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]

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 liquid viscosity
• ρ is the liquid density
• 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 Ohnersorge number).^[1][4]

See also

• 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

1. McKinley, Gareth H.; Renardy, Michael (2011). "Wolfgang von Ohnesorge". Physics of Fluids. 23: 127101. Bibcode:2011PhFl...23l7101M. doi:10.1063/1.3663616.
2. Lefebvre, Arthur Henry (1989). Atomization and Sprays. New York and Washington, D.C.: Hemisphere Publishing Corp. ISBN 978-0-89116-603-0. OCLC 18560155.
3. Ohnesorge, W (1936). "Formation of drops by nozzles and the breakup of liquid jets". Journal of Applied Mathematics and Mechanics. 16: 355–358.
4. 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. doi:10.1146/annurev-matsci-070909-104502. ISSN 1531-7331.