Laminar Flow in Tubes
Poiseuille's law (also known as the Hagen-Poiseuille equation) describes the pressure drop in an incompressible and Newtonian fluid undergoing completely laminar flow through a long cylindrical pipe of constant cross-section.
Unlike Bernoulli's equation, which assumes a perfect frictionless fluid, Poiseuille's law accounts for the viscous friction between the fluid and the pipe walls. It reveals a profound relationship: the volumetric flow rate is directly proportional to the fourth power of the pipe's radius ($r^4$). This means that even a tiny change in a pipe's diameter results in a massive change in fluid flow.
Applications in Medicine and Engineering
- Blood Flow: The human cardiovascular system relies on this law. If plaque buildup decreases the radius of an artery by just 10%, the flow rate drops by over 34%, forcing the heart to pump dangerously harder to maintain blood pressure.
- Oil Pipelines: Heavy crude oil is highly viscous. Engineers use Poiseuille's law to determine exactly how powerful the pumping stations must be to push the thick oil across hundreds of miles of pipeline.
- Needles and Syringes: The gauge (radius) of an IV needle determines how quickly fluids can be injected. A slightly wider needle allows a drastically faster injection rate.
The Formula
Example Calculation
Calculate the flow rate of water (viscosity $\mu = 0.001 , ext{Pa}\cdot\text{s}$) flowing through a $10 , ext{m}$ garden hose with a radius of $0.01 , ext{m}$ (1 cm) under a pressure difference of $10,000 , ext{Pa}$.
- Numerator: $\pi \cdot 10000 \cdot (0.01)^4 = \pi \cdot 10000 \cdot 0.00000001 = 0.000314$.
- Denominator: $8 \cdot 0.001 \cdot 10 = 0.08$.
- Divide: $0.000314 / 0.08 \approx 0.0039 , ext{m}^3\text{/s}$.
The hose delivers about $3.9$ liters of water every second.