Eötvös number

In fluid dynamics the Eötvös number (Eo), also called the Bond number (Bo), is a dimensionless number measuring the importance of surface tension forces compared to body 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

The Eötvös or Bond number is given by

$\mathrm{Eo}=\mathrm{Bo}=\frac{\Delta\rho \,g \,L^2}{\sigma}$

• $\Delta\rho$: difference in density of the two phases, (SI units: kg/m³)
• g: gravitational acceleration, (SI units : m/s²)
• L: characteristic length, (SI units : m)
• $\sigma$: surface tension, (SI units : N/m)

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. 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:

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

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 0-12-176950-X.
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): 716–716. 1937. Bibcode:1937Natur.140Q.716.. doi:10.1038/140716a0.