Weber Number Calculator

Compute the Weber number (We) — the ratio of inertial force to surface tension force. Essential for droplet dynamics, atomization, and multiphase flow.

Weber Number Formula: We = ρ · v² · L / σ

Where: ρ = density (kg/m³), v = velocity (m/s), L = characteristic length (m), σ = surface tension (N/m)

Water droplet (1 m/s, 1mm)
Air jet over water
Oil droplet in water
Spray atomization
Calculating...

Understanding Weber Number

The Weber number (We) is a fundamental dimensionless quantity in fluid mechanics, characterizing the relative importance of fluid inertia compared to surface tension. It is defined as:

We = \frac{\text{Inertial force}}{\text{Surface tension force}} = \frac{\rho \, v^2 \, L}{\sigma}

where ρ = density, v = velocity, L = characteristic length (e.g., droplet diameter), σ = surface tension.

Physical derivation: Inertial force scales as ρ v² L² (dynamic pressure times area), while surface tension force scales as σ L (surface tension times length). Their ratio gives ρ v² L² / (σ L) = ρ v² L / σ.

Typical Regimes & Critical Values

Weber range Physical behavior Example
We < 1 Surface tension dominant, droplets oscillate but stay intact Slow dripping faucet
1 < We < 10 Transition, deformation visible Raindrops falling
10 < We < 100 Significant deformation, bag breakup Spray nozzles
We > 100 Catastrophic breakup, atomization Fuel injectors

The critical Weber number for droplet breakup depends strongly on the Ohnesorge number (Oh = μ / √(ρ σ L)), which accounts for viscosity. For inviscid droplets in a gas, bag breakup typically begins around We ≈ 12. As viscosity increases (higher Oh), the critical We rises. In many engineering correlations, the critical Weber number is expressed as a function of Oh.

Real‑world Example: Raindrop

A typical raindrop of diameter 2 mm (L = 0.002 m) falling at terminal velocity ~9 m/s in air (ρwater = 1000 kg/m³, σ ≈ 0.072 N/m) yields:

We = (1000 × 9² × 0.002) / 0.072 = 2250

Such a high Weber number explains why raindrops are strongly deformed and may break up into smaller droplets during free fall.

Applications in Science and Engineering

  • Atomization & sprays – fuel injection, painting, agricultural sprays (We >> 1 ensures fine droplets).
  • Droplet impact on surfaces – inkjet printing, rain erosion on aircraft, spray cooling.
  • Emulsions and bubble formation – in chemical reactors, the Weber number helps predict drop size in agitated vessels.
  • Microfluidics – droplet generation in microchannels often operates at low We to maintain stability.

Relation to Other Dimensionless Numbers

The Weber number often appears together with the Reynolds number (Re = ρ v L / μ) and the Ohnesorge number. For instance, the Capillary number Ca = μ v / σ = We / Re links viscous, inertial, and surface tension forces. In atomization research, the Ohnesorge diagram (Oh vs Re) is used to classify breakup regimes.

Frequently Asked Questions

There is no single critical value; it depends on the Ohnesorge number (Oh). For inviscid droplets in air, bag breakup starts around We ~ 12. For viscous droplets, critical We can be 20–100. Correlations like the Pilch & Erdman breakup model give Wecrit = 12(1 + 1.077 Oh1.6).

No, all parameters (density, velocity squared, length, surface tension) are positive, so Weber number is always ≥ 0. A zero value would imply either zero velocity or zero length (theoretical limit).

Use consistent SI units: kg/m³, m/s, m, N/m. The calculator assumes these. Any coherent unit system works (e.g., g/cm³, cm/s, cm, dyn/cm) but you must ensure units are consistent (1 dyn/cm = 0.001 N/m).

Common methods include the pendant drop method, Du Noüy ring, Wilhelmy plate, and maximum bubble pressure. For common fluids, tabulated values are available (e.g., water-air ~0.072 N/m at 20°C).

Capillary number Ca = μ v / σ compares viscous force to surface tension. Weber number compares inertial force to surface tension. They are related by Ca = We / Re (where Re is Reynolds number). In low‑Re flows, Ca dominates; in high‑Re flows, We matters more.