Washburn's equation

In physics, Washburn's equation describes capillary flow in a bundle of parallel cylindrical tubes; it is extended with some issues also to imbibition into porous materials. The equation is named after Edward Wight Washburn;^[1] also known as Lucas–Washburn equation, considering that Richard Lucas^[2] wrote a similar paper three years earlier, or the Bell-Cameron-Lucas-Washburn equation, considering J.M. Bell and F.K. Cameron's discovery of the form of the equation fifteen years earlier.^[3]

Derivation

In case of a fully wettable capillary, it is

$L^2=\frac{\gamma Dt}{4\eta}$

where $t$ is the time for a liquid of dynamic viscosity $\eta$ and surface tension $\gamma$ to penetrate a distance $L$ into the capillary whose pore diameter is $D$. In case of a porous materials many issues have been raised both about the physical meaning of the calculated pore diameter $D$^[4] and the real possibility to use this equation for the calculation of the contact angle of the solid.^[5] The equation is derived for capillary flow in a cylindrical tube in the absence of a gravitational field, but is sufficiently accurate in many cases when the capillary force is still significantly greater than the gravitational force.

In his paper from 1921 Washburn applies Poiseuille's Law for fluid motion in a circular tube. Inserting the expression for the differential volume in terms of the length $l$ of fluid in the tube $dV=\pi r^2 dl$, one obtains

$\frac{\delta l}{\delta t}=\frac{\sum P}{8 r^2 \eta l}(r^4 +4 \epsilon r^3)$

where $\sum P$ is the sum over the participating pressures, such as the atmospheric pressure $P_A$, the hydrostatic pressure $P_h$ and the equivalent pressure due to capillary forces $P_c$. $\eta$ is the viscosity of the liquid, and $\epsilon$ is the coefficient of slip, which is assumed to be 0 for wetting materials. $r$ is the radius of the capillary. The pressures in turn can be written as

$P_h=h g \rho - l g \rho\sin\psi$

$P_c=\frac{2\gamma}{r}\cos\phi$

where $\rho$ is the density of the liquid and $\gamma$ its surface tension. $\psi$ is the angle of the tube with respect to the horizontal axis. $\phi$ is the contact angle of the liquid on the capillary material. Substituting these expressions leads to the first-order differential equation for the distance the fluid penetrates into the tube $l$:

$\frac{\delta l}{\delta t}=\frac{[P_A+g \rho (h-l\sin\psi)+\frac{2\gamma}{r}\cos\phi](r^4 +4 \epsilon r^3)}{8 r^2 \eta l}$

Washburn's constant

The Washburn constant may be included in Washburn's equation.

It is calculated as follows:

$\frac{10^4 \left[ \mathrm{\frac{\mu m}{cm}} \right] \left[ \mathrm{\frac{N}{m^2}} \right]}{68947.6 \left[ \mathrm{\frac{dynes}{cm^2}} \right]} = 0.1450(38)$^[6][7]

Applications

Inkjet printing

The penetration of a liquid into the substrate flowing under its own capillary pressure can be calculated using a simplified version of Washburn's equation:^[8][9]

$l = \left[ \frac{r\cos\theta}{2} \right]^{\frac{1}{2}} \left[ \frac{\gamma}{\eta} \right]^{\frac{1}{2}} t^{\frac{1}{2}}$

where the surface tension-to-viscosity ratio $\left[ \tfrac{\gamma}{\eta} \right]^{\frac{1}{2}}$ represents the speed of ink penetration into the substrate.

Food

According to physicist and igNobel prize winner Len Fisher, the Washburn equation can be extremely accurate for more complex materials including biscuits.^[10][11] Following an informal celebration called national biscuit dunking day, some newspaper articles quoted the equation as Fisher's equation.^[12]

References

1. Edward W. Washburn (1921). "The Dynamics of Capillary Flow". Physical Review 17 (3): 273. Bibcode:1921PhRv...17..273W. doi:10.1103/PhysRev.17.273.
2. Lucas, R. (1918). "Ueber das Zeitgesetz des Kapillaren Aufstiegs von Flussigkeiten". Kolloid Z. 23: 15. doi:10.1007/bf01461107.
3. Bell, J.M. and Cameron, F.K. (1906). "The flow of liquids through capillary spaces". J. Phys. Chem. 10: 658–674. doi:10.1021/j150080a005.
4. Dullien, F. A. L. (1979). Porous Media: Fluid Transport and Pore Structure. New York: Academic Press. ISBN 0-12-223650-5.
5. Marco, Brugnara; Claudio, Della Volpe; Stefano, Siboni (2006). "Wettability of porous materials. II. Can we obtain the contact angle from the Washburn equation?". In Mittal, K. L. Contact Angle, Wettability and Adhesion. Mass. VSP.
6. Micromeritics, "Autopore IV User Manual", September (2000). Section B, Appendix D: Data Reduction, page D-1. (Note that the addition of 1N/m2 is not given in this reference, merely implied)
7. Micromeritics, "A new method of interpolation and smooth curve fitting based on local procedures", Journal of the Association of Computing Machinery (1970). Volume 17(4), pp.589-602.
8. Oliver, J. F. (1982). "Wetting and Penetration of Paper Surfaces" 200: 435–453. doi:10.1021/bk-1982-0200.ch022. ISSN 1947-5918.
9. Leelajariyakul, S.; Noguchi, H.; Kiatkamjornwong, S. (2008). "Surface-modified and micro-encapsulated pigmented inks for ink jet printing on textile fabrics". Progress in Organic Coatings 62 (2): 145–161. doi:10.1016/j.porgcoat.2007.10.005. ISSN 0300-9440.
10. "The 1999 Ig Nobel Prize Ceremony". http://www.improbable.com. Improbable Research. Retrieved 2015-10-07. Len Fisher, discoverer of the optimal way to dunk a biscuit. External link in |website= (help)
11. Barb, Natalie (25 November 1998). "No more flunking on dunking". http://news.bbc.co.uk. BBC News. Retrieved 2015-10-07. External link in |website= (help)
12. Fisher, Len (11 February 1999). "Physics takes the biscuit". http://www.nature.com. Nature. Retrieved 2015-10-07. Washburn will be turning in his grave to learn that the media have renamed his work the "Fisher equation". External link in |website= (help)