Which forces matter at an interface?
Droplets, jets and free surfaces add surface tension to inertia, viscosity and gravity. Assume a Newtonian liquid with constant properties. Surfactants, evaporation, moving contact lines and the surrounding gas are excluded, except where Example 3 adds the gas.
The note does not predict breakup speeds or resonance; it fixes a shared length, speed and measured frequency before comparing forces and times.
Fix the length convention first
| Variable | SI unit | Meaning |
|---|---|---|
| ρ, μ | kg/m³, Pa·s | Liquid density and viscosity |
| σ | N/m | Surface tension of a single interface |
| L, R | m | Representative diameter and spherical radius |
| v | m/s | Specified relative speed |
| g, f | m/s², Hz | Gravity and periodic frequency |
| Δp | Pa | Pressure jump across an interface |
Aerodynamic deformation needs a gas-density, slip-speed Weber number. Examples 1 and 2 use liquid inertia, Example 3 the gas form; never compare both with one threshold.
Derive the groups from stress scales
Construct inertial pressure ρv², viscous stress μv/L, curvature pressure σ/L and gravitational pressure ρgL. Their ratios give Re = ρvL/μ, We = ρv²L/σ and Bo = ρgL²/σ. Since Fr = v/√(gL), Bo = We/Fr².
Oh = μ/√(ρσL) = √We/Re
Bo = We/Fr² = (L/ℓc)², ℓc = √(σ/(ρg))
Ca compares viscosity with surface tension; Oh combines all three stresses without speed. The MIT surface-tension slides put the capillary length ℓc at about 2 mm for air and water. These identities check definitions, not interface shape or wetting.
Exact spherical pressure versus characteristic time
For a stationary spherical single interface, the Young–Laplace jump is Δp = 2σ/R = 4σ/L. σ/L was only a scale; the sphere needs the factor 4. A two-interface soap bubble differs again.
The advection time is tadv = L/v. Balancing ρL²/t² ≈ σ/L gives the inertio-capillary time tcap ≈ √(ρL³/σ), so We = (tcap/tadv)². The visco-capillary time tvc = μL/σ, used in the arXiv review of Ohnesorge units, gives tvc/tcap = Oh. These are scales from a chosen length, not the exact period of a droplet oscillation mode.
For period T = 1/f, St = fL/v = tadv/T. Record what f measures; equal St does not mean equal mode shapes. The NASA Glenn cylinder study at Re = 150 compares with an experimental St of 0.179–0.182, a value specific to that flow.
A shared-input numerical check
Example 1. Four calculators, one input set
Given (round values close to water, chosen to make identities visible): ρ = 1000 kg/m³, μ = 0.001 Pa·s, σ = 0.072 N/m, L = 0.002 m, v = 0.5 m/s, g = 9.81 m/s² and f = 50 Hz. The frequency is an imagined measurement, not predicted from surface tension.
Re = 1000; We = 6.94444; Fr = 3.56961; St = 0.2
Ca = 0.00694444; Oh = 0.00263523; Bo = 0.545
R = 0.001 m → single-sphere Δp = 144 Pa
tadv = 0.004 s; tcap ≈ 0.010541 s; T = 0.020 s
Cross-check (0.010541/0.004)² ≈ 6.94444 and 0.004/0.020 = 0.2. A failed identity usually means mixed diameter and radius or inconsistent definitions.
Real water drops: 2 mm and 100 µm
Example 2. Which drop does gravity flatten?
Given: water at 20 °C with ρ = 998.2 kg/m³ and μ = 1.0000 × 10⁻³ Pa·s (IAEA Table 3.3, saturated liquid used as low-pressure water) and σ = 0.0728 N/m (UCF/OpenStax). The diameters 2 mm and 100 µm are assumed, roughly a raindrop and a spray droplet.
Capillary length ℓc = √(0.0728/(998.2 × 9.81)) = 2.727 mm
| Quantity | L = 2 mm | L = 100 µm |
|---|---|---|
| Bo = ρgL²/σ | 0.538 | 0.001345 |
| Oh = μ/√(ρσL) | 0.002623 | 0.01173 |
| Δp = 4σ/L (Pa) | 145.6 | 2912 |
| tcap = √(ρL³/σ) (s) | 0.010473 | 1.171 × 10⁻⁴ |
| tvc = μL/σ (s) | 2.747 × 10⁻⁵ | 1.374 × 10⁻⁶ |
At 2 mm, close to ℓc, Bo = 0.538 is not negligible and a resting or slowly falling drop departs visibly from a sphere. At 100 µm Bo is about 400 times smaller, surface tension holds the shape and the Laplace pressure rises twentyfold to 2912 Pa. Oh grows as L−1/2 but stays far below one, so viscosity barely damps shape oscillations. Real properties shift tcap by under 1% from Example 1 (0.010473 against 0.010541 s): fine for scaling, not for five-digit checks.
