How to calculate Eotvos Bond number Eo gravity surface tension

Calculate Eotvos number Eo (Bond) = (rho_h-rho_l)*g*L^2/sigma gravity to surface tension ratio bubbles droplets multiphase flow. Eo<0.1 spherical; >100 cap large flat. Lorand Eotvos 1886 Hungary. NIST + MIT + Stanford + ExxonMobil + Dow.

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How to calculate Eotvos number (Eo) also known as Bond number (Bo) dimensionless multiphase fluid dynamics ratio measuring gravitational forces (differential weight between two phases at different densities) to surface tension forces (interfacial capillarity): Eo = (rho_h - rho_l) * g * L^2 / sigma. rho_h heavy phase density kg/m^3 (water 1000 at 20°C); rho_l light phase kg/m^3 (air 1.2 at 20°C); g 9.81 m/s^2 Earth; L characteristic length m; sigma interfacial surface tension N/m (water-air 0.0728 N/m at 20°C; mercury-air 0.486 N/m; oil-water 0.025-0.05 N/m).

Capillary length L_c = sqrt(sigma/(delta_rho*g)) natural length scale; water-air 20°C L_c=2.71 mm. Eo = (L/L_c)^2.

Regime classification: Eo100 strong gravity flat interface.

Lorand Eotvos (Hungary 1848-1919) Budapest University Hungarian physicist pioneer geophysics + capillarity 1886 Wiedemann's Annalen Eotvos rule sigma*V^(2/3) = K_E*(T_c-T).

Wilfrid Norman Bond (UK 1897-1937) Cambridge University 1928 Phil Mag 'Bubble in a viscous liquid' Bond number eponym. Clift-Grace-Weber 1978 'Bubbles, Drops, and Particles' Academic Press classic textbook.

USA universities + national labs: NIST National Institute Standards Technology Boulder CO + Gaithersburg MD 1901 3000 thermophysical properties + REFPROP database surface tension; MIT Massachusetts Institute Technology Cambridge MA 1861 11000 + Department Mechanical Engineering + Department Chemical Engineering + Department Materials Science + Howard Brenner transport phenomena + Patrick Doyle microfluidics + David Hardt microfluidics; Stanford University Palo Alto CA 1885 17000 + Mechanical Chemical Engineering + Juan Santiago microfluidics + Howard Stone fluid dynamics; Caltech California Institute Technology Pasadena CA 1891 2300 + GALCIT + John Brady transport phenomena + Mark Davis catalysis; Harvard School Engineering Applied Sciences SEAS 1847 + Howard Stone fluid dynamics + David Weitz soft matter physics; UC Berkeley Lawrence Berkeley National Lab LBNL 1868 45000 Chemical Engineering + David Liem; Princeton 1746 Chemical Biological Engineering + Howard Stone soft matter; University Wisconsin Madison 1848 Chemical Engineering Bird Stewart Lightfoot 1960 textbook + Stuart Berg modern; Cornell Ithaca NY 1865 + School Chemical Biological Engineering + Lynden Archer; University Pennsylvania 1740 + Bioengineering + School Engineering Applied Science + Howard Stone microfluidics;

University Minnesota Twin Cities 1851 + Chemical Engineering Materials Science + Lanny Schmidt + Frank Bates; Carnegie Mellon Pittsburgh 1900 + Chemical Engineering Lorenz Biegler optimization.

USA national labs: NREL Golden CO 1977 3000 renewable + biorefinery + biomass conversion catalysis; ORNL Oak Ridge TN 1943 5000 + Spallation Neutron Source SNS + UHV beamlines + bioenergy; ANL Argonne National Lab Lemont IL 1946 + Advanced Photon Source APS synchrotron + catalysis + chemistry; LLNL Lawrence Livermore CA 1952 + High Performance Computing + CFD multiphase flow; SNL Sandia Albuquerque NM + Livermore CA 14000 + Combustion Research Facility CRF Livermore 100 hydrogen flames + droplet research; LANL Los Alamos NM 1943 12000 nuclear + materials + UHV; PNNL Pacific Northwest Richland WA 1965 5000 environmental + chemistry + catalysis.

