Weber Number (Inertia against Surface Tension)

The Weber number asks one question: is the flow trying to tear this interface apart faster than surface tension can pull it back? The numerator $\rho v^2$ is a dynamic pressure, a stress in pascals. The denominator $\sigma/L$ is also a stress — surface tension has units of force per length, so dividing by a length gives force per area, and that is the Laplace pressure scale of a curved interface. Two stresses, one ratio, no fudge factors. That is all it is.

What it is not is a measured force balance. Nobody has ever put a load cell on the inertia of a droplet and another on its surface tension. $\rho v^2$ is an order-of-magnitude estimate of the disrupting stress, off by a factor of two the moment you ask whether it should have a half in front of it; $\sigma/L$ is an order-of-magnitude estimate of the restoring stress, off by a factor of two the moment you ask whether $L$ is a radius or a diameter. The group is useful because it is crude. If it depended on getting those factors right it would not survive contact with a real nozzle.

Two conventions decide the answer more than any measurement does. Which density. A drop being torn apart in a gas stream takes the gas density, because the gas supplies the momentum flux doing the tearing — this is the aerodynamic Weber number, and it is what a critical value of about 12 refers to. A liquid jet breaking up under its own inertia takes the liquid density. For water and air those differ by a factor near 830, so a Weber number quoted without saying which is not a number at all. Which length. Drop diameter is usual, nozzle orifice diameter and film thickness both appear, and Weber scales linearly with whichever you chose.

The critical value deserves the same scepticism. The number 12 comes from experiments on drops suddenly exposed to a gas stream, and the published band runs from roughly 10 to 20 depending on the release, the acceleration history and who was measuring. Below it the drop deforms without breaking; from about 12 to 50 it goes through bag breakup, where the flattened drop blows out like a windsock and bursts; higher still it strips ligaments from its equator; above a few hundred it disintegrates outright. Those boundaries are descriptions of a continuum, not switches.

And 12 assumes a low-viscosity liquid. A viscous drop resists breakup well past it, because viscosity damps the very surface waves that would otherwise grow — which is exactly the effect the Ohnesorge number was invented to carry. Weber tells you whether breakup happens; Ohnesorge tells you how. Neither is complete on its own.

The most everyday consequence: raindrops have a maximum size near 5 to 6 mm because a bigger one, falling at its own terminal velocity, exceeds the critical Weber number and breaks. That ceiling is not a property of water alone — it is a property of water, air and Earth's gravity together, which is a good reminder that a dimensionless group is a statement about a whole situation and not about a substance.