Weber number#

Named after: Moritz Weber (1871-1951).

$$\text{We} \stackrel{\text{def}}{=} \frac{\rho U^{2}}{\sigma/L} \sim \frac{\text{inertia}}{\text{surface tension}}$$

Measures inertial stress relative to surface-tension stress. It indicates whether interfaces, jets, and droplets remain cohesive or break up.

Name	Symbol	SI units	Dimension
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
surface tension	$\sigma$	$\mathrm{N}\,\mathrm{m}^{-1}$	$\text M\,\text T^{-2}$
characteristic length	$L$	$\mathrm{m}$	$\text L$

Liquid jet breakup

Range	Regime	Description
0 – 8	Rayleigh	Surface tension drives axisymmetric capillary instability. Jet breaks into droplets larger than the jet diameter at a distance far downstream.
8 – 40	first wind induced	Aerodynamic forces enhance Rayleigh instability. Droplets form near the jet diameter; breakup length begins to decrease.
40 – 350	second wind induced	Short wavelength surface waves stripped by aerodynamic pressure produce droplets smaller than the jet diameter.
350 – ∞	atomization	Jet disintegrates immediately at the nozzle exit into a dense spray of very fine droplets. Dominant in fuel injectors.

Secondary droplet breakup

Range	Regime	Description
0 – 12	vibrational	Droplet oscillates but remains intact. Breakup is marginal and occurs only after many oscillation cycles.
12 – 50	bag	Droplet deforms into a thin bag inflated by the flow, which ruptures into fine droplets with a coarser rim fragment.
50 – 100	multimode	Combined bag and stamen breakup. A liquid stamen forms inside the bag, producing a bimodal fragment size distribution.
100 – 350	sheet stripping	Liquid is continuously stripped from the droplet equator as thin sheets, forming many small satellite droplets.
350 – ∞	catastrophic	Droplet shatters almost instantaneously into a cloud of fine fragments. Relevant to high speed impacts and sprays.

Name	Symbol	SI units	Dimension
mass density	\(\rho\)	\(\mathrm{kg}\,\mathrm{m}^{-3}\)	\(\text L^{-3}\,\text M\)
velocity	\(U\)	\(\mathrm{m}\,\mathrm{s}^{-1}\)	\(\text L\,\text T^{-1}\)
surface tension	\(\sigma\)	\(\mathrm{N}\,\mathrm{m}^{-1}\)	\(\text M\,\text T^{-2}\)
characteristic length	\(L\)	\(\mathrm{m}\)	\(\text L\)