How to calculate Ohnesorge number Oh atomization inkjet

Calculate Ohnesorge number Oh = mu/sqrt(rho*sigma*L) viscous vs inertia + surface tension droplet atomization. Z=1/Oh inkjet printability. Oh<0.1 printable; 0.1-1 optimal; Oh>1 viscous. Wolfgang von Ohnesorge 1936 + IBM Fromm 1984. HP Canon Epson Xerox.

Calculation parameters
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Ohnesorge number Oh dimensionless ratio of viscous forces to inertia and surface tension forces in droplet formation, atomization and liquid jet breakup: Oh = mu / sqrt(rho * sigma * L). mu dynamic viscosity Pa*s (water 25°C 1e-3; ethanol 1.1e-3; glycerol 1.5; engine oils 0.1-1); rho density kg/m^3 (water 1000; mercury 13534); sigma surface tension N/m (water 25°C 0.072; ethanol 0.022; mercury 0.485); L characteristic length m (inkjet nozzle 20-50 micron; spray atomizer 100-500 micron). Equivalent Oh = sqrt(We)/Re.

Z printability number: Z = 1/Oh = sqrt(rho*sigma*L)/mu Fromm 1984 IBM Research criterion droplet ejection.

Regimes: Oh10 clean droplet ejection + Rayleigh-Plateau capillary breakup; 0.1<Oh<1 inkjet optimal Z=1-10 DOD piezoelectric + thermal bubble jet; 1<Oh<10 viscous regime ligaments long + satellite drops; Oh>10 highly viscous no droplet formation.

Related numbers: Reynolds Re = rho*v*L/mu; Weber We = rho*v^2*L/sigma; Capillary Ca = mu*v/sigma; Bond Bo = rho*g*L^2/sigma.

Plateau-Rayleigh instability: cylindrical liquid jet capillary breakup wavelength lambda_max=9.02*R + Tomotika 1935 viscous + Eggers 1993 self-similar pinch-off.

Inkjet technologies: Continuous Inkjet CIJ Hertz Sweden 1965 KTH + Domino Cambridge UK + Linx + Videojet + industrial coding 100k droplets/s; Drop-On-Demand DOD piezoelectric Spectra + thermal bubble jet HP 1979 + Canon BJ 1981 + Epson MicroPiezo 1984 + droplet 1-100 picoliter; industrial DOD Xaar Cambridge UK 1990 + Seiko + KYOCERA + Konica Minolta; 3D printing inkjet Stratasys + 3D Systems + ProJet MultiJet Polyjet photopolymer; OLED inkjet Samsung Display + LG Display + KATEEVA San Jose YIELDjet RGB pixel; printed electronics Plastic Logic Cambridge + Heraeus silver ink + Dupont. History: Wolfgang von Ohnesorge (Berlin Germany 1901 - Stuttgart 1976) Universitat Gottingen PhD 1936 Ludwig Prandtl supervisor + AVA Gottingen + Stuttgart MTU Aero Engines; 1936 PhD thesis 'Die Bildung von Tropfen an Dusen und die Auflosung flussiger Strahlen' ZAMM 16:355-358 classic jet breakup + 4 regimes (Rayleigh + first wind-induced + second wind-induced + atomization) + Reynolds-Ohnesorge diagram log-log; pioneer atomization.

Lord Rayleigh (Strutt 3rd Baron Essex UK 1842-1919) Cambridge Cavendish + Royal Society President + Nobel Prize 1904; 1879 paper Proc London Math Soc 10:4 'On the instability of jets' Plateau-Rayleigh capillary instability + lambda_max=9.02*R + 1916 Phil Mag Rayleigh-Benard.

Joseph Plateau (Brussels Belgium 1801-1883) Universite Etat Gand Ghent 1835; 1873 textbook 'Statique experimentale et theorique des liquides' classic capillarity + Plateau border foam + Plateau laws minimal surface; blind from 1843 after sun gazing experiments.

Susumu Tomotika (Japan 1908-1989) Tokyo Imperial University; 1935 paper Proc Roy Soc A 150:322 viscous Rayleigh-Plateau extension.

Jens Eggers (Wuppertal 1962-) MIT + Saarbrucken + Bristol; 1993 Phys Rev Lett 71:3458 'Universal pinching of 3D axisymmetric free-surface flow' classic self-similar pinch-off + Eggers-Villermaux 2008 Rep Prog Phys 71:036601 review classic.

