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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]

It is defined as:

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

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.[2][3]

In inkjet printing, liquids whose Ohnesorge number is less than 1 and greater than 0.1 are jettable (1<Z<10 where Z is the reciprocal of the Ohnesorge number).[1][4]

Laplace number - There is an inverse relationship, $${\mathrm {Oh}}=1/{\sqrt {{\mathrm {La}}}}$$, 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

McKinley, Gareth H.; Renardy, Michael (2011). "Wolfgang von Ohnesorge". Physics of Fluids. 23 (12): 127101. Bibcode:2011PhFl...23l7101M. doi:10.1063/1.3663616.
Lefebvre, Arthur Henry (1989). Atomization and Sprays. New York and Washington, D.C.: Hemisphere Publishing Corp. ISBN 978-0-89116-603-0. OCLC 18560155.
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". doi:10.26153/tsw/3391.

Derby, Brian (2010). "Inkjet Printing of Functional and Structural Materials: Fluid Property Requirements, Feature Stability, and Resolution". Annual Review of Materials Research. 40 (1): 395–414. Bibcode:2010AnRMS..40..395D. doi:10.1146/annurev-matsci-070909-104502. ISSN 1531-7331.

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Dimensionless numbers in fluid mechanics

Archimedes Atwood Bagnold Bejan Biot Bond Brinkman Capillary Cauchy Chandrasekhar Damköhler Darcy Dean Deborah Dukhin Eckert Ekman Eötvös Euler Froude Galilei Graetz Grashof Görtler Hagen Iribarren Kapitza Keulegan–Carpenter Knudsen Laplace Lewis Mach Marangoni Morton Nusselt Ohnesorge Péclet Prandtl
magnetic turbulent Rayleigh Reynolds
magnetic Richardson Roshko Rossby Rouse Schmidt Scruton Sherwood Shields Stanton Stokes Strouhal Stuart Suratman Taylor Ursell Weber Weissenberg Womersley

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