# Exercise 11

The Hagen-Poiseuille equation, also known as the Hagen-Poiseuille law, Poiseuille Law or Poiseuille equation, is a physical law that describes the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. The assumptions of the equation are that the fluid is incompressible and Newtonian; the flow is laminar through a pipe of constant circular cross-section that is substantially longer than its diameter; and there is no acceleration of fluid in the pipe. For velocities and pipe diameters above a threshold, actual fluid flow is not laminar but turbulent, leading to larger pressure drops than calculated by the Hagen-Poiseuille equation as shown here.

where:

Δp |
= | pressure difference between the two ends (ML^{−1}T^{−2}) |

L |
= | length of pipe (L) |

μ |
= | dynamic viscosity (ML^{−1}T^{−1}) |

Q |
= | volumetric flow rate (L^{3}T^{−1}) |

R |
= | pipe radius (L) |

A |
= | cross section of pipe (L^{2}) |

The equation does not hold close to the pipe entrance. The equation fails in the limit of low viscosity, wide and/or short pipe. Low viscosity or a wide pipe may result in turbulent flow, making it necessary to use more complex models, such as the Darcy–Weisbach equation. The ratio of length to radius of a pipe should be greater than one forty-eighth of the Reynolds number for the Hagen-Poiseuille law to be valid.

- What are the assumptions associated with the Hagen-Poiseulle equation and the Poiseuille law?
- How does the viscosity of the fluid change the relationship between pressure gradient and flow?
- How is the equation for laminar flow in a full pipe similar to Darcy’s law?

Click here for solution to Exercise 11

Return to where text linked to Exercise 11