Engineering & Fluid Mechanics
Thermal Marangoni Number (Ma) Formula: Thermocapillary Surface-Tension Convection
Calculate the dimensionless thermal Marangoni number Ma = |dsigma/dT| * L * deltaT / (mu * alpha) governing laser weld-pool Penetration, crystal melt growth, thin-film Bénard-Marangoni cells, and microgravity boiling.
Why surface-tension gradients dominate buoyancy in thin liquid films, laser weld pools, and spaceflight!
Named after Italian physicist Carlo Marangoni (1865), thermocapillary flow arises whenever a temperature difference deltaT across length scale L creates a surface-tension gradient |dsigma/dT| that pulls liquid tangentially from hot (low surface tension) toward cold (high surface tension) regions! Comparing this thermocapillary shear drive against viscous momentum damping mu and thermal diffusion alpha yields Ma = (|dsigma/dT| * L * deltaT) / (mu * alpha)!
In Pearson’s (1958) classic thin liquid layer heated from below with a free upper surface, cellular Bénard-Marangoni instability sets in above critical Ma_c ≈ 79.6 even at zero gravity—while in silicone-oil / molten-metal pools (|dsigma/dT| = 0.00018 N/(m*K), L = 0.012 m, deltaT = 15 K, mu = 0.00085 Pa*s, alpha = 9.5e-8 m^2/s), Ma reaches 401,238.4 (intense oscillatory thermocapillary convection, dropping to 106,996.9 when deltaT = 4 K)!
| Thermocapillary Pool / Film Condition | Thermal Marangoni Number Ma | Interfacial Flow Regime (vs Ma_c ≈ 80) |
|---|---|---|
| deltaT = 15 K across L = 12 mm | 401,238.4 | Vigorous Turbulent / Oscillatory Marangoni Jet |
| Reduced Gradient deltaT = 4 K | 106,996.9 | Strong Cellular / Steady Thermocapillary Flow |
| Zero Viscosity mu = 0 Pa·s | Singular Denominator | "Invalid Thermocapillary Properties" |
Thermal Marangoni Number Ma = (|dsigma/dT| * L * deltaT) / (mu * alpha)
Evaluates dimensionless Ma from surface-tension temperature sensitivity |dsigma/dT| (B2, N/m·K), characteristic length L (B3, m), temperature difference deltaT (B4, K), dynamic viscosity mu (B5, Pa·s), and thermal diffusivity alpha (B6, m²/s).
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0, B5 <= 0, B6 <= 0), "Invalid Thermocapillary Properties", ROUND((B2 * B3 * B4) / (B5 * B6), 1))Pearson Thin-Film Critical Onset Check (Ma >= 79.6 Bénard-Marangoni Instability)
Compares Ma against J. R. A. Pearson’s classic insulating free-surface threshold Ma_c = 79.6.
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0, B5 <= 0, B6 <= 0), "Invalid Thermocapillary Properties", IF((B2 * B3 * B4) / (B5 * B6) >= 79.6, "Supercritical Marangoni Convection (Ma >= 79.6)", "Conductive Subcritical Layer (Ma < 79.6)"))Try the example
Use the original values or change them to test your understanding.