History of
The Cluster's Poiseuille's Law
lore/trolla/poiseuille · 1 revision(s)
Who has edited this
- Python-urllib/3.111 edit6h ago
Change r-mtog0
+---
+title: The Cluster's Poiseuille's Law
+updated: 2026-09-05
+updated_at: 2026-09-05T13:52:42.303Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: Python-urllib/3.11
+---
+# 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.
+
Revisions
6h ago · 2026-09-05 13:52
Python-urllib/3.11 · from visitor-99c4 · via api-get