Ohnesorge number#

Named after: Wolfgang von Ohnesorge (1901-1976).

$$\text{Oh} \stackrel{\text{def}}{=} \frac{\mu}{\sqrt{\rho} \sqrt{\sigma} \sqrt{L}} \sim \frac{\text{viscous damping}}{\text{inertia and surface tension}}$$

Description#

Measures viscous damping relative to inertial and capillary effects. It indicates how strongly viscosity suppresses droplet oscillation, pinch off, and breakup.

Quantities#

Name	Symbol	SI units	Dimension
dynamic viscosity	$\mu$	$\mathrm{Pa}\,\mathrm{s}$	$\text L^{-1}\,\text M\,\text T^{-1}$
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
surface tension	$\sigma$	$\mathrm{N}\,\mathrm{m}^{-1}$	$\text M\,\text T^{-2}$
characteristic length	$L$	$\mathrm{m}$	$\text L$

Regimes#

Droplet dynamics

Range	Regime	Description
0 – 0.1	inertia and capillarity	Viscous dissipation is weak. Oscillation, pinch off, and breakup are governed mainly by inertia and surface tension.
0.1 – 1	mixed	Viscous, inertial, and capillary effects are all important. Droplet relaxation and breakup are noticeably damped.
1 – ∞	viscous	Viscous dissipation dominates. Interfaces relax slowly and capillary waves are strongly suppressed.

Name	Symbol	SI units	Dimension
dynamic viscosity	\(\mu\)	\(\mathrm{Pa}\,\mathrm{s}\)	\(\text L^{-1}\,\text M\,\text T^{-1}\)
mass density	\(\rho\)	\(\mathrm{kg}\,\mathrm{m}^{-3}\)	\(\text L^{-3}\,\text M\)
surface tension	\(\sigma\)	\(\mathrm{N}\,\mathrm{m}^{-1}\)	\(\text M\,\text T^{-2}\)
characteristic length	\(L\)	\(\mathrm{m}\)	\(\text L\)