Hagen–Poiseuille equation

Hagen–Poiseuille equation describes the pressure drop caused by the laminar flow of a Newtonian fluid through a long, cylindrical pipe of constant cross section. It is a fundamental principle in the field of fluid mechanics and plays a crucial role in various applications, including medical sciences, particularly in understanding blood flow through vessels.

Overview

The equation is named after Gotthilf Hagen and Jean Léonard Marie Poiseuille, who independently derived the relationship in the 19th century. It mathematically expresses how the volumetric flow rate of a fluid passing through a pipe relates to the fluid's viscosity, the pipe's length and radius, and the pressure difference across the pipe.

Mathematical Formulation

The Hagen–Poiseuille equation is given by:

\[ Q = \frac{\pi r^4 \Delta P}{8 \mu L} \]

where:

• \(Q\) is the volumetric flow rate,
• \(r\) is the radius of the pipe,
• \(\Delta P\) is the pressure difference between the two ends of the pipe,
• \(\mu\) is the dynamic viscosity of the fluid, and
• \(L\) is the length of the pipe.

Applications

In medicine, the Hagen–Poiseuille equation is used to model the flow of blood through the circulatory system, particularly in small blood vessels or in situations where blood flow is steady and laminar. It helps in understanding conditions like atherosclerosis and the effects of blood viscosity changes on circulation.

Limitations

The equation assumes a number of ideal conditions: laminar flow, Newtonian fluid, a rigid and circular pipe, and a fully developed flow profile. These assumptions limit its direct application in complex biological systems, where factors like pulsatile blood flow, non-Newtonian behavior of blood, and irregular vessel geometry come into play.

Related Equations

• Bernoulli's principle: Describes the relationship between velocity, pressure, and potential energy in fluid flow.
• Navier-Stokes equations: A more general set of equations governing the motion of viscous fluid substances.

See Also

• Fluid mechanics
• Viscosity
• Laminar flow
• Circulatory system

External Links

• Link to further resources