Marangoni number Ma dimensionless ratio of thermocapillary surface tension driven convection to viscous and thermal diffusion forces: Ma = -(dsigma/dT) * dT * L / (mu * alpha). dsigma/dT surface tension gradient with temperature N/m/K (water 25°C -1.5e-4; silicone oils -7e-5; molten steel -5e-4 pure or +1e-4 sulfur surfactant); dT characteristic temperature difference K; L characteristic length m (thin film 0.1-10 mm; welding pool 5 mm; Czochralski 100 mm); mu dynamic viscosity Pa*s; alpha thermal diffusivity m^2/s.

Concentration Marangoni: Ma_c = -(dsigma/dC)*dC*L/(mu*D).

Regimes: Ma<80 sub-critical stable + Pearson 1958 critical Ma_c=80 J Fluid Mech 4:489 free-rigid boundary; 80<Ma<1000 onset Benard-Marangoni cellular hexagonal + thin films + microgravity dominant; 1000<Ma<1e5 strong thermocapillary plumes welding Czochralski thin coating; Ma>1e5 turbulent chaotic surface flow.

Boundary conditions: free-rigid Ma_c=80 Pearson 1958; free-free Ma_c=48 Nield 1964; cylindrical Ma_c~70-100.

Bond number: Bo = Ra/Ma = rho*g*L^2/sigma + Bo>1 gravity Rayleigh-Benard; Bo<1 Marangoni microgravity ISS. History: Carlo Giuseppe Matteo Marangoni (Pavia Italy 1840 - Florence 1925) Universita Pavia + Pisa + Florence Liceo Dante Alighieri physics teacher 1870-1899; 1865 PhD thesis 'Sull'espansione delle goccie d'un liquido galleggianti sulla superficie di altro liquido' + 1869 paper Annalen Physik 143:337 'Uber die Ausbreitung der Tropfen' classic + Marangoni effect + Marangoni number eponym + Florence 30 years teaching + spreading oil-water experiments.

James Thomson (Belfast Ireland 1822-1892) Glasgow University Regius Professor Engineering 1873-1889 elder brother of Lord Kelvin; 1855 paper Phil Mag (4) 10:330 'On certain curious motions observable at the surfaces of wine and other alcoholic liquors' wine tears + ethanol evaporation alcohol gradient + first observation 10 years before Marangoni.

John Royston Anthony Pearson UK Imperial College London + Cambridge; 1958 J Fluid Mechanics 4:489 'On convection cells induced by surface tension' classic + Pearson Marangoni number critical Ma_c=80 + linear stability free-rigid + thin films + microgravity dominant.

D A Nield 1964 + free-free Ma_c=48.

L E Scriven + C V Sternling Carnegie Mellon Pittsburgh 1959-1960; 1959 Nature 187:186 + 1960 Chem Eng Sci 12:98 surface tension instability Sternling-Scriven.

Otto Heiple + W F Roper 1982 Welding Journal 61:97s + sulfur surfactant welding pool + dsigma/dT sign reversal + steel weld penetration Heiple-Roper. Legacy: Velarde-Christov 1995 + Pismen 'Patterns and Interfaces' 2006 Springer + Davis 'Theory of Solidification' 2001 Cambridge + Probstein 'Physicochemical Hydrodynamics' 1994 Wiley.

USA research Marangoni + thermocapillary + crystal growth + ISS microgravity: NASA Glenn Cleveland OH 1941 1500 + Microgravity Science Laboratory + ISS experiments + thermocapillary; NASA Marshall Space Flight Center Huntsville AL 1960 6000 + Spacelab Microgravity Sciences and Materials Lab; NASA Johnson Space Center Houston TX 1961 11000 + ISS operations; NIST Gaithersburg MD 1901 3500 + surface tension reference data; Sandia Albuquerque NM 1949 14000 + Combustion Research Facility + thermocapillary; LBNL Berkeley CA 4400; NREL Golden CO 2900 + thin film solar cell coating.

USA universities Marangoni: MIT Cambridge MA 1861 11700 + Mech Eng + ChemE + materials; Stanford CA 1885 17000 + Mech Eng + George Karniadakis CFD + Eric Shaqfeh; Caltech Pasadena 1891 2400 + GALCIT 1928 + Tim Colonius CFD + John Brady microhydrodynamics; UC Berkeley 1868 45000 + ChemE Cussler heritage + microfluidics; Princeton NJ 1746 8000 + MAE + ChemE Howard Stone microfluidics + Howard Stone microfluidics; Cornell Ithaca NY 1865 25000 + ChemE; Georgia Tech Atlanta 1885 36000; UIUC Urbana 1867 56000; Penn State 1855 50000; UT Austin TX 1883 51000; Univ Michigan Ann Arbor 1817 50000 + ChemE; Northwestern Evanston IL 1851 22000 + ChemE Stephen Davis solidification crystal growth Marangoni; RPI Rensselaer Polytechnic Institute Troy NY 1824 7000 + ChemE; Lehigh Bethlehem PA 1865 7000; Case Western Reserve Cleveland OH 1826 11000.

