Poiseuilles Law Calculator
Calculate volumetric flow rate through a pipe using Poiseuilles law from pressure drop, radius, length, and dynamic viscosity.
What Is Poiseuilles Law?
Poiseuilles law (also called the Hagen-Poiseuille equation) predicts volumetric flow rate for laminar, incompressible fluid flow through a long cylindrical pipe with constant cross section. It is widely used in plumbing, biomedical engineering, and microfluidics.
Flow Rate Equation
The volumetric flow rate depends on pressure drop, pipe geometry, and fluid viscosity:
$$Q = \frac{\pi \cdot \Delta P \cdot r^4}{8 \cdot \mu \cdot L}$$where \(Q\) is flow rate in m³/s, \(\Delta P\) is pressure change in pascals, \(r\) is pipe radius in meters, \(\mu\) is dynamic viscosity in Pa·s, and \(L\) is pipe length in meters.
Flow resistance is defined as \(R = 8 \mu L / (\pi r^4)\), so \(Q = \Delta P / R\). Note that flow rate scales with the fourth power of radius, so small radius changes have a large effect.
Related tools: Pipe Velocity Calculator and Poise to Stokes Converter.
Frequently Asked Questions
When does Poiseuilles law apply?
It applies to steady, laminar, incompressible flow in a straight pipe with circular cross section. Turbulent flow or short pipes with entrance effects require different models.
Why does radius appear to the fourth power?
Velocity profile in laminar pipe flow is parabolic. Integrating that profile over the circular area produces an \(r^4\) dependence in the resulting flow rate formula.
How is Poiseuilles law used in medicine?
It models blood flow in vessels, airway resistance in the lungs, and fluid transport in kidney tubules. Vessel constriction dramatically reduces flow because radius is raised to the fourth power.
What units should I use?
Use SI units: meters for radius and length, pascals for pressure, Pa·s for viscosity, and m³/s for flow rate. The calculator also shows flow in mm³/s for small volumes.
What is flow resistance in this context?
Resistance \(R = 8\mu L / (\pi r^4)\) has units of Pa·s/m³. Higher viscosity, longer pipes, or smaller radii all increase resistance and reduce flow for a given pressure drop.
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