John Eggers Fromm IBM Research Yorktown Heights 1984; 1984 IBM J Res Develop 28:322 Z = 1/Oh inkjet printability index DOD numerical simulation.

USA research atomization + inkjet + spray: NIST Gaithersburg MD 1901 3500 drop measurements + ink standards; NASA Glenn Cleveland OH 1941 1500 + Lewis Research microgravity atomization + ISS experiments; NREL Golden CO 1977 2900 + spray combustion biofuels; ORNL Oak Ridge TN 1943 6500 + nuclear spray atomizer + emergency core cooling ECCS; Sandia Albuquerque NM 1949 14000 Combustion Research Facility CRF Livermore + Direct injection spray atomization research; LBNL Berkeley 1931 4400; Lawrence Livermore LLNL 1952 8000 ICF Inertial Confinement Fusion + ignition spray.

USA universities atomization: MIT Cambridge MA 1861 11700 Mech Eng + Bergman-Lavine textbook; Stanford CA 1885 17000 Mech Eng + George Karniadakis CFD spray; Caltech Pasadena 1891 2400 + GALCIT Liepmann Roshko + Tim Colonius CFD; UC Berkeley 1868 45000 + Carlos Fernandez-Pello + spray combustion; Princeton NJ 1746 8000 MAE Lex Smits + Howard Stone Microfluidics; Cornell Ithaca NY 1865 25000 MAE; Georgia Tech Atlanta 1885 36000 Wayne Whiteman + Suresh Menon spray combustion; UIUC Urbana 1867 56000 + Vanka multi-phase; Penn State University Park 1855 50000 + Daniel Haworth + Domenic Santavicca; Univ Michigan Ann Arbor 1817 50000 + Volker Sick GDi engines + Margaret Wooldridge; UT Austin TX 1883 51000; Purdue West Lafayette IN 1869 47000 + Robert Lucht spray; Univ Wisconsin-Madison 1848 49000 Engine Research Center ERC Reitz; UCLA 1919 45000; UC Davis 1908 40000.

USA inkjet industry: HP Hewlett-Packard Palo Alto CA 1939 51000 + thermal bubble jet 1979 ThinkJet + DeskJet + DesignJet + commercial PageWide industrial; Canon USA Melville NY (Tokyo Japan) 11000 + BubbleJet 1981 BJ + PIXMA imageRUNNER + commercial; Epson Long Beach CA (Seiko Epson Japan) 8000 + MicroPiezo 1984 + EcoTank + commercial Workforce + SureColor + Stylus; Xerox Norwalk CT 1906 21000 + Phaser solid ink + iGen production inkjet + Trivor + Brenva HD CF Continuous Feed; Lexmark Lexington KY 1991 6000 + thermal inkjet + commercial; Kodak Rochester NY 1888 4500 + Stream + Prosper inkjet + KODAK NEXFINITY commercial.

USA automotive injection: Bosch USA Farmington Hills MI 36000 + GDi gasoline + diesel common rail + Bosch Engineering; Continental Auburn Hills MI 28000; Denso Long Beach CA Southfield MI (Toyota) 24000; BorgWarner Auburn Hills MI 1928 50000 + injection systems; Eaton Cleveland OH 1911 92000; Delphi Technologies (Aptiv) Troy MI 1994 22000; Cummins Columbus IN 1919 73000 + diesel common rail + heavy duty; Caterpillar Peoria IL 1925 105000 + diesel injection.

USA pharmaceutical MDI nebulizers: Pfizer NY 1849 80000 + MDI inhalers + nebulizers; Merck Kenilworth NJ 1891 73000; Johnson&Johnson NJ 1886 152000; Eli Lilly Indianapolis IN 1876 39000; AbbVie Lake Bluff IL 2013 50000; Bristol-Myers Squibb NY 1858 33000; Amgen Thousand Oaks CA 1980 24000; Gilead Foster City CA 1987 16000.

USA cosmetic spray: P&G Cincinnati OH 1837 107000 + Olay Pantene Old Spice; Estee Lauder NY 1946 62000; Coty NY 1904 11000; Revlon NY 1932 5800; Avon NY 1886 30000; Mary Kay Addison TX 1963 5500.

USA food spray drying: Nestle USA Solon OH 47000 + coffee instant Nescafe spray dry; Kraft Heinz Pittsburgh PA 38000; General Mills Minneapolis MN 1928 35000; Kellogg's Battle Creek MI 1906 31000 + cereal spray drying; PepsiCo Purchase NY 1965 318000; Coca-Cola Atlanta GA 1886 79000.

