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.

\displaystyle \Delta p=\frac{8\mu LQ}{\pi R^{4}}=\frac{8\pi \mu LQ}{A^{2}}


Δp = pressure difference between the two ends (ML−1T−2)
L = length of pipe (L)
μ = dynamic viscosity (ML−1T−1)
Q = volumetric flow rate (L3T−1)
R = pipe radius (L)
A = cross section of pipe (L2)

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 DarcyWeisbach equation. The ratio of length to radius of a pipe should be greater than one forty-eighth \left ( \textrm{i.e.,}> \frac{1}{48} \right ) of the Reynolds number for the Hagen-Poiseuille law to be valid.

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

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Introduction to Karst Aquifers Copyright © 2022 by Eve L. Kuniansky, Charles J. Taylor, and Frederick Paillet. All Rights Reserved.