The rate of flow of liquid in a capillary tube is precisely described by Poiseuille's Formula, which quantifies the volume of liquid flowing per unit time. The formula, as provided, is:
$V = \frac{\pi p r^4}{8 \eta L}$
This fundamental equation in fluid dynamics helps understand how various factors influence the flow of viscous fluids through narrow, cylindrical tubes.
Understanding Poiseuille's Law
Poiseuille's Law (also known as the Hagen-Poiseuille equation) is a physical law that gives the pressure drop in an incompressible Newtonian fluid undergoing laminar flow, through a long cylindrical pipe of constant circular cross-section. In the context of the question, it describes the volume flow rate (V).
Components of Poiseuille's Formula
Each variable in the formula plays a critical role in determining the flow rate:
| Symbol | Description | Unit (SI) |
|---|---|---|
| V | Volume Flow Rate | cubic meters per second (m³/s) |
| π | Pi (mathematical constant) | Dimensionless |
| p | Pressure Difference | Pascal (Pa) |
| r | Radius of the Capillary Tube | meter (m) |
| η | Dynamic Viscosity of the Liquid | Pascal-second (Pa·s) |
| L | Length of the Capillary Tube | meter (m) |
Key Factors Affecting Flow Rate
The formula reveals several crucial relationships between the physical properties of the tube and the liquid, and the resulting flow rate:
- Pressure Difference (p): The rate of flow (V) is directly proportional to the pressure difference across the ends of the tube. This means if you double the pressure difference, you double the flow rate.
- Radius of the Tube (r): This is the most significant factor. The flow rate is directly proportional to the fourth power of the radius ($r^4$). This implies that even a small increase in the tube's radius leads to a disproportionately large increase in the flow rate. For example, doubling the radius increases the flow rate by a factor of $2^4 = 16$ times! This is why tiny blockages in arteries can drastically reduce blood flow.
- Viscosity of the Liquid (η): The flow rate is inversely proportional to the dynamic viscosity of the liquid. Highly viscous liquids (like honey) flow much slower than less viscous liquids (like water) under the same conditions.
- Length of the Tube (L): The flow rate is inversely proportional to the length of the capillary tube. A longer tube offers more resistance to flow, thus reducing the flow rate.
Practical Insights and Examples
- Medical Applications: Understanding Poiseuille's Law is vital in medicine. For instance, the flow of blood through arteries and veins, or the administration of intravenous fluids, is governed by these principles. A slight narrowing of an artery due to plaque buildup (reducing 'r') can severely impede blood flow.
- Industrial Applications: In chemical engineering, this law helps design pipelines for transporting fluids, ensuring optimal flow rates by adjusting pipe diameter, pump pressure, and considering the fluid's viscosity.
- Filtration Systems: The effectiveness of filters with fine pores (acting as capillary tubes) depends heavily on the size of these pores and the viscosity of the fluid being filtered.
In summary, Poiseuille's formula provides the exact mathematical relationship for the rate of liquid flow in a capillary tube, highlighting the dominant role of the tube's radius and the liquid's viscosity.
