Capillary number#

$$\text{Ca} \stackrel{\text{def}}{=} \frac{\mu U}{\sigma} \sim \frac{\text{viscous stress}}{\text{surface tension}}$$

Measures viscous stress relative to surface-tension stress. It indicates whether flow can deform interfaces against capillary restoring forces.

Name	Symbol	SI units	Dimension
dynamic viscosity	$\mu$	$\mathrm{Pa}\,\mathrm{s}$	$\text L^{-1}\,\text M\,\text T^{-1}$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
surface tension	$\sigma$	$\mathrm{N}\,\mathrm{m}^{-1}$	$\text M\,\text T^{-2}$

Multiphase flow

Range	Regime	Description
0 – 0.001	capillary dominated	Surface tension forces dominate. Droplets and bubbles maintain compact, nearly spherical shapes.
0.001 – 1	transitional	Both surface tension and viscous forces are significant. Interfaces are noticeably deformed.
1 – ∞	viscous dominated	Viscous forces dominate. Interfaces deform freely and droplets may be highly elongated or broken up.

Name	Symbol	SI units	Dimension
dynamic viscosity	\(\mu\)	\(\mathrm{Pa}\,\mathrm{s}\)	\(\text L^{-1}\,\text M\,\text T^{-1}\)
velocity	\(U\)	\(\mathrm{m}\,\mathrm{s}^{-1}\)	\(\text L\,\text T^{-1}\)
surface tension	\(\sigma\)	\(\mathrm{N}\,\mathrm{m}^{-1}\)	\(\text M\,\text T^{-2}\)