Weber Number Formula
Calculate the Weber number using We = rho v squared L / sigma.
Ratio of inertia to surface tension for droplets, sprays, and atomization.
The Formula
The Weber number is a dimensionless number in fluid mechanics that compares the relative importance of a fluid's inertia to its surface tension. It was named after Moritz Weber, a German engineer who developed the theory of dimensional analysis in the early 1900s. When the Weber number is low, surface tension dominates and holds the fluid together in droplets or smooth streams. When the Weber number is high, inertial forces overcome surface tension and the fluid breaks apart into smaller droplets or spray.
Understanding the Weber number is essential in any situation where liquid surfaces deform, break up, or merge. Consider a raindrop falling through the air. At low speeds, surface tension keeps it roughly spherical. As the drop grows larger or falls faster, aerodynamic forces increase and the Weber number rises. When the Weber number exceeds a critical value (typically around 10-12 for a free-falling drop), the drop becomes unstable and breaks apart into smaller droplets.
In engineering, the Weber number plays a critical role in fuel injection systems. Internal combustion engines, jet turbines, and rocket engines all rely on breaking liquid fuel into a fine mist (atomization) to achieve efficient combustion. A higher Weber number at the nozzle exit means finer atomization and better fuel-air mixing. Spray painting, agricultural spraying, and pharmaceutical aerosol delivery all depend on controlling the Weber number to produce the desired droplet size distribution.
The Weber number is also important in the study of bubble dynamics, wave breaking, and liquid jet stability. When two droplets collide, the outcome (merging, bouncing, or shattering) depends on the Weber number of the collision. In inkjet printing, the Weber number must be carefully controlled to produce uniform droplets that travel straight and land precisely.
Like the Reynolds number, the Weber number enables engineers to scale experimental results from small models to full-size systems, as long as the Weber numbers are matched.
Variables
| Symbol | Meaning |
|---|---|
| We | Weber number (dimensionless) |
| ρ | Fluid density (kg/m³) |
| v | Velocity of the fluid or droplet (m/s) |
| L | Characteristic length, typically droplet diameter (m) |
| σ | Surface tension of the fluid (N/m) |
Critical Weber Number Thresholds
| Weber Number Range | Behavior |
|---|---|
| We < 1 | Surface tension dominates, droplets remain stable |
| We ≈ 1 | Transition zone, minor deformation begins |
| We ≈ 10-12 | Critical breakup threshold for free drops |
| We > 12 | Droplet breakup, bag breakup mode |
| We > 100 | Catastrophic breakup, fine atomization |
Example 1
A water droplet (ρ = 1000 kg/m³, σ = 0.0728 N/m) with diameter 3 mm falls at 8 m/s. Will it break apart?
We = ρv²L / σ
We = 1000 × 8² × 0.003 / 0.0728
We = 1000 × 64 × 0.003 / 0.0728
We = 192 / 0.0728
We ≈ 2637. This is far above the critical value of 12, so the droplet will undergo catastrophic breakup.
Example 2
A fuel injector produces diesel droplets (ρ = 850 kg/m³, σ = 0.025 N/m) with diameter 50 μm at 100 m/s. What is the Weber number?
We = ρv²L / σ
We = 850 × 100² × 50 × 10⁻⁶ / 0.025
We = 850 × 10000 × 0.00005 / 0.025
We = 0.425 / 0.025
We = 17. Above the critical breakup threshold, so secondary atomization is likely.
When to Use It
Use the Weber number when you need to:
- Predict whether droplets will remain intact or break apart
- Design fuel injection nozzles for engines and turbines
- Optimize spray systems for agriculture, painting, or coating
- Analyse inkjet printing parameters for consistent droplet formation
- Study wave breaking, bubble dynamics, or liquid jet stability
- Scale fluid experiments between model and full-size systems
The Weber number is often used together with the Reynolds number and Ohnesorge number (Oh = √We / Re) to fully characterize droplet behavior in practical applications.