FLD-05

Poiseuille discharge

Q = π r⁴ Δp / (8 μ L). Laminar flow in a circular tube.

Reading speed

FluidsPoiseuille

Governing equation

Q=\dfrac{\pi r^4\Delta p}{8\mu L}

$r \in \left[0.05, 20\right]$

where

r: Radius (mm)
\Delta p: Pressure drop (kPa)
\mu: Viscosity (Pa·s)
L: Length (m)
Q: Discharge (mL/s)

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

Q=\dfrac{\pi r^4\Delta p}{8\mu L}

$r \in \left[0.05, 20\right]$

$r = 0.05$ $Q = 4.91\times 10^{-6}$

Along the curve

②

Domain of the variable — blue

r_{\min} = 0.05,\quad r_{\max} = 20

$r \in \left[0.05, 20\right]$

$r = 0.05$ $r = 0.05$

Along the curve

③

Response curve — green

Q=\dfrac{\pi r^4\Delta p}{8\mu L}

$r \in \left[0.05, 20\right]$

$r = 0.05$ $Q = 4.91\times 10^{-6}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}Q}{\mathrm{d}r}

$r \in \left[0.05, 20\right]$

$r = 0.175$ $Q^{\prime} = 0.026$

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-05 — Poiseuille discharge) is the form associated with Poiseuille. Working symbols: $r$ , $\Delta p$ , $\mu$ , $L$ $\rightarrow$ $Q$ . The r⁴ law: halving the radius cuts the flow sixteenfold. Blood vessels live here.

Purpose

Purpose: compute

Q

from

r

\Delta p

\mu

L

in Fluid physics via

Q=\dfrac{\pi r^4\Delta p}{8\mu L}

Q = π r⁴ Δp / (8 μ L). Laminar flow in a circular tube. 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

r = 1.000\,\mathrm{mm}

\Delta p = 2.000\,\mathrm{kPa}

\mu = 0.001\,\mathrm{Pa·s}

L = 1.000\,\mathrm{m}

, the governing relation

Q=\dfrac{\pi r^4\Delta p}{8\mu L}

yields

Q = 0.7854\,\mathrm{mL/s}

. A capillary, a parabolic profile, a volume flux. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Radius r1 mmPressure drop \Delta p2 kPaViscosity \mu0.001 Pa·sLength L1 m

Outputs

Discharge Q0.7854 mL/s

Reading speed

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Narration of this film

A capillary, a parabolic profile, a volume flux.

The r⁴ law: halving the radius cuts the flow sixteenfold. Blood vessels live here.

Reading speed

Watch on YouTube