Weber Number Calculator

The Weber number is what tells you whether a droplet will hold itself together or fly apart. It compares the inertia trying to deform a liquid surface to the surface tension trying to keep it intact. Get the Weber number wrong by an order of magnitude in a fuel injector and the engine misfires; get it wrong in an inkjet head and your printout is a smudge. Moritz Weber, a German engineer working on ship hydrodynamics in the early 1900s, formalized this dimensionless group as part of the dimensional-analysis framework that made modern fluid mechanics possible.

The formula:

We = ρ · v² · L / σ

Where ρ is fluid density (kg/m³), v is velocity (m/s), L is a characteristic length (typically droplet diameter, m), and σ is surface tension (N/m, equivalent to J/m²). The Weber number is dimensionless.

Why surface tension and inertia?

A drop of fluid in motion is fighting two things: aerodynamic forces (or, in shear flow, viscous forces) trying to stretch and deform it, and surface tension trying to minimize its surface area. The inertial pressure on a moving drop scales as ρv², and the restoring stress from surface tension scales as σ/L. The ratio of these is the Weber number. When We is small, surface tension wins and the drop stays roughly spherical. When We is large, inertia wins and the drop deforms and eventually breaks apart.

Critical breakup thresholds:

For a free droplet exposed to a uniform airflow, established breakup regimes:

These thresholds vary by 20-50% depending on the source’s experimental conditions and the Ohnesorge number Oh = μ/√(ρσL) (which captures viscous effects). For low-viscosity fluids like water and ethanol, Oh < 0.1 and the values above hold well. For viscous fluids like glycerin, Oh > 1 and the critical We values shift higher.

Worked example, raindrops:

A raindrop falling at terminal velocity. For a 3 mm raindrop, v_terminal ≈ 8 m/s. Water: ρ = 1000 kg/m³, σ = 0.072 N/m.

We = 1000 · 8² · 0.003 / 0.072 = 192 / 0.072 = 2,667

That’s well above the catastrophic threshold. So why don’t raindrops shred mid-air? Because at the same time the drop is decelerating: the aerodynamic force balances gravity at terminal velocity, but it’s the relative velocity to the surrounding air that drives breakup, and at terminal velocity that relative velocity is constant rather than growing. Real raindrops do break up at sizes above 4-5 mm — which is why you never see truly enormous raindrops; they explode into smaller ones in flight.

Worked example, fuel injection:

A diesel injector spray with 200 m/s droplet velocity, 20 μm droplet diameter. Diesel: ρ ≈ 820 kg/m³, σ ≈ 0.025 N/m.

We = 820 · 200² · 20×10⁻⁶ / 0.025 = 656 / 0.025 = 26,240

Strongly catastrophic. Which is what you want in a diesel engine: the injector pressure pushes 250 MPa fuel through a tiny orifice, achieving We values in the thousands so the spray breaks into μm-sized droplets that burn quickly and cleanly. Lower-We sprays (e.g., from a port fuel injector at 5 MPa) produce coarser droplets and dirtier combustion.

Worked example, inkjet printing:

A piezoelectric inkjet drop: v ≈ 5 m/s, droplet diameter 25 μm, ink density ≈ 1000 kg/m³, σ ≈ 0.035 N/m.

We = 1000 · 5² · 25×10⁻⁶ / 0.035 = 0.625 / 0.035 = 17.9

In the bag-breakup range — not what an inkjet wants. To produce a single clean droplet rather than a bursting bag, the printhead must operate just above the ejection threshold (We ≈ 4-12) but below the disruption threshold. This narrow window of acceptable Weber numbers is one of the harder optimizations in inkjet printing, and explains why printheads are characterized by Oh-We “operability diagrams” in product specs.

Other dimensionless companions:

The Weber number rarely tells the whole story alone. In multi-phase flow, it works with:

• Reynolds number Re = ρvL/μ (viscous vs inertial). High Re means turbulence; Weber breakup criteria above assume Re is moderately high (Re > 100 typically).
• Ohnesorge number Oh = μ/√(ρσL) = √(We)/Re, captures viscous effects on breakup. For Oh > 1, viscosity smooths out breakup and shifts critical We higher.
• Capillary number Ca = μv/σ (viscous vs surface tension). Used in microfluidics where Re is small.
• Bond number Bo = ρgL²/σ (gravity vs surface tension). Tells you if a drop will stay attached to a ceiling or fall.

For pure aerodynamic breakup at moderate viscosity, We alone is the controlling parameter. For viscous fluids or strong shear, Oh and Ca matter too.

Where else We shows up:

• Atomizers and sprays: pesticides, paint, perfume, asthma inhalers all have target droplet sizes designed by controlling We.
• Combustion: gas turbines, rocket engines, IC engines all rely on Weber-controlled fuel atomization.
• Coastal engineering: wave breaking at the beach is partially a Weber-criteria problem (with gravity dominating via Froude).
• Microfluidics: droplet generation in PDMS chips for biological assays uses controlled We to make uniform droplets.
• Pharmaceutical: nasal sprays target droplet sizes (typically We ≈ 200) for optimal nasal mucosa deposition.

Limits:

For very small drops (sub-100 μm in air), viscous effects can dominate before We breakup occurs. For very high-pressure or supersonic flows, compressibility matters (Mach number takes over). For non-Newtonian fluids like polymer solutions, the breakup physics is qualitatively different and We is insufficient.