Capillary number
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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. Alongside the Bond number, commonly denoted <math>\mathrm{Bo}</math>, this term is useful to describe the forces acting on a fluid front in porous or granular media, such as soil.<ref name="granular">Dynamics of viscous entrapped saturated zones in partially wetted porous media. Transport in Porous Media (2018), 125(2), 193-210 </ref> The capillary number is defined as:<ref name="CAH">Template:Cite journal</ref><ref>Template:Cite web</ref>
- <math>\mathrm{Ca} = \frac{\mu V}{\sigma} </math>
where <math>\mu</math> is the dynamic viscosity of the liquid, <math>V</math> is a characteristic velocity and <math>\sigma</math> 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 <math>N_c</math> instead of <math>\mathrm{Ca}</math>.<ref name="Petropedia">Template:Cite web</ref>
For low capillary numbers (a rule of thumb says less than 10−5), flow in porous media is dominated by capillary forces,<ref>Ding, M., Kantzas, A.: Capillary number correlations for gas-liquid systems, SEP 2004-062 (2004)</ref> whereas for high capillary numbers the capillary forces are negligible compared to the viscous forces. Flow through the pores in an oil reservoir has capillary number values in the order of 10−6, whereas flow of oil through an oil well drill pipe has a capillary number in the order of unity.<ref name="Petropedia" />
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.<ref name="Lambert2013">Template:Cite book</ref>
Multiphase formulation
Multiphase flows forms when two or more partially or immiscible fluids are brought in contact.<ref name=":0">Template:Cite journal</ref> 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: Template:Clarification needed
<math>\mathrm{Ca} = \frac{\mu V}{\sigma}, \frac{\mu }{\hat{\mu}} </math>
where <math>\mu </math> and <math>\hat{\mu} </math> are the viscosity of the continuous and the dispersed phases respectively.<ref name=":0" />
Multiphase microflows are characterized by the ratio of viscous to surface forces, the capillary number (Ca), and by the ratio of fluid viscosities:<ref name=":0" />
<math>\mathrm{Ca} = \frac{\mu V}{\sigma}</math> and <math>\frac{\mu }{\hat{\mu}}.</math>