FLD-06

Hagen pressure drop

Δp = 32 μ L v / D². The mean-speed form of Hagen–Poiseuille.

Reading speed

FluidsHagen

Governing equation

\Delta p=32\mu L v/D^2

$v \in \left[0.01, 8\right]$

where

\mu: Viscosity (Pa·s)
L: Length (m)
v: Mean speed (m/s)
D: Diameter (mm)
\Delta p: Pressure drop (kPa)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\Delta p=32\mu L v/D^2

$v \in \left[0.01, 8\right]$

$v = 0.01$ $\Delta p = 0.032$

Along the curve

②

Domain of the variable — blue

v_{\min} = 0.01,\quad v_{\max} = 8

$v \in \left[0.01, 8\right]$

$v = 0.01$ $v = 0.01$

Along the curve

③

Response curve — green

\Delta p=32\mu L v/D^2

$v \in \left[0.01, 8\right]$

$v = 0.01$ $\Delta p = 0.032$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\Delta p}{\mathrm{d}v}

$v \in \left[0.01, 8\right]$

$v = 0.06$ $\Delta p^{\prime} = 3.2$

Along the curve

Lecture brief

Historical brief

Bernoulli, Navier–Stokes, Reynolds (1883) and Stokes drag made continuum flow a dimensionless craft. The sheets compute head, drag, Re and a pedagogical NS snapshot. This sheet (FLD-06 — Hagen pressure drop) is the form associated with Hagen. Working symbols: $\mu$ , $L$ , $v$ , $D$ $\rightarrow$ $\Delta p$ . Same laminar circular pipe as Poiseuille, written for the engineer who measures v.

Purpose

Purpose: compute

\Delta p

from

\mu

L

v

D

in Fluid physics via

\Delta p=32\mu L v/D^2

Δp = 32 μ L v / D². The mean-speed form of Hagen–Poiseuille. Use it when a real fluid physics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\mu = 0.001\,\mathrm{Pa·s}

L = 10.000\,\mathrm{m}

v = 0.500\,\mathrm{m/s}

D = 10.000\,\mathrm{mm}

, the governing relation

\Delta p=32\mu L v/D^2

yields

\Delta p = 1.600\,\mathrm{kPa}

. A pipe, a mean arrow, a falling pressure. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Viscosity \mu0.001 Pa·sLength L10 mMean speed v0.5 m/sDiameter D10 mm

Outputs

Pressure drop \Delta p1.600 kPa

Reading speed

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Narration of this film

A pipe, a mean arrow, a falling pressure.

Same laminar circular pipe as Poiseuille, written for the engineer who measures v.

Reading speed

Watch on YouTube