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What Is Poiseuille's Law?
Poiseuille's Law describes the volumetric flow rate of an incompressible, Newtonian fluid through a long cylindrical pipe with constant circular cross-section under laminar flow conditions. Derived independently by Gotthilf Hagen and Jean Poiseuille in the 1840s, it is one of the most fundamental equations in fluid mechanics and hemodynamics.
The law reveals that flow rate is proportional to the fourth power of the radius, meaning even small changes in pipe diameter dramatically affect flow. Doubling the radius increases flow by a factor of 16. This fourth-power relationship is why atherosclerosis (arterial narrowing) can have such devastating effects on blood circulation, and why pipe diameter is the most critical design parameter in fluid systems.
The Hagen-Poiseuille Equation
Where Q is volumetric flow rate (m³/s), r is the pipe inner radius (m), ΔP is the pressure difference between the two ends (Pa), μ is the dynamic viscosity (Pa·s), and L is the pipe length (m).
Assumptions and Limitations
- The fluid must be incompressible and Newtonian (constant viscosity regardless of shear rate).
- Flow must be laminar with Reynolds number below approximately 2100.
- The pipe must be rigid, straight, and have a constant circular cross-section.
- The pipe length must be much greater than its diameter (fully developed flow).
- No slip condition at the pipe wall (fluid velocity is zero at the wall).
- Steady-state conditions with no time-dependent acceleration.
Applications
| Field | Application | Typical Fluid |
|---|---|---|
| Medicine | Blood flow in arteries and veins | Blood (~3-4 cP) |
| Chemical Engineering | Pipe network design | Various chemicals |
| Microfluidics | Lab-on-chip devices | Reagents, buffers |
| Oil & Gas | Pipeline transport | Crude oil, natural gas |
Frequently Asked Questions
Why does radius have such a large effect on flow rate?
Flow rate depends on radius to the fourth power because a larger radius provides both more cross-sectional area and allows higher velocities. The parabolic velocity profile means fluid near the center moves much faster than fluid near the wall. A larger pipe has proportionally more fluid in the fast-moving central region, creating a compounding effect.
What happens when flow becomes turbulent?
Poiseuille's law is only valid for laminar flow (Re below ~2100). In turbulent flow, the pressure drop increases faster than linearly with flow rate. The Darcy-Weisbach equation with Moody friction factor must be used instead, and the relationship between pressure and flow becomes approximately quadratic rather than linear.
How is this law used in medicine?
In cardiovascular medicine, Poiseuille's law explains why a 50% reduction in arterial radius leads to a 94% reduction in blood flow. It is used to estimate vascular resistance, understand hypertension, and design IV fluid delivery systems. However, blood is non-Newtonian and arteries are elastic, so corrections are needed for precise calculations.