Viscous Flow Rate Calculator Formula
Understand the math behind the viscous flow rate calculator. Each variable explained with a worked example.
Formulas Used
Flow Rate
flow_rate = pi * pow(pipe_radius, 4) * pressure_drop / (8 * viscosity * pipe_length)Flow Rate (L/min)
flow_rate_lpm = pi * pow(pipe_radius, 4) * pressure_drop / (8 * viscosity * pipe_length) * 60000Variables
| Variable | Description | Default |
|---|---|---|
pipe_radius | Pipe Radius(m) | 0.01 |
pressure_drop | Pressure Drop(Pa) | 1000 |
viscosity | Dynamic Viscosity(Pa s) | 0.001 |
pipe_length | Pipe Length(m) | 1 |
How It Works
Hagen-Poiseuille Flow
For steady, laminar, incompressible flow through a long circular pipe, the flow rate depends strongly on the pipe radius.
Formula
Q = pi R^4 dP / (8 mu L)
The R^4 dependence means doubling the radius increases flow by 16 times.
Worked Example
Water (mu = 0.001 Pa s) through a 1 cm radius, 1 m long pipe with 1 kPa pressure drop.
- 01Q = pi * (0.01)^4 * 1000 / (8 * 0.001 * 1)
- 02R^4 = 1e-8
- 03Q = pi * 1e-8 * 1000 / 0.008
- 04Q = pi * 1e-5 / 0.008 = 3.927e-3 m3/s
- 05Q = 235.6 L/min
Frequently Asked Questions
Why does flow rate depend on R to the fourth power?
The velocity profile is parabolic: fluid moves fastest at the centre and is stationary at the walls. Integrating the parabolic profile over the circular cross-section gives the R^4 dependence.
When does this equation apply?
Only for laminar flow (Re < 2300 in the pipe), incompressible fluid, and fully developed flow far from the pipe entrance.
How does this relate to blood flow?
The Hagen-Poiseuille law governs blood flow in small vessels. Narrowing of arteries (stenosis) dramatically reduces flow because of the R^4 dependence.
Ready to run the numbers?
Open Viscous Flow Rate Calculator