USA semiconductor industry Czochralski silicon wafer: Intel Santa Clara CA 1968 131000 + silicon wafer fab Hillsboro OR + Chandler AZ + Albuquerque NM + Czochralski crystal; TSMC USA Phoenix AZ 11000 + 5 nm 3 nm wafer fab + Czochralski; GlobalFoundries Malta NY 13000; Micron Technology Boise ID 1978 49000 + DRAM NAND flash + Czochralski; Texas Instruments Dallas TX 1930 33000; Analog Devices Norwood MA 1965 24000; NXP Semiconductors USA 31000; Onsemi Phoenix AZ 1999 25000; Applied Materials Santa Clara CA 1967 33000 + Czochralski equipment furnace; Lam Research Fremont CA 1980 18000; SunEdison St Louis MO 1959 + GlobalWafers acquired 2016 1500 silicon wafer; SUMCO Phoenix AZ 1500 Japan-based silicon wafer; Wolfspeed Durham NC 4000 SiC silicon carbide.

USA welding industry: Lincoln Electric Cleveland OH 1895 11000 + welding equipment + electrodes; Miller Electric Appleton WI 1929 + ITW 4000; Esab Annapolis MD 12000; Hobart Welding Troy OH 1917 1500; Tweco Florence SC 200; Air Liquide Welding USA 4000; Lincoln Welding USA 11000; Praxair Connecticut Danbury CT 1907 27000 + Argon Helium welding gas; Air Products Allentown PA 1940 21000.

USA laser welding 3D printing: 3D Systems Rock Hill SC 1986 1700; Stratasys Eden Prairie MN 1989 2700; Desktop Metal Burlington MA 2015 1100; HP Multi Jet Fusion Palo Alto 51000; GE Additive Avio Aero collaboration; Trumpf USA Farmington CT 4000; IPG Photonics Oxford MA 1990 5000 fiber laser industrial.

USA wine industry tears: Constellation Brands Victor NY 1945 9000 + Robert Mondavi Napa Valley + Kim Crawford; E&J Gallo Modesto CA 1933 7000; Treasury Wine Estates St Helena CA 3500; Napa Valley Wine Train; Robert Mondavi Winery Napa Valley 1966; Kendall-Jackson Wine Estates Santa Rosa CA 1982 1500; Sonoma Cutrer Wines Windsor CA 1981 100.

Marangoni formula

Definition: Ma = -(dsigma/dT) * dT * L / (mu * alpha).

Concentration: Ma_c = -(dsigma/dC) * dC * L / (mu * D).

Critical: Ma_c=80 Pearson 1958 free-rigid + Ma_c=48 Nield 1964 free-free.

Bond: Bo = Ra/Ma + Bo>1 gravity dominant; Bo<1 Marangoni microgravity.

History

Carlo Marangoni (Pavia 1840 - Florence 1925) Universita Pavia + Florence physics teacher 1870-1899 + 1865 PhD thesis Pavia + 1869 Annalen Physik 143:337 classic + Marangoni effect + Marangoni number eponym. James Thomson (Belfast 1822-1892) Glasgow + 1855 Phil Mag 10:330 wine tears Lord Kelvin elder brother. Pearson 1958 J Fluid Mech 4:489 critical Ma_c=80. Nield 1964 free-free 48. Sternling-Scriven Carnegie Mellon 1959-1960. Heiple-Roper 1982 sulfur welding.

USA research: NASA Glenn Cleveland 1500 Microgravity Lab + ISS experiments; NASA Marshall Huntsville 6000 Spacelab; NASA Johnson Houston 11000; NIST Gaithersburg 3500; Sandia 14000; NREL 2900 thin film solar; LBNL 4400.

USA universities: MIT 11700; Stanford 17000 Karniadakis Shaqfeh; Caltech 2400 GALCIT Colonius Brady; UC Berkeley 45000; Princeton 8000 Stone microfluidics; Cornell 25000; Georgia Tech 36000; UIUC 56000; Penn State 50000; Michigan 50000; Northwestern 22000 Davis solidification Marangoni; RPI 7000; Lehigh 7000; Case Western 11000.

USA semiconductor: Intel 131000 Hillsboro Chandler Albuquerque; TSMC USA Phoenix 11000; GlobalFoundries Malta NY 13000; Micron Boise 49000; TI Dallas 33000; Analog Devices 24000; NXP USA 31000; Onsemi 25000; Applied Materials Santa Clara 33000 Czochralski; Lam Research 18000; SunEdison GlobalWafers 1500; SUMCO Phoenix 1500; Wolfspeed Durham 4000 SiC.

USA welding: Lincoln Electric 11000; Miller Electric ITW 4000; Esab 12000; Hobart 1500; Praxair 27000; Air Products 21000.

USA 3D printing: 3D Systems 1700; Stratasys 2700; Desktop Metal 1100; HP MJF 51000; Trumpf USA 4000; IPG Photonics 5000.