Ohnesorge formula

Definition: Oh = mu / sqrt(rho * sigma * L). Equivalent Oh = sqrt(We)/Re.

Z printability: Z = 1/Oh = sqrt(rho*sigma*L)/mu Fromm 1984. Z>10 printable; Z=1-10 optimal DOD; Z<1 viscous limited.

Typical coefficients: water 25°C mu=1e-3 Pa*s, rho=1000, sigma=0.072 N/m; ethanol mu=1.1e-3, sigma=0.022; glycerol mu=1.5, sigma=0.063; mercury mu=1.5e-3, sigma=0.485.

Ohnesorge-Reynolds diagram: 4 regimes Rayleigh + first wind-induced + second wind-induced + atomization.

History and context

Wolfgang von Ohnesorge (Berlin Germany 1901 - Stuttgart 1976) Universitat Gottingen PhD 1936 Ludwig Prandtl + AVA Gottingen + Stuttgart MTU 1955-1976; 1936 PhD thesis 'Die Bildung von Tropfen an Dusen' ZAMM 16:355-358 classic jet breakup + 4 regimes + Reynolds-Ohnesorge diagram. Lord Rayleigh (Essex UK 1842-1919) Cambridge Cavendish 1879-1884 + Nobel 1904 argon; 1879 Proc London Math Soc 10:4 'On the instability of jets' Plateau-Rayleigh capillary instability + 1916 Rayleigh-Benard + 4 volumes Theory of Sound 1877. Joseph Plateau (Brussels 1801-1883) Universite Etat Gand 1835 + 1873 textbook 'Statique experimentale theorique liquides' minimum surface + Plateau border. Tomotika Japan 1935 Proc Roy Soc A 150:322 viscous Rayleigh-Plateau. Eggers 1993 Phys Rev Lett 71:3458 self-similar pinch-off + 1997 Rev Mod Phys 69:865 + Eggers-Villermaux 2008 Rep Prog Phys 71:036601 review classic. Fromm IBM Yorktown Heights 1984 Z = 1/Oh printability + Hertz KTH Sweden 1965 continuous inkjet patent.

USA research atomization: NIST Gaithersburg 3500; NASA Glenn Cleveland 1500 microgravity ISS; NREL Golden 2900 spray biofuels; ORNL 6500 nuclear ECCS; Sandia CRF Livermore 14000 direct injection spray; LLNL 8000 ICF; LBNL 4400; Argonne 3400.

USA universities: MIT 11700 Mech Eng; Stanford 17000 Karniadakis CFD; Caltech 2400 GALCIT Colonius; UC Berkeley 45000 Fernandez-Pello combustion; Princeton 8000 Smits + Stone microfluidics; Cornell 25000; Georgia Tech 36000 Menon spray; UIUC 56000 Vanka; Penn State 50000 Haworth + Santavicca; Michigan 50000 Sick GDi + Wooldridge; UT Austin 51000; Purdue 47000 Lucht spray; Wisconsin-Madison ERC Reitz 49000; UCLA 45000; UC Davis 40000.

USA inkjet industry: HP Palo Alto 51000 ThinkJet 1979 thermal bubble + DeskJet + PageWide; Canon Melville NY 11000 BubbleJet 1981 + PIXMA imageRUNNER; Epson Long Beach 8000 MicroPiezo 1984 + EcoTank + Workforce; Xerox Norwalk CT 21000 Phaser solid ink + iGen + Trivor; Lexmark KY 6000 + commercial; Kodak Rochester 4500 Stream Prosper.

USA automotive: Bosch USA Farmington Hills 36000 GDi diesel; Continental Auburn Hills 28000; Denso Long Beach 24000; BorgWarner Auburn Hills 50000; Cummins Columbus IN 73000 diesel; Caterpillar Peoria IL 105000.

USA pharmaceutical: Pfizer NY 80000 MDI nebulizers; Merck NJ 73000; JNJ 152000; Lilly 39000; AbbVie 50000; BMS 33000; Amgen 24000.

USA cosmetic: P&G Cincinnati 107000 Olay Pantene; Estee Lauder NY 62000; Coty 11000; Revlon 5800.

USA food spray drying: Nestle USA Solon 47000 Nescafe; Kraft Heinz 38000; General Mills 35000; Kellogg's 31000 cereal.

