Ohnesorge number

The Ohnesorge number (Oh) is a dimensionless number that relates the viscous forces to inertial and surface tension forces. The number was defined by Wolfgang von Ohnesorge in his 1936 doctoral thesis.^[1][2]

It is defined as:

${\displaystyle \mathrm {Oh} ={\frac {\mu }{\sqrt {\rho \,\sigma \,L}}}={\frac {\sqrt {\mathrm {We} }}{\mathrm {Re} }}\sim {\frac {\mbox{viscous forces}}{\sqrt {{\mbox{inertia}}\cdot {\mbox{surface tension}}}}}}$

Where

• μ is the dynamic viscosity of the liquid
• ρ is the density of the liquid
• σ is the surface tension
• L is the characteristic length scale (typically drop diameter)
• Re is the Reynolds number
• We is the Weber number

Applications

The Ohnesorge number for a 3 mm diameter rain drop is typically ~0.002. Larger Ohnesorge numbers indicate a greater influence of the viscosity.

This is often used to relate to free surface fluid dynamics such as dispersion of liquids in gases and in spray technology.^[3][4]

In inkjet printing, liquids whose Ohnesorge number are in the range 0.1 < Oh < 1.0 are jettable (1<Z<10 where Z is the reciprocal of the Ohnesorge number).^[1][5]

See also

• Laplace number - There is an inverse relationship, ${\displaystyle \mathrm {Oh} =1/{\sqrt {\mathrm {La} }}}$, between the Laplace number and the Ohnesorge number. It is more historically correct to use the Ohnesorge number, but often mathematically neater to use the Laplace number.

References

1. McKinley, Gareth H.; Renardy, Michael (2011). "Wolfgang von Ohnesorge". Physics of Fluids. 23 (12): 127101–127101–6. Bibcode:2011PhFl...23l7101M. doi:10.1063/1.3663616. hdl:10919/24403. S2CID 50633355.
2. Fardin, Marc-Antoine; Hautefeuille, Mathieu; Sharma, Vivek (2022). "Spreading, pinching, and coalescence: the Ohnesorge units". Soft Matter. 18 (17): 3291–3303. arXiv:2112.06713. Bibcode:2022SMat...18.3291F. doi:10.1039/d2sm00069e. PMID 35416235. S2CID 245123849.
3. Lefebvre, Arthur Henry (1989). Atomization and Sprays. New York and Washington, D.C.: Hemisphere Publishing Corp. ISBN 978-0-89116-603-0. OCLC 18560155.
4. Ohnesorge, W (1936). "Die Bildung von Tropfen an Düsen und die Auflösung flüssiger Strahlen". Zeitschrift für Angewandte Mathematik und Mechanik. 16 (6): 355–358. Bibcode:1936ZaMM...16..355O. doi:10.1002/zamm.19360160611. English translation: Ohnesorge, Wolfgang von (2019). "The formation of drops by nozzles and the breakup of liquid jets". Texas Scholar Works. doi:10.26153/tsw/3391. S2CID 214403876.
5. Derby, Brian (2010). "Inkjet Printing of Functional and Structural Materials: Fluid Property Requirements, Feature Stability, and Resolution" (PDF). Annual Review of Materials Research. 40 (1): 395–414. Bibcode:2010AnRMS..40..395D. doi:10.1146/annurev-matsci-070909-104502. ISSN 1531-7331. S2CID 138001742.