Fluid Flow and Viscosity

Viscosity $\eta$ measures a fluid's resistance to flow. In laminar flow the layers slide smoothly; in turbulent flow they mix in eddies. A small sphere moving slowly through a fluid feels drag $F=6\pi\eta rv$ (Stokes' law). A falling sphere reaches terminal velocity when weight = upthrust + drag.

Edexcel YPH11 WPH11≈ 35 minRevision 1 · updated 2026-10-27

Section 1 of 14

Learning objectives

Describe laminar and turbulent flow.
Use upthrust $=$ weight of fluid displaced.
Use Stokes' law $F=6\pi\eta rv$ .
Find terminal velocity and use a falling-ball viscometer.

Pause and think

Why does honey pour slowly but water splashes?

Section 2 of 14

Core idea

Start hereHoney and water

Pour honey and water side by side. Water splashes; honey folds slowly onto itself. Both are liquids, but honey's layers grip one another far more strongly. That grip is viscosity, and it sets how fast things can fall through a fluid.

The big idea

A slow sphere in a fluid feels drag $F=6\pi\eta rv$ . Falling at constant speed, its weight equals upthrust plus drag, which lets us find $\eta$ or $v_t$ .

1. Forces on a falling sphere

Terminal velocity

$W=U+F$
Constant velocity: resultant force zero.
$\tfrac43\pi r^3\rho_sg=\tfrac43\pi r^3\rho_fg+6\pi\eta rv_t$
Weight, upthrust (displaced fluid), Stokes' drag.
$6\pi\eta rv_t=\tfrac43\pi r^3(\rho_s-\rho_f)g$
Collect weight terms.
$v_t=\frac{2r^2(\rho_s-\rho_f)g}{9\eta}$
Divide by $6\pi\eta r$ .

v_t\propto r^2

: a ball twice as wide falls four times as fast.

Worked exampleSteel in glycerol

$r=1.0$ mm, $\rho_s=7800$ , $\rho_f=1260$ kg m⁻³, $\eta=1.4$ Pa s.

Goal: $v_t$ .
Strategy: Use the result above.

$2(1.0\times10^{-3})^2(6540)(9.81)=0.128$
Numerator.
$9\times1.4=12.6$
Denominator.
$v_t=0.0102\text{ m s}^{-1}$
Divide.

AnswerAbout

1.0

cm s⁻¹.

✓ CheckSlow enough for laminar flow, so Stokes' law applies ✓.

2. Flow

Laminar or turbulent?

Drawing diagram…

Stokes’ law needs laminar flow.

Key takeawayTemperature Liquids become less viscous when heated; gases become more viscous.

Common misconceptionRadius, not diameter Measure the diameter with a micrometer, then halve it.

Pause and think

Why are the timing marks placed well below the surface in a viscometer?

Summary

$U=\rho_fVg$ .
$F=6\pi\eta rv$ .
$v_t=\frac{2r^2(\rho_s-\rho_f)g}{9\eta}$ .

Viscosity: Resistance of a fluid to flow.
Laminar flow: Smooth flow in parallel layers.
Terminal velocity: Constant speed when forces balance.

HistoryStokes. George Stokes derived his drag law in 1851. Millikan later used it to find the size of oil droplets, and so the charge on the electron.

Fun factThe pitch drop. Pitch looks solid, but it's a liquid with viscosity about

10^{8}

Pa s. In the Queensland pitch-drop experiment, started in 1927, only nine drops have fallen.

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