Definition
Poiseuille’s Law (or the Hagen-Poiseuille equation) describes the pressure drop required to maintain a steady, laminar flow of an incompressible, viscous fluid through a long, cylindrical pipe of constant cross-section. It dictates the relationship between flow rate, pressure gradient, fluid viscosity, and pipe geometry.
How to read: Volumetric flow rate Q equals pi times radius r to the fourth power, times the pressure difference delta P, all divided by eight times dynamic viscosity eta times the length of the pipe L. Meaning / when to use: Used to calculate the flow rate of a fluid given a specific pressure drop , or vice versa. The most critical feature is the profound dependency.
Why It Matters
This law is the governing principle behind microfluidics, plumbing, and cardiovascular physiology. Because flow rate scales with the fourth power of the radius, a tiny decrease in a tube’s diameter causes a catastrophic drop in flow. In medicine, if plaque reduces an artery’s radius by half, the heart must pump 16 times harder to maintain the same blood flow. Ignoring this non-linear geometric scaling leads to failed engineering designs and fatal medical misdiagnoses.
Core Concepts
- Fourth Power Dependency (): The dominant variable in the equation. Radius matters exponentially more than length, viscosity, or pressure.
- Laminar Flow Constraint: The law only applies to smooth, layered, non-turbulent flow (low Reynolds number). If the fluid moves too fast and becomes turbulent, resistance spikes and the equation fails.
- No-Slip Boundary Condition: The fluid velocity is assumed to be exactly zero right at the walls of the pipe, with maximum velocity in the direct center, creating a parabolic velocity profile.
- Viscosity (): The internal friction of the fluid. Thicker fluids (like honey) require proportionally more pressure to move than thin fluids (like water).