Surface tension and unsteady flow: connect force ratios to time ratios

Which forces matter at an interface?

A droplet, jet or free surface involves surface tension as well as inertia, viscosity and gravity. Assume a Newtonian liquid with constant density, viscosity and interfacial tension. Surfactant gradients, evaporation, moving contact lines and surrounding-gas effects are outside this simple scale comparison.

This article does not predict a breakup speed or a resonance frequency. It shows how to specify a shared length, relative speed and measured frequency before comparing forces and times.

Fix the length convention first

Aerodynamic deformation may instead require a Weber number based on gas density and relative speed. The example below uses a liquid-inertia convention. Do not compare it directly with a gas-based breakup threshold without changing the definitions.

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.

Ca compares viscosity with surface tension. Oh combines inertia, viscosity and surface tension without depending on speed. These identities check definitions; they do not establish interface shape or wetting conditions.

Exact spherical pressure versus characteristic time

For a stationary spherical single interface, the Young–Laplace jump is Δp = 2σ/R = 4σ/L. The earlier σ/L is only a scale: the spherical result needs its factor of 4. A thin soap bubble has two interfaces and cannot use this single-interface formula unchanged.

The advection time is tadv = L/v. Balancing inertial and restoring-pressure scales, ρL²/t² ≈ σ/L, gives tcap ≈ √(ρL³/σ). Consequently We = (tcap/tadv)². This capillary time is a scale based on a chosen length, not the exact period of a particular droplet oscillation mode.

For period T = 1/f, St = fL/v = tadv/T. Record what oscillation f measures. Equal St values in different geometries do not establish equal mode shapes or resonance behavior.

A shared-input numerical check

Take illustrative 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 supplied as an imagined measurement, not predicted from surface tension.

Cross-check (0.010541/0.004)² ≈ 6.94444 and 0.004/0.020 = 0.2. A failed identity can reveal mixed diameter/radius conventions or inconsistent density and speed definitions.

Size changes and interpretation limits

At fixed properties and speed, reducing L by a factor of 10 reduces Re and We by 10 and Bo by 100. Fr grows by √10, while tcap changes by 10⁻³ᐟ². A smaller geometry changes the gravity–capillarity balance as well. 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.