# Eötvös number

In fluid dynamics the Eötvös number (Eo), also called the Bond number (Bo), is a dimensionless number measuring the importance of gravitational forces compared to surface tension forces and is used (together with Morton number) to characterize the shape of bubbles or drops moving in a surrounding fluid. The two names commemorate the Hungarian physicist Loránd Eötvös (1848–1919)[1][2][3][4] and the English physicist Wilfrid Noel Bond (1897–1937),[3][5] respectively. The term Eötvös number is more frequently used in Europe, while Bond number is commonly used in other parts of the world.

## Definition

Describing the ratio of capillary forces to gravitational forces, the Eötvös or Bond number is given by the equation:[6]

${\displaystyle \mathrm {Eo} =\mathrm {Bo} ={\frac {\Delta \rho \,g\,L^{2}}{\gamma }}}$.
• ${\displaystyle \Delta \rho }$: difference in density of the two phases, (SI units: kg/m3)
• g: gravitational acceleration, (SI units : m/s2)
• L: characteristic length, (SI units : m) (for example the radii of curvature for a drop)
• ${\displaystyle \gamma }$: surface tension, (SI units : N/m)

The Bond number can also be written as

${\displaystyle \mathrm {Bo} =\left({\frac {L}{\lambda _{\rm {c}}}}\right)^{2}}$,

where ${\textstyle \lambda _{\rm {c}}={\sqrt {\frac {\gamma }{\rho g}}}}$ is the capillary length.

A high value of the Eötvös or Bond number indicates that the system is relatively unaffected by surface tension effects; a low value (typically less than one) indicates that surface tension dominates [6]. Intermediate numbers indicate a non-trivial balance between the two effects. It may be derived in a number of ways, such as scaling the pressure of a drop of liquid on a solid surface. It is usually important, however, to find the right length scale specific to a problem by doing a ground-up scale analysis. Other similar dimensionless numbers are:

${\displaystyle \mathrm {Bo} =\mathrm {Eo} =2\,\mathrm {Go} ^{2}=2\,\mathrm {De} ^{2}\,}$

where Go and De are the Goucher and Deryagin numbers, which are identical: the Goucher number arises in wire coating problems and hence uses a radius as a typical length scale while the Deryagin number arises in plate film thickness problems and hence uses a Cartesian length.

## References

1. Clift, R.; Grace, J. R.; Weber, M. E. (1978). Bubbles Drops and Particles. New York: Academic Press. p. 26. ISBN 978-0-12-176950-5.
2. Tryggvason, Grétar; Scardovelli, Ruben; Zaleski, Stéphane (2011). Direct Numerical Simulations of Gas–Liquid Multiphase Flows. Cambridge, UK: Cambridge University Press. p. 43. ISBN 9781139153195.
3. Hager, Willi H. (2012). "Wilfrid Noel Bond and the Bond number". Journal of Hydraulic Research. 50 (1): 3–9. doi:10.1080/00221686.2011.649839.
4. de Gennes, Pierre-Gilles; Brochard-Wyart, Françoise; Quéré, David (2004). Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. New York: Springer. p. 119. ISBN 978-0-387-00592-8.
5. "Dr. W. N. Bond". Nature. 140 (3547): 716. 1937. Bibcode:1937Natur.140Q.716.. doi:10.1038/140716a0.
6. Li, S (2018). "Dynamics of Viscous Entrapped Saturated Zones in Partially Wetted Porous Media". Transport in Porous Media. 125 (2): 193–210. arXiv:1802.07387. doi:10.1007/s11242-018-1113-3.