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Capillary number

## Summary

In fluid dynamics, the capillary number (Ca) is a dimensionless quantity representing the relative effect of viscous drag forces versus surface tension forces acting across an interface between a liquid and a gas, or between two immiscible liquids. For example, an air bubble in a liquid flow tends to be deformed by the friction of the liquid flow due to viscosity effects, but the surface tension forces tend to minimize the surface area. The capillary number is defined as:[1][2]

${\displaystyle \mathrm {Ca} ={\frac {\mu V}{\sigma }}}$

where ${\displaystyle \mu }$ is the dynamic viscosity of the liquid, ${\displaystyle V}$ is a characteristic velocity and ${\displaystyle \sigma }$ is the surface tension or interfacial tension between the two fluid phases.

Being a dimensionless quantity, the capillary number's value does not depend on the system of units. In the petroleum industry, capillary number is denoted ${\displaystyle N_{c}}$ instead of ${\displaystyle \mathrm {Ca} }$.[3]

For low capillary numbers (a rule of thumb says less than 10−5), flow in porous media is dominated by capillary forces,[4] whereas for high capillary numbers the capillary forces are negligible compared to the viscous forces. Flow through the pores in an oil field reservoir have capillary number on the order of 10−6, whereas flow of oil through an oil well drill pipe has a capillary number on the order of 1.[3]

The capillary number plays a role in the dynamics of capillary flow; in particular, it governs the dynamic contact angle of a flowing droplet at an interface.[5]

## Multiphase formulation

Multiphase flows forms when two or more partially or immiscible fluids are brought in contact.[6] The Capillary number in multiphase flow has the same definition as the single flow formulation, the ratio of viscous to surface forces but has the added(?) effect of the ratio of fluid viscosities:

${\displaystyle \mathrm {Ca} ={\frac {\mu V}{\sigma }}{\frac {\mu }{\hat {\mu }}},}$

where ${\displaystyle \mu }$ and ${\displaystyle {\hat {\mu }}}$ are the viscosity of the continuous and the dispersed phases respectively. [6]

Multiphase microflows are characterized by the ratio of viscous to surface forces, the capillary number (Ca), and by the ratio of fluid viscosities:[6]

${\displaystyle \mathrm {Ca} ={\frac {\mu V}{\sigma }}~{and}~{\frac {\mu }{\hat {\mu }}}.}$