When the surrounding gas does the deforming
The Ansys CFX Theory Guide defines the droplet Weber number with the surrounding-fluid density, the slip velocity and the particle diameter, We = ρF V²slip dP/σ, and Oh with the droplet properties, Oh = μP/√(ρP σ dP). It notes that some authors use the radius instead, that significant deformation starts near We of one, and lists typical regimes: vibrational below 12, bag breakup 12–50, bag-and-stamen 50–100, sheet stripping 100–350 and catastrophic above 350.
Example 3. A water drop in an air stream
Given: σ = 0.0728 N/m, air density 1.2043 kg/m³ (IAEA Table 3.1 value 1.166 kg/m³ at 98.1 kPa, scaled to 101.325 kPa by the ideal-gas law), drop diameter 2 mm and slip speed 10 m/s (assumed).
Weg = 1.2043 × 10² × 0.002 / 0.0728 = 3.309 (below the bag threshold)
Slip speed for Weg = 12: v = √(12σ/(ρg d)) = 19.04 m/s; for Weg = 50: 38.87 m/s
Same Weg = 12 for a 100 µm drop: 85.17 m/s
Liquid-to-gas density ratio 998.2/1.2043 = 829
A liquid-based We would be 829 times larger at the same speed and length, which is why Example 1's We = 6.94444 cannot be read against the 12 threshold. The limits are typical values from one cited classification; a more viscous liquid (larger Oh) or a radius-based definition shifts them.
Size changes and interpretation limits
At fixed properties and speed, reducing L tenfold reduces Re and We by 10 and Bo by 100. Fr grows by √10, while tcap changes by 10⁻³ᐟ². Shrinking size alone therefore does not preserve all force ratios.
- Breakup limits depend on geometry, surrounding fluid and deformation mode; We alone does not determine one.
- A static spherical pressure jump is not the pressure at every point of a moving, distorted droplet.
- Spatially varying σ requires attention to tangential Marangoni stresses.
- Use the identities only with consistent properties and length conventions.
Questions about interface force and time ratios
Should We use the liquid or the gas density?
Use the inertia that deforms the interface: gas density and slip speed for a drop in an air stream (as in the Ansys regimes), liquid density for a jet or a drop hitting a wall. For water and air at 20 °C they differ 829-fold.
Why is the spherical pressure 4σ/L and not σ/L?
σ/L is a scale. Young–Laplace for a sphere gives 2σ/R, which is 4σ/L with L as diameter. Using radius for L, or a two-surface bubble, changes the factor again.
Is tcap the oscillation period of the drop?
No. It is a scale. Actual mode periods carry mode-dependent constants, so use tcap to compare drops, not to predict a measured frequency.
Does Oh change with speed?
No. Oh = μ/√(ρσL) depends only on properties and size, so it classifies a liquid and drop size before the speed is known.
References
- MIT (J. W. M. Bush): Surface tension: capillary pressure, scaling, wetting. Bond number and capillary length of about 2 mm for air and water.
- arXiv 2112.06713: Spreading, Pinching, and Coalescence: The Ohnesorge Units. Oh and the visco-capillary time ηD/Γ.
- Ansys: CFX Theory Guide, Secondary Breakup Models. Gas-based We and Oh definitions and typical breakup regimes.
- UCF Pressbooks (OpenStax College Physics): Surface Tension. Water at 20 °C, 0.0728 N/m.
- IAEA (2008): Thermophysical Properties of Materials for Nuclear Engineering. Tables 3.1 (dry air) and 3.3 (saturated water).
- NASA Glenn / NPARC: Laminar Flow Over a Circular Cylinder. St = fD/U and experimental St 0.179–0.182 at Re = 150.
The round values of Example 1, the drop sizes and the slip speed are our assumptions; printed results are recomputed by the site tests. Checked 2026-10-22.