Library / Dimensional analysis and similarity / DIM-06
Dimensional analysis and similarity
Weber number
Ratio of inertial stress to surface tension.
- Ref
- DIM-06
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- We
We = ρU²L/σ compares the inertial stress ρU² with the capillary stress σ/L that surface tension can supply. When We is small, surface tension holds a drop, bubble or jet together; when it is large, inertia deforms and breaks it up. For a drop, use its diameter for L, and take the density and relative speed of the phase that supplies the inertial stress: for a drop in a gas stream, that is the gas. Critical values for break-up are of the order of ten, but depend on the viscosity and on how the load is applied.
Ratio of inertial stress to surface tension. A large value means inertia overcomes surface tension, so a drop or jet breaks up; a small value means surface tension keeps the interface intact.
Calculator
- Dynamic pressure, ½ρU²2,000 Pa
- Capillary pressure scale, σ/L24 Pa
- Speed at which We = 10.1549 m/s
- We with the speed doubled666.7
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A 3 mm water drop moving at 2 m/s relative to the fluid has We = 166.7: inertia is 167 times the surface-tension stress, so the drop would be strongly deformed or broken.
- Doubling the speed multiplies We by 4, while halving the drop diameter only halves it: a spray with small drops is stable, which is why the finest droplets survive.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| U Characteristic speed | 0.01 to 100 m/s | 0.004167 | 416,700 | 416,700 | +10% | +21 % |
| L Characteristic length (for a drop, its diameter) | 0.01 to 1,000 mm | 0.5556 | 55,560 | 55,560 | +10% | +10 % |
| σ Surface tension | 1 to 1,000 mN/m | 12 | 12,000 | 11,990 | +10% | −9.09 % |
| ρ Fluid density | 1 to 2,000 kg/m³ | 0.1667 | 333.3 | 333.2 | +10% | +10 % |
Sweep of Characteristic speed
| Characteristic speed (m/s) | Weber number |
|---|---|
| 0.01 | 0.004167 |
| 0.03162 | 0.04167 |
| 0.1 | 0.4167 |
| 0.3162 | 4.167 |
| 1 | 41.67 |
| 3.162 | 416.7 |
| 10 | 4,167 |
| 31.62 | 41,670 |
| 100 | 416,700 |
Sweep one variable
Variables and how each one contributes
Weber number
What the equation gives you. Everything else feeds it.
Fluid density
The density of the fluid that supplies the inertia, in kg/m³ (water about 1000, air about 1.2). Raises the Weber number in direct proportion. Raising it by 10 % (to 1,100) moves the result by +10 %, from 166.7 to 183.3.
Characteristic speed
The characteristic speed, in m/s: the relative speed between a drop and the surrounding gas, or the jet speed. Raises the Weber number with the square of the speed, so speed is the strongest lever. Raising it by 10 % (to 2.2) moves the result by +21 %, from 166.7 to 201.7.
Characteristic length (for a drop, its diameter)
The characteristic length, in mm: the drop or bubble diameter, or the jet diameter. Raises the Weber number in direct proportion: bigger drops break up more easily. Raising it by 10 % (to 3.3) moves the result by +10 %, from 166.7 to 183.3.
Surface tension
The surface tension of the interface, in mN/m (water in air about 72). Lowers the Weber number in inverse proportion: lower surface tension makes break-up easier. Raising it by 10 % (to 79.2) moves the result by −9.09 %, from 166.7 to 151.5.
Worked example with the values above
- Inertial term: ρU²L = 1,000 × 2² × 3 = 12,000 (with L in mm).
- Divide by the surface tension in mN/m: 12,000 / 72.
- We = 166.67.
Assumptions and limits
- A single interface between two fluids with a constant surface tension.
- L and U are defined with the same convention used for any critical value you compare against, and the density is that of the phase supplying the inertial stress.
Common mistakes
- Entering the surface tension in N/m instead of mN/m: 0.072 N/m is 72 mN/m.
- Using the liquid density for a drop in a gas stream when the gas supplies the inertial stress.
- Using the drop radius where the diameter is intended.
Statistics and uncertainty
The result is a product of powers, so to first order its relative error is the root sum of squares of each input’s relative error multiplied by its exponent (ρ × 1, U × 2, L × 1, σ × −1). The speed term dominates, and surface tension varies with temperature and with any contamination or surfactant, so measure it for the actual liquid.
Where it is used
- Predicting droplet break-up in sprays, atomisers, fuel injectors and rain.
- Matching surface-tension effects in a scale model of a jet, bubble or splash.
Maths you need
- Surface tension: force per length along a liquid interface
- Dynamic pressure: ½ρU², the inertial stress in a flow