Capillary number
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In
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
, the capillary number (Ca) is a
dimensionless quantity A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1 ...
representing the relative effect of
viscous The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the in ...
drag forces versus
surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) t ...
forces acting across an interface between a
liquid A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, ...
and a gas, or between two
immiscible Miscibility () is the property of two substances to mix in all proportions (that is, to fully dissolve in each other at any concentration), forming a homogeneous mixture (a solution). The term is most often applied to liquids but also appli ...
liquids. Alongside the Bond number, commonly denoted Bo, this term is useful to describe the forces acting on a fluid front in porous or granular media, such as soil.Dynamics of viscous entrapped saturated zones in partially wetted porous media
Transport in Porous Media (2018), 125(2), 193-210 The capillary number is defined as: :\mathrm = \frac where \mu is the
dynamic viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the int ...
of the liquid, V is a characteristic velocity and \sigma is the surface tension or
interfacial tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) t ...
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 N_c instead of \mathrm. For low capillary numbers (a rule of thumb says less than 10−5), flow in
porous media A porous medium or a porous material is a material containing pores (voids). The skeletal portion of the material is often called the "matrix" or "frame". The pores are typically filled with a fluid (liquid or gas). The skeletal material is usu ...
is dominated by
capillary force Capillary action (sometimes called capillarity, capillary motion, capillary rise, capillary effect, or wicking) is the process of a liquid flowing in a narrow space without the assistance of, or even in opposition to, any external forces l ...
s,Ding, M., Kantzas, A.: Capillary number correlations for gas-liquid systems, SEP 2004-062 (2004) 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. The capillary number plays a role in the dynamics of capillary flow; in particular, it governs the dynamic
contact angle The contact angle is the angle, conventionally measured through the liquid, where a liquid–vapor interface meets a solid surface. It quantifies the wettability of a solid surface by a liquid via the Young equation. A given system of solid, liq ...
of a flowing droplet at an interface.


Multiphase formulation

Multiphase flows forms when two or more partially or immiscible fluids are brought in contact. 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:   \mathrm = \frac \frac, where \mu and \hat are the viscosity of the continuous and the dispersed phases respectively. Multiphase microflows are characterized by the ratio of viscous to surface forces, the capillary number (Ca), and by the ratio of fluid viscosities: \mathrm = \frac ~~ \frac.


See also

* Bond number *
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dom ...
*
Capillary pressure In fluid statics, capillary pressure () is the pressure between two immiscible fluids in a thin tube (see capillary action), resulting from the interactions of forces between the fluids and solid walls of the tube. Capillary pressure can serve as bo ...
*
Froude number In continuum mechanics, the Froude number (, after William Froude, ) is a dimensionless number defined as the ratio of the flow inertia to the external field (the latter in many applications simply due to gravity). The Froude number is based on ...


References

Dimensionless numbers of fluid mechanics {{Fluiddynamics-stub