using diameter D = 2r = 0.002 m and assuming a density ρ of 1000 kg/m³:

Re = ρ · vmean · D / μ\n Re = 1000 · 0.00625 · 0.002 / 0.001 = 12.5

Interactive Experiment

Explore the parabolic speed profile of viscous pipe flow using the interactive module at https://riano.app/module/poiseuille-pipe-flow.

Follow these steps to see the fourth-power sensitivity of flow rate to radius:

1. Set Pipe radius r to 1, Pipe length L to 20, Viscosity mu to 1, and Pressure drop deltaP to 10. Observe the Flow Q is approximately 0.0196 mL/s.
2. Now, double the Pipe radius r to 2, keeping other parameters constant.
3. Observe that Flow Q increases to approximately 0.314 mL/s. Notice that doubling the radius increases the flow by a factor of 16 (since 2⁴ = 16).

Limitations and Assumptions

This model assumes steady, incompressible, and Newtonian flow through a straight, perfectly circular pipe. It relies on the no-slip condition at the boundary, meaning the fluid velocity is exactly zero at the pipe walls.

Crucially, Poiseuille's law is only valid for laminar flow, which typically occurs when the Reynolds number is below approximately 2000. In our entire supported parameter grid, the maximum Reynolds number reaches 200, which is safely within the laminar regime. If conditions resulted in a higher Reynolds number, a turbulent solver and considerations for entrance effects (which this model ignores) would be necessary.

References

This module is based on classical fluid mechanics as described in university physics resources. Read more about the laminar tube no-slip profile, Poiseuille's flow and resistance, and the diameter-based Reynolds number in:
OpenStax University Physics Volume 1: 14.7 Viscosity and Turbulence