# 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]

• 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.