The Weber Number — Inertia vs. Surface Tension

“Why are raindrops spherical? Why does an inkjet droplet land precisely without spattering? Why does diesel fuel shatter into micron-scale droplets inside a combustion chamber? The Weber number governs all of these — it is the gatekeeper of every fluid interface.”

1. Introduction

In large-scale flows — rivers, ocean currents, pipe networks — surface tension is negligible and can safely be ignored. But whenever the characteristic length scale shrinks to millimetres or below, or whenever high-speed flow impinges on a fluid interface, surface tension becomes a dominant force. The Weber number (We) quantifies the competition between inertial forces, which tend to deform and break interfaces, and surface tension, which resists deformation and drives interfaces toward minimum-area (spherical) shapes.

The Weber number answers: Does flow have enough inertia to overcome surface tension and rupture the interface? Its value governs droplet formation, jet stability, bubble dynamics, spray atomisation, inkjet printing, and the sinking-funnel problem of IYPT 2027.

2. Historical Note

Moritz Weber (1871–1951) was a German mechanical engineer and professor at the Technische Hochschule Berlin. In 1919 he systematised the theory of dynamic similarity for flows involving viscosity and surface tension, introducing what became known as the Weber number. Weber was motivated by the need to correctly scale model experiments of ship propellers and hydraulic machinery — situations where surface tension effects at model scale could corrupt the results if not properly accounted for.

3. Physical Interpretation

Consider a liquid jet of radius $r$ or a droplet of diameter $d$. Two competing forces act:

• Inertial pressure: $Delta p_{inertia} sim
  ho v^2$ — dynamic pressure driving deformation
• Laplace pressure: $Delta p_{Laplace} = 2sigma/r$ — excess pressure inside a curved interface resisting deformation

Their ratio for characteristic length $L = 2r = d$:

$$We = rac{
ho v^2 L}{sigma} sim rac{Delta p_{inertia}}{Delta p_{Laplace}}$$

We value	Regime	Interface behaviour	Examples
We < 1	Surface-tension dominated	Interface intact, droplet spherical	Morning dew, soap bubble
We ~ 1	Transitional	Droplet deforms but survives	Light drizzle
1 < We < 12	Moderate inertia	Deformation without breakup	Moderate rain, dripping tap
We ≈ 12	Critical breakup	Bag breakup onset for single droplet in airflow	Windscreen raindrop
We >> 12	Inertia dominated	Catastrophic breakup, atomisation	Diesel injector, spray nozzle

4. Rayleigh–Plateau Instability — Jet Breakup

A cylindrical liquid jet of radius $r_0$ is always unstable to perturbations with wavelength $lambda > 2pi r_0$. Rayleigh (1879) showed that the fastest-growing perturbation has wavenumber $kr_0 pprox 0.697$, producing droplets of diameter:

$$d_{drop} pprox 1.89, d_{jet}$$

The growth rate of perturbations scales as $omega sim sqrt{sigma/(
ho r_0^3)}$. The Weber number controls whether the jet breaks in the dripping regime (low We, large droplets form near the nozzle) or the jetting regime (moderate We, jet extends several diameters before breaking) or the atomisation regime (high We, chaotic breakup very close to the nozzle).

5. Droplet Breakup Mechanisms

When a droplet of liquid travels through a gas stream, the aerodynamic pressure distorts and ultimately fragments it. The breakup morphology depends on We:

We range	Breakup mode	Description
We < 12	None (stable)	Droplet deforms but remains intact
12–18	Bag breakup	Droplet inflates into a bag then bursts
18–45	Stripping / multimode	Thin sheets stripped from droplet rim
45–350	Wave crest stripping	Surface waves grow and shed ligaments
> 350	Catastrophic breakup	Explosive fragmentation into fine mist

6. The Ohnesorge Number — Adding Viscosity

When viscosity is also important (e.g., in inkjet printing, polymer sprays, biological fluids), a third parameter is needed. The Ohnesorge number combines We and the Reynolds number Re:

$$Oh = rac{mu}{sqrt{
ho sigma L}} = rac{sqrt{We}}{Re}$$

$Oh$ compares viscous damping to the inertia–surface-tension restoring force. The Ohnesorge diagram (log$,Oh$ vs. log$,We$) maps the complete phase space of drop and jet behaviour into distinct regimes: dripping, jetting, and atomisation. This diagram is the primary design tool for inkjet nozzles, agricultural sprayers, and pharmaceutical inhalers.