USA chemical industry petrochemical: ExxonMobil Spring TX 1999 62000 + ExxonMobil Chemical 22000 olefins + polymers + heat exchanger + bubble columns reactor + Baytown TX 590000 bpd refinery + Beaumont TX + Baton Rouge LA + Joliet IL refineries; Chevron San Ramon CA 1879 45000 oil + Chevron Phillips Chemical 8000 petrochemical bubble columns + Pascagoula MS + Richmond CA refineries; Phillips 66 Houston TX 14000; Marathon Petroleum 17000 13 refineries; Valero Energy 10000 15 refineries; Halliburton Houston TX 40000 oilfield services + multiphase flow analysis; Schlumberger Houston TX 99000 services + reservoir multiphase; Baker Hughes Houston TX 55000 oilfield services.

USA chemical companies: Dow Chemical Midland MI 1897 36000 + olefins + polymers + reactor design; DuPont Wilmington DE 1802 23000 + Kevlar Teflon + materials; LyondellBasell 19000 + polymers; Chemours Wilmington DE 6000 + TiO2 + Krytox; Sherwin-Williams 60000 paints + droplets atomization; PPG Industries Pittsburgh PA 50000 paints + glass + reactor; Eastman Chemical Kingsport TN 14000; Air Products Allentown PA 1940 19000 industrial gases + helium + cryogenic; Linde Danbury CT subsidiary Ireland parent 80000 gases + reactor; Praxair Danbury CT Linde subsidiary 2018; Honeywell Performance Materials Charlotte NC 13000 + UOP catalysts + bubble columns; Westinghouse Pittsburgh PA + nuclear boilers; GE Power + Steam boilers + heat exchanger; Trane Carrier Honeywell Johnson Controls + Daikin HVAC.

USA pharmaceutical drug delivery aerosol pMDI DPI: Pfizer NYC 1849 80000 + drug synthesis + research; Merck Kenilworth NJ 1891 70000; Johnson Johnson New Brunswick NJ 1886 130000 + JJ Janssen + Cilag; AbbVie 50000; Bristol Myers Squibb 30000; Eli Lilly 35000; Gilead Foster City CA 16000; Regeneron Tarrytown NY 12000; Moderna Cambridge MA 5000 mRNA + Lipid nanoparticle LNP; Vertex Pharmaceuticals Boston MA 1989 5000; USA respiratory pMDI DPI inhaler manufacturers: Boehringer Ingelheim USA Ridgefield CT pMDI Spiriva tiotropium + Atrovent ipratropium + Striverdi olodaterol; GSK USA Philadelphia PA pMDI Ventolin albuterol + Seretide DPI Diskus salmeterol fluticasone; AstraZeneca USA Wilmington DE + Cambridge UK + Symbicort budesonide formoterol DPI Turbuhaler; Glaxo Wellcome Research Triangle Park NC; Mylan Canonsburg PA generic inhalers; Teva USA Parsippany NJ generic + brand respiratory; Cipla USA generic; Lupin USA generic.

USA microfluidic + lab-on-chip droplet: 10x Genomics Pleasanton CA 2012 3000 droplet microfluidic single-cell sequencing Chromium platform; BioFire Diagnostics Salt Lake City UT 1990 1000 BioMerieux subsidiary FilmArray syndromic + Filmarray respiratory; Fluidigm South San Francisco CA 2003 microfluidic genomics; Bio-Rad Hercules CA 1952 8000 microfluidic + droplet PCR digital ddPCR; QIAGEN Germantown MD subsidiary German parent 6000 microfluidic + droplet PCR; Beckman Coulter Brea CA 1935 + microfluidic + flow cytometry.

USA microbubble ultrasound contrast agents: Lantheus Medical Imaging North Billerica MA + Definity perflutren + Optison; GE HealthCare Chicago IL 51000 + ultrasound + contrast research; Bracco Diagnostics USA Princeton NJ subsidiary Italian parent + Lumason perflutren contrast; Targeson La Jolla CA targeted microbubbles + Microbubbles Engineering BioCenter MA + UCB BioPharma + Lumason imaging.

Eotvos Bond formula

Eo = Bo = (rho_h - rho_l) * g * L^2 / sigma. L_c = sqrt(sigma/(delta_rho*g)) capillary length. Water-air 20°C L_c=2.71 mm; mercury-air 1.91 mm. Eo = (L/L_c)^2.

