Poiseuille's equation
Poiseuille's Equation
Poiseuille's equation, also known as the Hagen-Poiseuille equation, describes the flow of a viscous fluid through a cylindrical pipe. It is a fundamental principle in fluid mechanics and is particularly important in the study of blood flow in biology, oil flow in pipelines, and other similar situations where a viscous fluid flows through a confined space.
The Equation
The Poiseuille's equation for laminar flow of an incompressible and Newtonian fluid through a circular pipe is given by:
[ Q = \frac{\pi \Delta P r^4}{8 \mu L} ]
where:
- ( Q ) is the volumetric flow rate
- ( \Delta P ) is the pressure difference between the two ends of the pipe
- ( r ) is the radius of the pipe
- ( \mu ) is the dynamic viscosity of the fluid
- ( L ) is the length of the pipe
Derivation
The equation is derived from a balance between the pressure force driving the flow and the viscous resistance opposing it. It assumes a steady-state flow, incompressible fluid, Newtonian fluid behavior, no slip at the pipe wall, and a laminar flow regime.
Key Points
- Laminar Flow: Poiseuille's equation is only valid for laminar flow, which occurs at low Reynolds numbers (Re < 2000).
- Viscosity: The fluid must be Newtonian, meaning its viscosity does not change with the rate of shear.
- Cylindrical Pipe: The equation assumes flow through a circular pipe, although it can be modified for other cross-sectional shapes.
- No Slip Condition: The fluid is assumed to have zero velocity at the pipe wall (stick to the wall).
Table of Differences and Important Points
| Feature | Description |
|---|---|
| Flow Type | Laminar flow only |
| Fluid Type | Incompressible and Newtonian |
| Pipe Shape | Circular |
| Viscosity | Constant (( \mu )) across the flow |
| Flow Rate (( Q )) | Proportional to ( r^4 ) and ( \Delta P ), inversely to ( \mu ) and ( L ) |
| Pressure Difference (( \Delta P )) | Driving force for the flow |
| Pipe Radius (( r )) | A small increase in radius greatly increases flow rate due to the ( r^4 ) term |
| Pipe Length (( L )) | Longer pipes decrease flow rate |
Examples
Example 1: Basic Calculation
Given a pipe with a radius of 0.01 m, a length of 1 m, a viscosity of ( 1 \times 10^{-3} ) Pa·s, and a pressure difference of 100 Pa, calculate the flow rate.
[ Q = \frac{\pi \Delta P r^4}{8 \mu L} = \frac{\pi \cdot 100 \cdot (0.01)^4}{8 \cdot 1 \times 10^{-3} \cdot 1} \approx 3.93 \times 10^{-6} \text{ m}^3/\text{s} ]
Example 2: Effect of Radius on Flow Rate
Compare the flow rates for pipes with radii of 0.01 m and 0.02 m, keeping all other factors constant.
Using the Poiseuille's equation, if the radius is doubled, the flow rate increases by a factor of ( 2^4 = 16 ).
[ Q_{0.02} = 16 \cdot Q_{0.01} ]
This demonstrates the strong dependence of flow rate on the pipe radius.
Example 3: Clinical Application
In medical scenarios, Poiseuille's equation is used to understand blood flow. If a blood vessel narrows due to plaque, the radius may decrease, significantly reducing the flow rate of blood. For instance, if the radius is halved, the flow rate decreases by a factor of ( 2^4 = 16 ), which can lead to tissue ischemia.
Conclusion
Poiseuille's equation is a powerful tool in fluid mechanics for predicting the flow rate of a viscous fluid in a pipe. It highlights the importance of pipe radius and pressure difference in determining flow rate, while also showing the impact of viscosity and pipe length. Understanding this equation is crucial for engineers and medical professionals dealing with fluid flow in pipes and blood vessels.