Ohnesorge Number (Oh): Droplet Formation Regimes in Jets, Sprays, and Microfluidics
The Ohnesorge number (Oh) compares viscous forces to the combined action of inertia and surface tension in liquid jets and droplets. Expressed as Oh = μ / √(ρ σ L), where μ is dynamic viscosity, ρ density, σ surface tension, and L a characteristic length (often nozzle diameter), Oh reveals how easily a jet thins and breaks into droplets. Low-Oh flows behave inertially with rapid atomisation; high-Oh flows damp instabilities, producing thick ligaments or preventing breakup entirely. By uniting viscosity with the Weber number (We = ρU²L/σ) and the Reynolds number (Re = ρUL/μ), Oh frames droplet formation in a single similarity parameter because Oh = √(We)/Re.
This deep dive chronicles the history of Ohnesorge’s experiments, articulates the theory behind jet stability, presents measurement and data-reduction strategies, and surveys applications from aerospace fuel sprays to pharmaceutical inhalers and additive manufacturing. The article also highlights why modern standards (ISO 80000-11) emphasise consistent notation, and how Oh supports predictive simulations and data-driven optimisation of multiphase devices.
Definition, Dimensional Analysis, and Regime Map
The Ohnesorge number is dimensionless: viscosity μ (kg·m⁻¹·s⁻¹) divided by √(ρ σ L) carries the same units. For a given fluid and nozzle, Oh depends only on fluid properties and length, independent of velocity. Typical Oh < 0.1 flows exhibit violent atomisation with satellite droplets, common in diesel injectors. Oh ≈ 0.1–1 marks transitional regimes where viscosity moderates breakup and provides clean droplet streams, ideal for inkjet printing. Oh > 1 indicates viscous dominance; sprays may devolve into dribbling ligaments, while extreme Oh > 10 enables fibre drawing or prevents atomisation entirely.
Visual regime maps often plot Oh versus We or Re. Contours delineate Rayleigh breakup, first-wind-induced breakup, and atomisation zones. Because Oh = √(We)/Re, designers can estimate Oh from easily measured velocity-dependent numbers. For instance, a water jet (μ = 1 mPa·s, ρ = 1000 kg·m⁻³, σ = 0.072 N·m⁻¹) exiting a 100 µm nozzle yields Oh ≈ 0.012, implying negligible viscous damping. Replacing water with glycerol (μ ≈ 1 Pa·s) at identical geometry raises Oh near 12, dramatically altering breakup behaviour. Such comparisons underscore the sensitivity of spray performance to fluid formulation.
Historical Development and Standardisation
The number honours Wolfgang von Ohnesorge, whose 1936 doctoral dissertation at the Technical University of Berlin analysed inkjet breakup for telecommunications printing. Using high-speed photography of vibrating nozzles, he identified viscosity’s stabilising role and proposed the parameter Z = 1/Oh (still used in printing). Ohnesorge’s insights accelerated teleprinter development and laid groundwork for modern drop-on-demand systems. In the mid-20th century, Arthur Taylor and Hans Wierzba extended the concept to fuel sprays, embedding Oh into empirical correlations for breakup length and droplet size.
Standardisation followed as aviation and automotive industries pursued reliable atomisation. Organisations such as ASTM and SAE issued spray-characterisation protocols referencing Oh-derived correlations. ISO 80000-11 recognises Oh, prescribing symbol usage and relationships to Re and We. Contemporary research integrates Oh into open-source simulation platforms and measurement standards, ensuring reproducibility across laboratories investigating sprays, electrohydrodynamic jets, or aerosol generation for medical devices.
Conceptual Foundations and Jet Stability Theory
Linear stability analysis
Rayleigh’s classical analysis of inviscid jets predicts a most-unstable wavelength λ ≈ 4.5 L. Ohnesorge added viscosity, showing that growth rate diminishes with increasing μ. In linear theory, the dimensionless growth rate ω̂ depends on Oh and Re. For Oh ≪ 1, instability mirrors inviscid predictions; for Oh > 1, perturbations decay. This explains why honey forms long threads while water snaps into droplets quickly.