History and context

Lorand Eotvos Hungary 1848-1919 Budapest University 1886 Wiedemann Annalen Eotvos rule sigma*V^(2/3) = K_E*(T_c-T) + 1890 Eotvos pendulum gravity. Wilfrid Bond UK 1897-1937 Cambridge 1928 Phil Mag Bond number eponym. Clift-Grace-Weber 1978 Academic Press + Levich 1962 Prentice Hall + de Gennes-Brochard-Quere 2002 Springer + de Gennes Nobel 1991 Capillarity. USA universities: NIST Boulder + Gaithersburg 1901 3000 thermophysical + REFPROP surface tension; MIT 1861 11000 + Howard Brenner transport + Patrick Doyle microfluidics + David Hardt; Stanford 1885 17000 + Juan Santiago microfluidics + Howard Stone (Princeton); Caltech 1891 2300 + GALCIT + John Brady + Mark Davis; Harvard SEAS 1847 + David Weitz soft matter + Howard Stone; UC Berkeley LBNL 1868 45000 + David Liem; Princeton 1746 + Howard Stone soft matter; Wisconsin Madison 1848 + BSL textbook 1960 + Stuart Berg; Cornell 1865 + Lynden Archer; UPenn 1740 + Howard Stone microfluidics; UMN 1851 + Lanny Schmidt + Frank Bates; CMU 1900 + Lorenz Biegler. USA national labs: NREL Golden 3000 + biorefinery; ORNL Oak Ridge 5000 + SNS bioenergy; ANL Lemont 5000 + APS catalysis; LLNL Livermore + HPC multiphase CFD; SNL Albuquerque + CRF Combustion Research Facility 100 hydrogen droplet; LANL 12000; PNNL Richland 5000 catalysis. USA chemical industry: ExxonMobil 62000 + Chemical 22000 + Baytown 590k bpd + Beaumont + Baton Rouge + Joliet refineries; Chevron 45000 + Chevron Phillips 8000 + Pascagoula + Richmond;

Phillips 66 14000 + Marathon 17000 + Valero 10000 + Halliburton 40000 multiphase + Schlumberger 99000 reservoir + Baker Hughes 55000. USA chemical companies: Dow 36000 + DuPont 23000 + LyondellBasell 19000 + Chemours 6000 + Sherwin-Williams 60000 atomization + PPG 50000 + Eastman 14000 + Air Products 19000 + Linde 80000 + Praxair Linde 2018 + Honeywell 13000 + UOP catalysts + Westinghouse + GE Power + Trane Carrier HVAC. USA pharma: Pfizer 80000 + Merck 70000 + JJ 130000 Janssen + AbbVie 50000 + Bristol Myers 30000 + Eli Lilly 35000 + Gilead 16000 + Regeneron 12000 + Moderna 5000 mRNA LNP + Vertex 5000. USA respiratory pMDI DPI: Boehringer USA Ridgefield Spiriva + Atrovent + Striverdi; GSK USA Philadelphia Ventolin + Seretide Diskus DPI; AstraZeneca USA + Symbicort Turbuhaler; Mylan + Teva + Cipla + Lupin generics. USA microfluidic: 10x Genomics 3000 droplet single-cell sequencing Chromium; BioFire Diagnostics 1000 FilmArray respiratory; Fluidigm; Bio-Rad 8000 ddPCR digital; QIAGEN 6000; Beckman Coulter. USA ultrasound contrast: Lantheus Definity perflutren + Optison; GE HealthCare 51000; Bracco USA Princeton + Lumason perflutren; Targeson microbubbles La Jolla CA.