7. Bubble Formation

When gas is injected through an orifice into a liquid, the Weber number (based on gas velocity at the orifice) controls bubble size. For $We_{gas} < 2$, discrete spherical bubbles form at the orifice tip (bubbling regime). For $We_{gas} > 2$–$4$, a continuous gas jet penetrates the liquid and breaks into irregular bubbles (jetting regime). The transition is used to control bubble size in bioreactors, metallurgical furnaces, and aerators.

8. Drop Impact on Surfaces

When a droplet impacts a solid surface, the outcome depends on We and the surface wettability:

• Spreading: We > 1; kinetic energy converts to surface energy as a thin lamella spreads outward
• Bouncing: We << 1 on superhydrophobic surfaces; surface tension rebounds the droplet
• Splashing: We > We_cr ≈ 80–500 (depending on surface roughness); unstable rim ejects secondary droplets

The maximum spreading diameter follows empirically:

$$rac{d_{max}}{d_0} pprox 0.61(We + 12)^{1/3}$$

9. IYPT 2027 — Problem P7: Sinking Funnel

10. Surface Tension Values and We at Standard Conditions

Fluid	$sigma$ (N/m) at 20°C	We at v=1 m/s, L=1 mm
Pure water	0.0728	13.7
Soapy water	0.025–0.040	25–40
Ethanol	0.0223	35.9
Mercury	0.485	27.6
Gasoline	0.021	34.3
Diesel fuel	0.027	31.5
Blood plasma	0.050	21.3

11. Diesel Fuel Atomisation

Modern diesel injectors operate at pressures of 2000–3000 bar, producing exit velocities of 300–600 m/s through orifices of 100–300 µm. The Weber number:

$$We = rac{
ho_{fuel} v^2 d_{nozzle}}{sigma_{fuel}} = rac{850 imes 300^2 imes 2 imes10^{-4}}{0.027} pprox 5.7 imes 10^5$$

This enormous Weber number (far exceeding the catastrophic breakup threshold of ~350) ensures complete atomisation into droplets of 5–20 µm within 1–2 mm of the nozzle exit. The resulting high surface area (relative to volume) enables complete combustion in the short time available during the engine cycle.

12. Weber Number in Pharmaceutical Aerosols

Inhaled drug delivery requires droplets of 1–5 µm to reach the alveoli. Pressurised metered-dose inhalers (pMDIs) and nebulisers exploit high We to atomise drug solutions. The challenge: We must be large enough to shatter the liquid into micron-scale droplets, but not so large that the droplets are too small (below 0.5 µm, droplets are exhaled without deposition). This narrow design window is navigated using the Ohnesorge diagram.

13. Related Dimensionless Numbers

Number	Formula	Physical meaning
Ohnesorge Oh	$mu/sqrt{ ho sigma L}$	Viscous damping vs surface tension
Bond/Eötvös Bo	$ ho g L^2/sigma$	Gravity vs surface tension (bubble rise)
Capillary Ca	$mu v/sigma$	Viscous flow vs surface tension (porous media)
Morton Mo	$gmu^4/( ho sigma^3)$	Bubble shape in viscous liquids
Laplace La	$ ho sigma L/mu^2 = We/Oh^2$	Surface tension vs viscosity²/inertia

14. Key Equations Summary

15. Quick-Reference: Weber Number Across Applications

Application	Typical We	Regime / outcome
Morning dew on leaf	< 0.1	Perfect sphere, no deformation
Raindrop (2 mm)	3–8	Stable, slightly oblate
Raindrop (5 mm)	30–60	Strongly deformed, near breakup
Dripping faucet	1–20	Rayleigh breakup, regular drops
Inkjet nozzle	10–100	Design operating window
Agricultural sprayer	50–300	Atomisation, 100–500 µm drops
Diesel injector	$10^5$–$10^6$	Catastrophic atomisation, ~10 µm
Sinking funnel (IYPT P7)	~ 1 (transition)	Discrete bubble vs. jetting