Eötvös number

In fluid dynamics the Eötvös number (Eo) is a dimensionless number named after Hungarian physicist Loránd Eötvös (1848–1919). It is also known as the Bond number. The term Eötvös number is more frequently used in Europe, while Bond number is commonly used in other parts of the world.

Together with Morton number it can be used to characterize the shape of bubbles or drops moving in a surrounding fluid. Eötvös number may be regarded as proportional to buoyancy force divided by surface tension force.

\[\mathrm{Eo}=\frac{\Delta\rho \,g \,L^2}{\sigma}\]

• \({\rm Eo}\) is the Bond Number or Eötvös number
• \(\Delta\rho\): 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 different statement of the equation is as follows:

\[{\rm Bo} = \frac{\rho a L^2}{\gamma}\]

where

• \({\rm Bo}\) is the Bond Number or Eötvös number
• \(\rho\) is the density, or the density difference between fluids.
• \(a\) the acceleration associated with the body force, almost always gravity.
• \(L\) the 'characteristic length scale', e.g. radius of a drop or the radius of a capillary tube.
• \(\gamma\) is the surface tension of the interface.

The Bond number is a measure of the importance of surface tension forces compared to body forces. A high Bond number indicates that the system is relatively unaffected by surface tension effects; a low number (typically less than one is the requirement) indicates that surface tension dominates. Intermediate numbers indicate a non-trivial balance between the two effects.

One over the Bond Number, i.e. \({ 1/Bo }\) is sometimes referred to as the Jesus number, \({ Je }\), since bodies can walk on water if they have a large Jesus number, that is if \({ Je >> 1 }\).

The Bond number is the most common comparison of gravity and surface tension effects and 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 dimensionless numbers are related to the Bond number:

\[\rm Bo = Eo = 2 Go^2 = 2 De^2\,\]

Where \(\rm Eo, Go,\) and \(\rm De\) are respectively the Eötvös, Goucher, and Deryagin numbers. The "difference" between the Goucher and Deryagin numbers is that the Goucher number (arises in wire coating problems) uses the letter \(R\) to represent length scales while the Deryagin number (arises in plate film thickness problems) uses \(L\).

References

bs:Eötvösov broj