USA wine: Constellation Brands 9000; E&J Gallo 7000; Treasury Wine Estates 3500; Kendall-Jackson 1500.

Marangoni values table

• Ma = -(dsigma/dT)*dT*L/(mu*alpha)
• Ma_c = -(dsigma/dC)*dC*L/(mu*D): concentration
• Ma<80 sub-critical
• Ma=80 Pearson 1958: free-rigid
• Ma=48 Nield 1964: free-free
• 80-1000 onset: cellular
• 1000-1e5 strong: plumes
• Ma>1e5 turbulent
• Water 25°C dsigma/dT=-1.5e-4
• Silicone oils -7e-5
• Molten steel -5e-4
• Steel sulfur +1e-4: surfactant
• Silicon melt -1.5e-4
• Mercury -2.3e-4
• Bo = Ra/Ma
• Bo>1 Rayleigh
• Bo<1 Marangoni microgravity
• Marangoni Italy 1840-1925
• Thomson 1855: wine tears
• Pearson 1958: Ma_c=80
• Heiple-Roper 1982: sulfur
• USA NASA Glenn 1500
• USA Intel 131000: Czochralski
• USA TSMC Phoenix 11000
• USA Lincoln Electric 11000
• USA Constellation 9000: wine

Numerical examples

Example 1 - Water thin film. Ma=5357 strong plumes. USA: MIT Mech Eng + Stanford labs.

Example 2 - Silicone oil microgravity. Ma=7000. USA: NASA Glenn Microgravity Lab + ISS experiments.

Example 3 - Welding pool steel. Ma=5.6e7 turbulent. USA: Lincoln Electric 11000 + Miller Electric + Esab + Praxair.

Example 4 - Silicon Czochralski. Ma=7.1e7. USA: Intel 131000 + TSMC Phoenix + Micron + Applied Materials Czochralski.

Example 5 - ISS microgravity water. Ma=7.1e8. USA: NASA Marshall Spacelab + ISS Microgravity Sciences.

Limits and considerations

1. Boundary conditions: free-rigid Pearson Ma_c=80; free-free Nield 48.
2. Bond number Bo: Bo>1 gravity Rayleigh-Benard; Bo<1 Marangoni microgravity.
3. Crispation Cr: surface deformation modify Ma_c.
4. Surfactant: surface-active CMC reverse dsigma/dT sign.
5. Dynamic surface tension: time-dependent + Maxwell viscoelastic.
6. Multi-component evaporation: solvent + concentration Marangoni.
7. Non-uniform heating: laser pulse + Czochralski hot zone.
8. Solidification: Czochralski striations + ribbon defects.
9. MHD magnetic damping: magnetic field reduce Marangoni.
10. Buoyancy-Marangoni coupling: Bo~1 + Nield 1964.

Common errors

Sign dsigma/dT: pure liquids negative + surfactant reverse positive.

Boundary conditions: free-rigid 80 vs free-free 48.

Marangoni vs Rayleigh: Bo>1 gravity vs Bo<1 microgravity.

Concentration vs thermal: dsigma/dT vs dsigma/dC.

Welding sulfur: Heiple-Roper 1982 sulfur surfactant sign reversal.

Practical applications

1. Czochralski silicon wafer: USA Intel 131000 Hillsboro Chandler + TSMC Phoenix 11000 + Micron Boise 49000 + Applied Materials Santa Clara Czochralski + GlobalWafers SUMCO + Wolfspeed SiC.
2. Welding pool TIG MIG laser: USA Lincoln Electric 11000 + Miller ITW 4000 + Esab 12000 + Hobart + Praxair 27000 + Air Products 21000 Argon Helium gas.
3. 3D printing metal SLM additive manufacturing: USA 3D Systems 1700 + Stratasys 2700 + Desktop Metal 1100 + HP MJF + Trumpf USA + IPG Photonics 5000 fiber laser.
4. Microgravity ISS experiments: USA NASA Glenn Microgravity Lab 1500 + NASA Marshall Spacelab 6000 + NASA Johnson 11000 + Constellation Mars Artemis programs.
5. Wine tears alcoholic beverages: USA Constellation Brands 9000 + E&J Gallo 7000 + Treasury Wine Estates 3500 + Robert Mondavi Napa + Kendall-Jackson + Sonoma Cutrer.
6. Soldering electronics PCB: USA Intel + TI + Analog Devices + automotive electronics.
7. Coating thin film solar cell CIGS perovskite: USA NREL Golden + First Solar Tempe AZ CIGS + SunPower CdTe.
8. Laser welding marine: USA Newport News Shipbuilding HII + Bath Iron Works General Dynamics.
9. Solar cells thin film deposition: USA First Solar 7400 CdTe Tempe + SunPower San Jose 4000.
10. Universities thermocapillary research: MIT + Stanford + Caltech + UC Berkeley + Princeton Stone microfluidics + Cornell + Georgia Tech + UIUC + Penn State + Michigan + Northwestern Davis solidification Marangoni + RPI + Lehigh + Case Western.

Updated July 2026