Nonlinear breakup and satellite droplets
Beyond the linear regime, Oh controls pinch-off dynamics and the emergence of satellite droplets. Numerical simulations using Navier–Stokes solvers show that intermediate Oh ≈ 0.05–0.3 minimises satellites—a key requirement for inkjet accuracy. Very low Oh flows exhibit ligament coalescence generating satellites; high Oh flows delay pinch-off, producing elongated tails. These effects translate into droplet size distributions measured by laser diffraction or phase-Doppler analyzers.
Coupling with electric and acoustic forcing
In electrohydrodynamic (EHD) jets and acoustically actuated printers, external fields modify effective surface tension or induce oscillations. Oh remains pivotal because it sets the baseline viscous damping. Designers adjust drive frequencies, voltages, or waveform shapes to compensate for fluids with high Oh, ensuring stable droplet ejection. Comparing Oh-based predictions with capillary-number models guides transitions between steady coating and pulsed ejection.
Measurement Strategies and Data Quality
Calculating Oh requires accurate property data. Laboratories measure viscosity with rotational rheometers or microfluidic viscometers, carefully reporting shear rate because non-Newtonian fluids exhibit shear-thinning that alters μ. Density derives from pycnometers or oscillating U-tube meters. Surface tension is obtained via pendant-drop or Wilhelmy plate methods, with attention to temperature and surfactant dynamics that can depress σ during fast jetting. Characteristic length L often equals nozzle diameter; microscopy verifies manufacturing tolerances since small deviations significantly affect √(σ L).
Documenting uncertainty is essential. Because Oh depends on the square root of σ and L, relative uncertainty contributions halve compared with linear dependencies, but viscosity measurement errors transfer directly. Researchers typically state Oh ± δOh using propagation-of-error formulas. When evaluating droplet size distributions, coupling Oh with Re and We contextualises data across varying velocities. The Reynolds number calculator supports this documentation workflow.
Applications in Technology and Research
Additive manufacturing and printed electronics
Inkjet and aerosol jet printers tune Oh via solvent selection, polymer loading, and temperature control. Printable electronics demand Z = 1/Oh between 1 and 10 to avoid nozzle clogging and ensure precise droplet placement. Emerging metal-organic inks for flexible circuits require balancing Oh with Weber-number constraints to prevent overspray.
Thermal management and aerospace sprays
Spray cooling of electronics and turbine blades relies on droplet sizes predicted from Oh. Low-Oh fluids (liquid nitrogen, water) produce fine mists for rapid heat removal, while higher-Oh coolants (glycol mixtures) need higher velocities or assistive atomisation. Fuel injectors in aviation maintain Oh in regimes that avoid coking while ensuring complete combustion, often pairing with Nusselt-number analyses for downstream heat transfer.
Biomedical aerosols and public health
Nebulisers, metered-dose inhalers, and intranasal vaccines depend on Oh to govern aerosolised droplet size, which influences deposition in respiratory tracts. Formulators adjust viscosity with excipients to tune Oh, balancing stability and patient comfort. Studies of pathogen-laden respiratory droplets also analyse Oh to model breakup during coughing or sneezing, informing ventilation design and infection-control protocols.
Energy transition and sustainability
Renewable energy technologies exploit Oh-guided sprays for catalyst deposition in fuel cells, electrolyzers, and batteries. In carbon capture, spray scrubbers rely on intermediate Oh solutions to maximise gas–liquid contact without flooding. Agricultural sprayers targeting precision application consider Oh to reduce drift and chemical waste, linking droplet metrics with environmental compliance.
Strategic Importance and Future Outlook
As industries adopt data-driven design, Oh enables high-fidelity surrogate models and optimisation loops. Machine-learning frameworks train on Oh-based descriptors to predict droplet size or breakup length, reducing experimental burden. Advanced diagnostics such as x-ray phase-contrast imaging and femtosecond shadowgraphy capture transient jet dynamics, validating Oh-governed simulations.
Future work explores complex fluids—non-Newtonian, viscoelastic, or particle-laden systems—where effective Oh depends on extensional viscosity and interfacial rheology. Researchers develop modified definitions incorporating elastic stresses or surfactant transport, while standards bodies refine reporting templates to include rheological spectra. Mastery of the Ohnesorge number, alongside capillary and Weber numbers, equips teams to innovate in printing, propulsion, healthcare, and climate-tech applications with confidence.