Engineering & Science
Capillary Number (Ca) & Bond / Eötvös Number (Bo) Interfacial Scaling
Calculate the viscous-to-interfacial Capillary Number Ca = (mu * U) / sigma and buoyancy-to-capillary Bond (Eötvös) Number Bo = (delta_rho * g * L^2) / sigma for microfluidics, droplets, and enhanced oil recovery.
Diagnose whether surface tension traps droplets—or viscous and buoyancy forces mobilize them!
In microfluidic droplet generators, inkjet nozzles, coating flows, and reservoir enhanced oil recovery (EOR), surface/interfacial tension sigma (N/m) fights to pin menisci and hold droplets spherical against two competing forces: viscous drag quantified by the Capillary Number Ca = (mu * U) / sigma, and gravitational buoyancy quantified by the Bond (Eötvös) Number Bo = (delta_rho * g * L^2) / sigma!
Look at water-oil pore displacement (dynamic viscosity mu = 0.001 Pa*s, velocity U = 0.02 m/s, interfacial tension sigma = 0.030 N/m, density difference delta_rho = 250 kg/m^3, droplet diameter L = 2.0 mm = 0.002 m): Ca = 0.000667 (capillary-dominated below Ca ~ 0.01 unless surfactant flooding slashes sigma 100-fold to 0.0003 N/m -> Ca = 0.066667!), while Bo = 0.3269 (< 1, confirming surface tension keeps 2 mm droplets nearly spherical against gravity)!
| Interfacial Regime (mu = 1 mPa*s, U = 0.02 m/s, d_rho = 250 kg/m^3, L = 2 mm) | Capillary Number Ca | Bond / Eötvös Number Bo | Physical Meniscus Behavior |
|---|---|---|---|
| Untreated Oil-Water Interface (sigma = 0.030 N/m) | 0.000667 (<< 1) | 0.3269 (< 1) | Capillary pinning & spherical drops |
| Surfactant-Flooded Interface (sigma = 0.0003 N/m) | 0.066667 | 32.6888 (>> 1) | Ganglia mobilized & gravity-deformed |
| Zero or Negative Surface Tension sigma <= 0 | Guarded Singularity | "Invalid Interfacial Inputs" | Requires sigma > 0, mu > 0, L > 0 |
Dimensionless Capillary Number Ca = (mu * U) / sigma
Compares viscous shear stress mu * U against interfacial tension sigma (B4, N/m).
=IF(OR(B2 <= 0, B3 < 0, B4 <= 0), "Invalid Interfacial Inputs", ROUND((B2 * B3) / B4, 6))Dimensionless Bond / Eötvös Number Bo = (delta_rho * 9.80665 * L^2) / sigma
Compares hydrostatic buoyancy across characteristic droplet/pore length L (m, cell B6) against surface tension.
=IF(OR(B4 <= 0, B5 < 0, B6 <= 0), "Invalid Interfacial Inputs", ROUND((B5 * 9.80665 * (B6 ^ 2)) / B4, 4))Try the example
Use the original values or change them to test your understanding.