Ohnesorge values table

  • Oh = mu/sqrt(rho*sigma*L): definition
  • Oh = sqrt(We)/Re: equivalent
  • Z = 1/Oh: printability Fromm 1984
  • Oh10
  • 0.1<Oh<1 optimal Z=1-10
  • 1<Oh<10 viscous limited
  • Oh>10 highly viscous
  • Water 25°C Oh~0.012: 50 micron
  • Glycerol Oh~150: viscous
  • Mercury Oh~6e-4
  • Diesel Oh~0.025: 200 micron
  • HP DOD 1979: thermal bubble
  • Canon BJ 1981
  • Epson MicroPiezo 1984
  • Ohnesorge Germany 1936: Gottingen Prandtl
  • Rayleigh 1879 jets
  • Plateau Belgium 1873
  • Tomotika Japan 1935: viscous
  • Eggers 1993: pinch-off
  • Fromm IBM 1984: Z printability
  • USA NIST 3500
  • USA NASA Glenn 1500
  • USA Sandia CRF 14000
  • USA HP 51000
  • USA Bosch 36000
  • USA Pfizer 80000

Numerical examples

Example 1 - Water 30 micron nozzle. mu=1e-3, rho=1000, sigma=0.072, L=30e-6. Oh=6.8e-3, Z=147. Inkjet printable. USA: HP ThinkJet + Canon BubbleJet + Epson MicroPiezo.

Example 2 - Inkjet ink. mu=3e-3, rho=1050, sigma=0.030, L=50e-6. Oh=0.076, Z=13. Optimal DOD UV-curable. USA: HP PageWide + Xerox iGen + Kodak Stream Prosper.

Example 3 - Glycerol viscous. mu=1.5, rho=1260, sigma=0.063, L=1e-3. Oh=168, Z=6e-3. Highly viscous + no droplet. USA: MIT + Stanford labs.

Example 4 - Pharma atomizer. mu=5e-3, rho=1100, sigma=0.040, L=100e-6. Oh=0.075, Z=13. Pharmaceutical optimal. USA: Pfizer + Merck + AbbVie MDI nebulizers.

Example 5 - Diesel injection. mu=3e-3, rho=830, sigma=0.027, L=200e-6. Oh=0.044, Z=22. Atomization regime. USA: Bosch USA + Cummins + Caterpillar diesel common rail.

Limits and considerations

  1. Nozzle geometry: round vs slot vs pintle vs pressure-swirl modify atomization spray cone.
  2. Newtonian: shear-thinning ink + Carreau-Yasuda + viscoelastic Maxwell.
  3. Compressibility: high-pressure injection 1500-2500 bar diesel + cavitation in-nozzle.
  4. Ambient gas: spray atomizer + Weber gas We_g = rho_g*v^2*L/sigma.
  5. Surface tension temperature: dσ/dT modify Marangoni Ma.
  6. Surfactant: CMC critical micelle + Marangoni + Plateau-Rayleigh modify.
  7. Multi-component evaporation: polymer ink + solvent + coffee ring.
  8. Electric field EHD: Coulomb + Taylor cone + electrospinning + electrospray.
  9. Acoustic forcing: piezoelectric DOD + 1-100 microsec + 1-50 kHz.
  10. Particle laden: nano-particle ink + sedimentation + clogging + 3D printing.

Common errors

Confusing Oh + We + Re: Oh = sqrt(We)/Re. All correlated.

Z = 1/Oh printability: Fromm 1984 IBM + 1<Z<10 optimal DOD inkjet.

L characteristic length: nozzle vs jet vs droplet diameter convention matters.

Newtonian assumption fail: shear-thinning + viscoelastic + DLP polymer inks.

Compressibility cavitation: high-pressure diesel + cavitation in-nozzle.

