Capillary number

{{short description|Ratio of viscous drag forces to surface tension in fluids}}

{{use dmy dates|date=April 2020}}

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 $\backslash mathrm\{Bo\}$, this term is useful to describe the forces acting on a fluid front in porous or granular media, such as soil.[https://hal.archives-ouvertes.fr/hal-02371151/document#page=4 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:{{cite journal|last1= Shi| first1= Z. |display-authors=et al | title= Dynamic contact angle hysteresis in liquid bridges.| journal= Colloids and Surfaces A: Physicochemical and Engineering Aspects | year=2018 | volume=555 | pages= 365–371| doi=10.1016/j.colsurfa.2018.07.004| arxiv= 1712.04703 | s2cid= 51916594 }}{{Cite web |url=http://myweb.clemson.edu/~jsaylor/paperPdfs/aichej.2012.SaylorBounds.pdf |title=Archived copy |access-date=2 July 2013 |archive-date=24 December 2013 |archive-url=https://web.archive.org/web/20131224102812/http://myweb.clemson.edu/~jsaylor/paperPdfs/aichej.2012.SaylorBounds.pdf |url-status=dead }}

:$\backslash mathrm\{Ca\}\; =\; \backslash frac\{\backslash mu\; V\}\{\backslash sigma\}$

where $\backslash mu$ is the dynamic viscosity of the liquid, $V$ is a characteristic velocity and $\backslash sigma$ 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 $N\_c$ instead of $\backslash mathrm\{Ca\}$.{{cite web |title=What is Capillary Number? - Definition from Petropedia |url=https://www.petropedia.com/definition/609/capillary-number |website=Petropedia |accessdate=5 October 2018 |language=en |archive-url=https://web.archive.org/web/20190327101131/https://www.petropedia.com/definition/609/capillary-number |archive-date=27 March 2019 |url-status=live}}

For low capillary numbers (a rule of thumb says less than 10⁻⁵), flow in porous media is dominated by capillary forces,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 of a flowing droplet at an interface.{{cite book |last1=Lambert |first1=Pierre |title=Surface Tension in Microsystems: Engineering Below the Capillary Length |date=2013 |publisher=Springer Science & Business Media |isbn=9783642375521 |pages=8–11 |url=https://books.google.com/books?id=tNq4BAAAQBAJ&q=Surface+Tension+in+Microsystems |language=en}}