# 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.