Weber Number — Physica

Equivalent forms

We = \frac{\text{inertial force}}{\text{surface tension force}}

We = Ca \cdot Re

Inertia versus the skin of a liquid — the number that decides droplets, sprays, and splashes.

Symbol	Name	SI unit	Dimension	Range
$\rho$	Density Fluid mass density	kg/m³	M·L⁻³	0.1 – 2000
$v$	Velocity Relative velocity between fluid and surroundings	m/s	L·T⁻¹	0 – 100
$L$	Characteristic Length Droplet diameter or jet width	m	L	0.00001 – 0.1
$\sigma$	Surface Tension Interfacial surface tension	N/m	M·T⁻²	0.01 – 0.5
$We$	Weber Numberoutput Dimensionless ratio of inertia to surface tension	dimensionless	1	0 – 100

Unit systems

Symbol	SI	CGS	Imperial
$\rho$	kg/m³	g/cm³	slug/ft³
$v$	m/s	cm/s	ft/s
$L$	m	cm	ft
$\sigma$	N/m	dyn/cm	lbf/ft
$We$	dimensionless	dimensionless	dimensionless

Where it holds

Applies wherever inertia competes with surface tension: droplet breakup, spray atomization, bubble formation, splashing. Critical breakup Weber numbers depend on viscosity (Ohnesorge number) for very viscous liquids.

Dimensional analysis

$[\rho v^{2}L/\sigma ] = (M\cdot L^{-3})(L^{2}\cdot T^{-2})(L)/(M\cdot T^{-2}) = 1 \checkmark$ (dimensionless)

Discovery

Moritz Weber · 1919

Moritz Weber, a German professor of naval mechanics, formalized this group while developing the theory of dynamic similarity for capillary and droplet phenomena, naming it after the work he systematized.

Try this

How fast must a raindrop fall before it shatters into a mist?

A water droplet 0.002 m across moves through air (ρ = 1.2 kg/m³) at 15 m/s. With surface tension 0.072 N/m, what is the Weber number — and will it break up?

Research status: stable

Real-world applications

Fuel injection and spray atomization in engines
Inkjet printing drop formation
Raindrop size limits and breakup in the atmosphere
Agricultural and pharmaceutical spray design

Common misconceptions

A large Weber number does not mean high surface tension — it means inertia overwhelms it
Weber number alone does not fix breakup for viscous liquids; the Ohnesorge number $Oh = \mu /\sqrt{\rho \sigma L}$ also matters
It is not the same as the capillary number, which compares viscous (not inertial) forces to surface tension

Experimental verification

High-speed imaging of droplets in airflow shows the bag, multimode, and shear breakup regimes at the predicted critical Weber numbers

(We \approx 12

, 50, 100+). Inkjet and fuel-spray studies use We to set drop-on-demand conditions.

Derivation

Compare the disrupting inertial (dynamic) pressure

\rho v^{2}

acting over the cross-section with the restoring capillary pressure

\sim \sigma /L

from surface tension.

The ratio

(\rho v^{2})/(\sigma /L) = \rho v^{2}L/\sigma is

the Weber number, expressing how strongly inertia tries to overcome the cohesive skin.

Limiting cases

We ≪ 1⟶ Surface tension dominatesThe drop stays spherical and intact; capillary forces win.

We \approx 12

⟶ Onset of bag breakupAerodynamic forces begin to deform and rupture the droplet (critical Weber number for breakup in a gas stream).

We ≫ 100⟶ Catastrophic atomizationInertia shreds the fluid into a fine spray of much smaller droplets.

What if…

What if the drop were twice as large?

Weber number doubles $(We \propto L)$ ; large drops break up more easily, which is why raindrops have a maximum stable size around 4-5 mm.

What if surface tension were halved (add surfactant)?

Weber number doubles, promoting breakup and finer sprays — exactly why surfactants are added to spray formulations.

1

Will a raindrop break up?

Given ·

\rho:: 1.2
v:: 15
L:: 0.002
\sigma:: 0.072

Find · We

Steps

Use the air density relative to the drop: $\rho = 1.2\,\mathrm{kg/m}^{3}$
$We = (1.2)(225)(0.002)/0.072$
$We \approx 7.5$ — below the critical $\sim 12$ , so the drop deforms but holds together

Result ·

We = \rho v^{2}L/\sigma = 1.2 \times 15^{2} \times 0.002 / 0.072 = 1.2 \times 225 \times 0.002 / 0.072 = 7.5

Young-Laplace Equation→Reynolds Number→Jurin's Law (Capillary Rise)→