Practical applications

  1. Inkjet printing 2D: USA HP 51000 + Canon 11000 + Epson 8000 + Xerox 21000 + Kodak 4500 + Lexmark 6000.
  2. 3D printing inkjet photopolymer: USA Stratasys + 3D Systems + ProJet MultiJet Polyjet.
  3. Automotive injection GDi diesel: USA Bosch 36000 + Continental 28000 + Denso 24000 + BorgWarner 50000 + Cummins 73000 + Caterpillar 105000.
  4. Aerospace jet engine spray combustor: USA GE Aviation + Pratt&Whitney + Boeing + Lockheed + NASA Glenn.
  5. Pharmaceutical MDI inhalers nebulizers: USA Pfizer 80000 + Merck 73000 + JNJ 152000 + Lilly 39000 + AbbVie 50000 + BMS 33000 + Amgen 24000.
  6. Cosmetic perfume spray: USA P&G 107000 + Estee Lauder 62000 + Coty 11000 + Revlon 5800 + Avon 30000 + Mary Kay 5500.
  7. Food spray drying: USA Nestle 47000 Nescafe + Kraft Heinz 38000 + General Mills 35000 + Kellogg 31000 cereal + PepsiCo 318000 + Coca-Cola 79000.
  8. Agricultural pesticide sprayers: USA John Deere + Case New Holland + ASABE standards.
  9. Combustion turbine power plant: USA GE Power + Siemens Westinghouse + Mitsubishi Power Americas.
  10. Universities atomization research: MIT + Stanford + Caltech + UC Berkeley + Princeton + Cornell + Georgia Tech + UIUC + Penn State + Michigan + Purdue + Wisconsin-Madison ERC Reitz.

Updated July 2026

Frequently Asked Questions

What is the Ohnesorge number?

Oh = mu/sqrt(rho*sigma*L) dimensionless ratio of viscous to inertia + surface tension forces in atomization + jet breakup + droplet formation. Equivalent Oh = sqrt(We)/Re. Z = 1/Oh inkjet printability Fromm IBM 1984. Oh<0.1 printable; 0.1<Oh<1 optimal DOD; Oh>1 viscous limited. Wolfgang von Ohnesorge Germany 1936 PhD Gottingen Prandtl. USA: HP 51000 + Canon 11000 + Epson 8000 + Pfizer 80000.

How do I calculate Ohnesorge?

Procedure: (1) Oh = mu/sqrt(rho*sigma*L); (2) mu Pa*s (water 25°C 1e-3); (3) rho kg/m^3; (4) sigma N/m (water 0.072); (5) L m (nozzle 50 micron); (6) Z=1/Oh printability; (7) regime. Water 30 micron: Oh=6.8e-3 + Z=147 printable. USA: NIST + Stanford CFD + MIT Mech Eng.

What is the Z printability number?

Z = 1/Oh = sqrt(rho*sigma*L)/mu Fromm 1984 IBM Research Yorktown Heights criterion droplet ejection + DOD numerical simulation. Z>10 printable clean ejection; Z=1-10 optimal DOD inkjet HP Canon Epson MicroPiezo Xerox Lexmark Kodak; Z<1 viscous limited ligaments long satellite. USA: HP Palo Alto + Xerox Norwalk + Kodak Rochester research.

What regimes does Oh classify?

Oh10; 0.1<Oh<1 optimal DOD piezoelectric thermal bubble; 1<Oh<10 viscous limited ligaments long; Oh>10 highly viscous no droplet. USA: HP DOD 1979 + Canon BubbleJet 1981 + Epson MicroPiezo 1984 + Bosch GDi 36000 + Pfizer MDI 80000.

Who was Wolfgang von Ohnesorge?

Wolfgang von Ohnesorge (Berlin Germany 1901 - Stuttgart 1976) Universitat Gottingen PhD 1936 Ludwig Prandtl supervisor + AVA Gottingen + Stuttgart MTU Aero Engines 1955-1976; 1936 PhD thesis ZAMM 16:355-358 'Tropfenbildung Dusen Auflosung Strahlen' classic jet breakup + 4 regimes atomization + Reynolds-Ohnesorge diagram log-log standard reference; pioneer atomization.

What is Plateau-Rayleigh instability?

Plateau-Rayleigh capillary instability Lord Rayleigh 1879 Proc London Math Soc 10:4 + Joseph Plateau Belgium 1873 textbook minimum surface area + cylindrical jet break-up wavelength lambda_max=9.02*R + linear stability + Tomotika 1935 Proc Roy Soc A 150:322 viscous extension + Eggers 1993 Phys Rev Lett 71:3458 self-similar pinch-off. USA: MIT + Stanford + Princeton.

What are Ohnesorge applications?

Applications: inkjet printing HP 51000 + Canon 11000 + Epson 8000 + Xerox 21000 + Kodak 4500; 3D printing Stratasys + 3D Systems Polyjet; automotive injection Bosch 36000 + Cummins 73000 + Caterpillar 105000; jet engine GE Aviation + Pratt&Whitney; pharmaceutical MDI Pfizer 80000 + Merck 73000 + JNJ 152000 + AbbVie 50000; cosmetic P&G 107000 + Estee Lauder 62000; food Nestle Nescafe 47000 + Kellogg cereal 31000.

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