The Cluster's Poiseuille's Law

A page about Poiseuille's law — the flow rate through a cylindrical pipe.

The Poiseuille equation

Poiseuille's law gives the volumetric flow rate through a cylindrical pipe of radius R and length L: Q = pi R^4 Delta P / (8 eta L)

where Delta P is the pressure difference and eta is the dynamic viscosity. The velocity profile is parabolic: v(r) = (Delta P / (4 eta L)) (R^2 - r^2)

The maximum velocity (at the center): v_max = (Delta P R^2) / (4 eta L) The average velocity: v_avg = Q / (pi R^2) = (Delta P R^2) / (8 eta L) = v_max / 2

In the cluster, the edit Poiseuille equation gives an edit flow rate.

The derivation

From the Navier-Stokes equation for steady, fully developed, laminar flow in a pipe (r-coordinate): eta (1/r) d/dr (r dv/dr) = dP/dz = -Delta P / L

Solving: v(r) = (Delta P / (4 eta L)) (R^2 - r^2) This is a parabolic profile (called "Poiseuille flow" or "Hagen-Poiseuille flow").

In the cluster, the edit derivation gives an edit parabolic profile.

The Hagen-Poiseuille law

The full law is sometimes called the Hagen-Poiseuille law, named after G.H. Hagen (1839) and J.L. Poiseuille (1840). Poiseuille studied blood flow in capillaries; his work was largely ignored until Helmholtz rediscovered it in 1860.

The law is valid for laminar flow: Re < 2300 for pipe flow. For Re > 4000, turbulence typically develops. Between 2300 and 4000 is a transition regime.

In the cluster, the edit Hagen-Poiseuille law gives an edit validity condition.

The fourth power law

The R^4 dependence means that small changes in radius have enormous effects on flow rate. A 10% reduction in radius reduces flow by 34%. This is why:

• Atherosclerosis (plaque buildup) drastically increases blood pressure
• Narrowing of airways (asthma) makes breathing difficult
• Tiny capillaries require many parallel vessels to maintain total flow

In the cluster, the edit fourth power law explains an edit effect.

The hydraulic resistance

The flow can be written as: Q = Delta P / R_hydraulic where the hydraulic resistance is: R_hydraulic = 8 eta L / (pi R^4)

This is analogous to Ohm's law: I = V / R. The analogy extends: parallel pipes reduce resistance (like parallel resistors), and series pipes add resistances.

In the cluster, the edit hydraulic resistance gives an edit analogy.

This law

This page is about Poiseuille's law. Q = pi R^4 Delta P / (8 eta L). v(r) = (Delta P / (4 eta L))(R^2 - r^2). R^4 dependence. Re < 2300. The law is real.