Eotvos Bond table

  • Eo = Bo = delta_rho * g * L^2 / sigma
  • L_c = sqrt(sigma/(delta_rho*g))
  • Water-air 20°C L_c=2.71 mm
  • Mercury-air L_c=1.91 mm
  • Water-air sigma=0.0728 N/m
  • Mercury-air sigma=0.486
  • Ethanol-air sigma=0.0223
  • Eo<0.1 surface tension
  • 1<Eo<10 mixed ellipsoidal
  • 10<Eo<100 gravity cap
  • Eo>100 strong gravity flat
  • Bubble 1 mm air-water Eo=0.13
  • Bubble 5 mm Eo=3.4
  • Bubble 50 mm Eo=336
  • Eotvos 1886 Wiedemann Hungary
  • Bond 1928 Phil Mag UK Cambridge
  • Clift-Grace-Weber 1978
  • de Gennes Nobel 1991
  • USA NIST 1901: 3000 REFPROP
  • USA MIT 1861: 11000 Brenner Doyle Hardt
  • USA Stanford: Santiago microfluidics
  • USA Harvard SEAS: Weitz Stone
  • USA Wisconsin BSL 1960 textbook
  • USA SNL CRF Livermore: 100 droplet combustion
  • USA ExxonMobil 62000: Baytown 590k bpd
  • USA Dow 36000: bubble columns
  • USA UOP Honeywell 13000: catalysts
  • USA 10x Genomics 3000: droplet single-cell
  • USA Lantheus Definity: perflutren contrast
  • USA Boehringer Spiriva: pMDI tiotropium

Numerical examples

Example 1 - Water droplet 1 mm air. Eo=0.135. Surface tension dominated. USA: rain droplet + spray atomization + SNL CRF Combustion Research Facility 100.

Example 2 - Air bubble 5 mm water. Eo=3.36. Mixed ellipsoidal. USA: bubble columns ExxonMobil refineries + Dow + DuPont reactor.

Example 3 - Large bubble 50 mm. Eo=336. Strong gravity. USA: pipelines multiphase Halliburton 40000 + Schlumberger reservoir + Chevron offshore Big Foot.

Example 4 - Mercury droplet 5 mm. Eo=6.83. Mixed regime. USA: NIST + research labs national.

Example 5 - Microfluidic 10 micron. Eo=7.85e-6. Strong surface tension. USA: 10x Genomics Chromium + Bio-Rad ddPCR + MIT + Stanford microfluidics.

Limits

  1. Two-phase assumption multiple Eo per pair.
  2. Static vs dynamic Re Weber Capillary Morton.
  3. Surface tension temperature Eotvos rule.
  4. Surfactants reduce sigma factor 30.
  5. Bubble shape Grace 1976 Eo-Re-Mo.
  6. Capillary length L_c separation.
  7. Wettability contact angle theta.
  8. Reduced gravity ISS Mars Moon.
  9. Non-Newtonian polymer biological.
  10. Pressure dependence sigma drop.

Common errors

Eo vs Bo: same number convention chemical engineering Bo geophysics Eo.

L definition: D vs R bubble convention.

delta_rho heavy-light: phase order.

Surfactants reduce sigma: factor 30.

Reduced gravity micro-g: ISS factor 100-1000.

Practical applications

  1. Bubble columns reactors: USA ExxonMobil 62000 + Dow Chemical 36000 + DuPont 23000 + LyondellBasell 19000 + Chemours 6000 + UOP Honeywell catalysts.
  2. Multiphase pipelines: Halliburton 40000 + Schlumberger 99000 + Baker Hughes 55000 + Chevron deepwater + ExxonMobil offshore.
  3. Spray atomization combustion: SNL CRF Combustion Research Facility 100 hydrogen + GE Power + Westinghouse boilers.
  4. Microfluidic + lab-on-chip: 10x Genomics 3000 + Bio-Rad 8000 ddPCR + BioFire 1000 + Fluidigm + MIT Doyle Hardt + Stanford Santiago + Harvard Stone Weitz.
  5. Pharmaceutical pMDI + DPI inhalers: Boehringer USA Spiriva + GSK Ventolin + Seretide + AstraZeneca Symbicort Turbuhaler + Mylan + Teva generics.
  6. Microbubble ultrasound contrast: Lantheus Definity perflutren + GE HealthCare + Bracco Lumason + Targeson microbubbles La Jolla.
  7. Boilers + steam generators: Westinghouse + GE Power + Babcock & Wilcox + nuclear research labs ORNL ANL.
  8. Electrochemistry green H2: Air Products 19000 + Linde 80000 + Westinghouse hydrogen production.
  9. Marine engineering: USA naval shipbuilding General Dynamics + HII Newport News + cruise.
  10. University research multiphase: NIST + MIT + Stanford + Caltech + Harvard SEAS + UC Berkeley + Princeton + Wisconsin + UMN + Cornell + UPenn + CMU.

