Stokes' Law — Physica

Equivalent forms

v_t = \frac{2 r^2 (\rho_s - \rho_f) g}{9 \eta}

C_d = \frac{24}{Re}

Six-pi-eta-r-v: the most quoted drag formula in physics.

Symbol	Name	SI unit	Dimension	Range
$\eta$	Dynamic Viscosity Fluid viscosity	Pa·s	M·L⁻¹·T⁻¹	0.0001 – 10
$r$	Sphere Radius Radius of the sphere	m	L	0.0001 – 0.01
$v$	Velocity Velocity of sphere relative to fluid	m/s	L·T⁻¹	0 – 1
$F_d$	Drag Forceoutput Viscous drag on the sphere	N	M·L·T⁻²	0 – 0.01

Unit systems

Symbol	SI	CGS	Imperial
$\eta$	Pa·s	P	lb/(ft·s)
$r$	m	cm	ft
$v$	m/s	cm/s	ft/s
$F_d$	N	dyn	lbf

Where it holds

Valid for low Reynolds number (Re < 1), rigid sphere, no-slip boundary, unbounded fluid. Fails for turbulent wakes, non-spherical particles, or near walls.

Dimensional analysis

$[\eta ][r][v] = (M\cdot L^{-1}\cdot T^{-1})(L)(L\cdot T^{-1}) = M\cdot L\cdot T^{-2} \checkmark$ (force)

Discovery

George Gabriel Stokes · 1851

Stokes derived the drag on a sphere in viscous fluid while studying pendulum motion, and used it to compute the settling of fog droplets in air.

Try this

Why does a tiny dust speck fall so slowly through air, while a marble drops like a stone?

A 1 mm radius steel ball (density 7850 kg/m³) falls through glycerin (viscosity 1.5 Pa·s, density 1260 kg/m³). What is its terminal velocity?

Research status: stable

Real-world applications

Millikan oil-drop experiment
Sedimentation rates of clay and silt in water
Centrifuge analysis of cells and macromolecules
Fog and cloud droplet settling in atmospheric physics

Common misconceptions

Stokes' law is NOT valid for everyday-sized objects in air — Re quickly exceeds 1
The drag is linear in v only in the creeping-flow regime, not in general
The factor $6\pi$ assumes a no-slip boundary; slip surfaces give a smaller coefficient $(4\pi for$ perfect slip)

Experimental verification

Millikan's oil-drop experiment (1909) used Stokes' law to determine the charge of the electron by measuring terminal velocities of micron-sized oil droplets in air. Modern sedimentation analyses in colloidal science routinely confirm Stokes' drag to <1%.

Derivation

Starting from the Navier-Stokes equation in the creeping-flow limit (Re ≪ 1), the inertial terms drop, leaving

\nabla p = \eta \nabla ^{2}v

with

\nabla \cdot v = 0

.

Solving in spherical coordinates with no-slip on the sphere and uniform flow at infinity yields the Stokeslet solution.

Integrating the resulting pressure and viscous stress over the sphere surface gives

F_d = 6\pi \eta rv

.

Two-thirds comes from viscous shear, one-third from pressure.

Limiting cases

r \to 0

⟶

F_d \to 0

Vanishingly small sphere feels no viscous drag.

v \to 0

⟶

F_d \to 0

Stationary sphere has no drag.

Re > 1⟶ Formula breaks downAt higher Reynolds numbers, inertial effects dominate and drag scales as

v^{2}

instead of v.

What if…

What if the sphere is twice as big?

Drag doubles at the same velocity, but terminal velocity rises $4\times$ (since gravity scales as $r^{3} and$ drag as r).

What if you use a non-spherical particle?

Drag depends on shape and orientation; for many objects the formula is replaced by $F_d = 6\pi \eta R_h\cdot v$ where R_h is the hydrodynamic radius.

What if Re exceeds 1?

You must use the Oseen correction or fully empirical drag curves; beyond $Re \approx 1000$ , drag scales as $v^{2}$ with $C_d \approx 0.47$ .

1

Steel ball in glycerin

Given ·

\eta:: 1.5
r:: 0.001
v:: 0.01

Find · F_d

Steps

Apply $F_d = 6\pi \eta rv$
$F_d = 6 \times 3.14159 \times 1.5 \times 0.001 \times 0.01$
$F_d \approx 2.83 \times 10^{-4} N$

Result ·

F_d = 6\pi \times 1.5 \times 0.001 \times 0.01 \approx 2.83 \times 10^{-4} N

2

Terminal velocity of a fog droplet

Given ·

\eta:: 0.000018
r:: 0.00001

Find · v_t

Steps

Balance gravity and Stokes drag at terminal velocity
$v_t = 2r^{2}\Delta \rho \cdot g/(9\eta )$
$v_t \approx 1.2\,\mathrm{cm/s}$ — explains why fog hangs in air

Result ·

v_t = 2r^{2}(\rho _w - \rho _air)g/(9\eta ) \approx 2(10^{-5})^{2}(1000)(9.8)/(9 \times 1.8\times 10^{-5}) \approx 0.012\,\mathrm{m/s}

Reynolds Number→Poiseuille's Law→Drag Force (Quadratic)→