Weber Number (We): Surface-Tension Scaling for Droplets, Sprays, and Multiphase Flows

Definition, Formula, and Physical Interpretation

The Weber number is defined as We = ρ V² L / σ, where ρ is fluid density, V characteristic velocity, L characteristic length (often droplet diameter), and σ surface tension. When We ≪ 1, surface tension dominates and droplets remain spherical. When We ≫ 1, inertial forces overcome surface tension, promoting deformation and breakup.

Because We combines density (kg·m⁻³), velocity (m·s⁻¹), length (m), and surface tension (N·m⁻¹), dimensional analysis confirms that all units cancel, yielding a dimensionless quantity. The Weber number appears alongside the Froude, Reynolds, and Bond numbers when formulating similarity criteria for fluid systems.

Alternative forms adjust L and σ to suit specific configurations. For example, in thin-film flows the hydraulic diameter may serve as L, while in multi-phase jets the interfacial tension between phases sets σ. Engineers also define a gas Weber number using gas density when gas inertia drives droplet breakup.

Historical Background and Development of Similarity Laws

The Weber number honors German physicist Moritz Weber, who introduced the parameter in the early 20th century while studying capillary waves and ship hydrodynamics. Weber’s work built on Lord Rayleigh’s analysis of liquid jets, which linked surface tension to instability growth. As dimensional analysis matured through the Buckingham π theorem, the Weber number became a standard π-group in multiphase flow research.

Subsequent investigations during the mid-20th century refined We-based correlations for spray breakup, droplet impact, and atomization. NASA, aerospace firms, and combustion researchers used Weber scaling to design fuel injectors capable of producing fine sprays for efficient combustion. In meteorology, We informed hailstone and raindrop breakup models that improve precipitation forecasting.

The dimensionless framework also influenced standardization bodies. ISO 80000-11, covering characteristic numbers, documents We alongside other similarity parameters. Academic curricula still emphasize Weber scaling as part of the foundation for fluid dynamics and heat transfer laboratories.

Conceptual Connections to Other Dimensionless Numbers

Weber, Reynolds, and Froude numbers jointly describe the balance of inertia, viscosity, gravity, and surface tension. For a liquid jet, the Reynolds number determines whether flow is laminar or turbulent, the Weber number gauges surface-tension resistance to breakup, and the Froude number measures gravitational effects. Plots of We versus Re delineate regimes such as Rayleigh breakup, first-wind-induced breakup, and atomization.

The Capillary number Ca = μV/σ, which compares viscous to surface tension forces, complements We. Their ratio defines the Ohnesorge number Oh = √We/Re, often used in inkjet printing to identify stable droplet generation windows. Combined with the Prandtl number and mass transfer coefficients, these groups enable holistic modeling of multiphase transport.

In microgravity environments, Weber and Bond numbers dictate droplet shape and oscillation behavior. Designers of spacecraft propellant management devices rely on We-derived scaling to maintain stable liquid interfaces in the absence of strong gravitational forces.

Critical Weber Numbers

Empirical studies define threshold Weber numbers for specific phenomena. Droplet impact on solid surfaces exhibits transitions between deposition, splash, and rebound at We values between ~10 and ~100, depending on surface roughness and wettability. In gas–liquid atomization, critical We values of 12–50 mark the onset of ligament formation and bag breakup. Understanding these thresholds aids engineers in selecting operating conditions that achieve desired droplet sizes without excessive overspray.

Measurement Techniques and Modeling Approaches

Determining We requires accurate measurements of velocity, length scale, and surface tension. High-speed imaging and laser Doppler velocimetry capture droplet velocities, while optical or X-ray techniques measure diameters. Surface tension is obtained using tensiometers such as pendant drop or Du Noüy ring instruments. Temperature control is essential because σ varies with temperature and composition.

Computational models incorporate Weber number criteria to simulate spray breakup and multiphase flows. Volume-of-fluid and level-set methods track interfaces, while Lagrangian particle models use We-dependent breakup kernels. Engineers calibrate these models against experimental data to ensure accurate predictions. The Reynolds number calculator assists with preliminary parameter estimation before executing detailed simulations.

Scale-model testing employs Weber similarity to replicate full-scale behavior. For example, fire suppression researchers design reduced-scale sprinkler tests by matching We, ensuring droplet breakup dynamics remain representative. Likewise, inkjet printer developers prototype nozzles using fluids with tuned viscosity and surface tension to achieve target We ranges.

Applications Across Industries

Combustion systems depend on Weber scaling to produce fine sprays that mix rapidly with oxidizers. Gas turbine fuel injectors, diesel engines, and rocket propulsion systems all target specific We ranges to balance atomization with droplet penetration. In agriculture, spray nozzles use Weber-based guidelines to achieve droplet sizes that maximize coverage while minimizing drift.

In additive manufacturing and printed electronics, inkjet nozzles must maintain Weber numbers that prevent satellite droplets yet ensure reliable deposition. The Ohnesorge diagram, plotting We against Re, guides material formulation for conductive inks and biomaterials. Microfluidic droplet generators for diagnostics similarly tune We to control droplet size and frequency.

Meteorology and climate science apply Weber scaling to raindrop breakup, which influences radar reflectivity and precipitation measurements. In icing research, We helps determine whether supercooled droplets will shatter upon impact with aircraft surfaces, informing anti-icing system design. Biomedical applications, including inhalation therapies and vaccine sprays, rely on Weber-guided atomization to deliver consistent dose distributions.

Environmental and Safety Considerations

Understanding Weber-number-driven breakup aids environmental protection. Spill response teams evaluate We to predict aerosol formation during high-pressure releases, informing mitigation strategies. Fire safety engineers assess We-driven spray behavior when designing sprinkler heads and water mist systems for energy-efficient suppression.

Strategic Importance and Future Research

Advances in additive manufacturing, electric propulsion, and sustainable aviation fuels heighten the need for precise Weber-number control. Tailoring nozzle geometries, pulse waveforms, and fluid formulations enables efficient atomization while reducing waste and emissions. Machine learning models trained on high-speed imaging datasets promise improved prediction of Weber-dependent breakup regimes.

Emerging microgravity research on the International Space Station explores Weber scaling for propellant management, life support systems, and biological experiments. Improved understanding will support long-duration missions and in-space manufacturing. On Earth, smart spraying systems integrate sensors and control algorithms that monitor We in real time to maintain optimal droplet sizes.

Continue building your dimensionless analysis toolkit by studying the Darcy–Weisbach friction factor, Prandtl number, and Strouhal number explainers, and leverage the Reynolds number calculator to evaluate flow regimes before estimating Weber numbers.