Updated July 2026

Frequently Asked Questions

What is the Eotvos Bond number?

Eo = Bo = (rho_h-rho_l)*g*L^2/sigma dimensionless multiphase fluid dynamics ratio gravity to surface tension forces. delta_rho density difference kg/m^3; g 9.81; L characteristic m; sigma N/m. Eo<0.1 surface tension dominates spherical; 1-10 ellipsoidal; >100 gravity dominates flat. Lorand Eotvos Hungary 1886 + Bond UK 1928. USA: NIST Boulder REFPROP + MIT Brenner Doyle + Stanford Santiago + Harvard Stone Weitz.

How to calculate Eotvos Bond?

Procedure: (1) Eo = (rho_h - rho_l) * g * L^2 / sigma; (2) rho_h heavy kg/m^3; (3) rho_l light; (4) g 9.81 m/s^2; (5) L characteristic m; (6) sigma N/m (water-air 0.0728 N/m). Example 5 mm air bubble water: Eo = 998.8*9.81*0.005^2/0.0728 = 3.36 mixed ellipsoidal. USA: bubble columns ExxonMobil + Dow + DuPont + Halliburton multiphase pipelines.

What is capillary length?

L_c = sqrt(sigma / (delta_rho * g)) natural length scale surface tension. Water-air 20°C L_c=2.71 mm; mercury-air 1.91 mm; oil-water 1-2 mm. Eo = (L/L_c)^2 squared ratio. Eo<1 L1 L>L_c gravity dominates. USA: NIST Boulder REFPROP fluid properties + MIT + Stanford + Harvard SEAS soft matter David Weitz.

What flow regimes does Eo classify?

Eo<0.1 surface tension: spherical droplet/bubble small Laplace 2*sigma/R; 0.1-1 capillary: small distortion; 1-10 mixed: ellipsoidal; 10-100 gravity: cap/spherical-cap Davies-Taylor; Eo>100 strong gravity: large flat interface industrial boilers. USA: research SNL CRF Combustion Research Facility 100 hydrogen + GE Power Westinghouse boilers + Honeywell UOP.

Who was Lorand Eotvos?

Lorand Eotvos (Hungary 1848-1919) Budapest University Hungarian physicist pioneer geophysics + capillarity; 1886 Wiedemann's Annalen 'Uber den Zusammenhang der Oberflachenspannung der Flussigkeiten mit ihrem Molekularvolumen' Eotvos rule sigma*V^(2/3) = K_E*(T_c-T) constant K_E=2.1e-7; 1890 Eotvos pendulum gravity. Wilfrid Bond UK 1897-1937 Cambridge 1928 Phil Mag Bond number eponym. USA: research MIT + Wisconsin BSL textbook + Princeton.

Typical surface tension values?

Water-air 20°C sigma=0.0728 N/m; 80°C 0.0626; Mercury-air 0.486; Ethanol-air 0.0223; Engine oil-air 0.030; Steam-water 100°C 0.0589; CO2 supercritical-water 0.001-0.005. Surfactants reduce sigma 0.030 → 0.001 factor 30. USA: research NIST + Harvard Weitz soft matter + Lantheus Definity perflutren ultrasound contrast + Boehringer pMDI Spiriva.

Applications of Eo Bond?

Applications: bubble columns reactors USA ExxonMobil 62000 Baytown 590k bpd + Dow 36000 + LyondellBasell 19000 + UOP Honeywell catalysts; multiphase pipelines Halliburton 40000 + Schlumberger 99000 + Baker Hughes 55000 + Chevron deepwater; spray atomization SNL CRF 100 hydrogen droplet + GE Power Westinghouse; microfluidic droplet 10x Genomics 3000 Chromium + Bio-Rad ddPCR; pMDI DPI inhalers Boehringer Spiriva + GSK Ventolin + AstraZeneca Symbicort; microbubble ultrasound Lantheus Definity + GE HealthCare 51000.

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