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UNIVERSITY “POLITEHNICA” of BUCHAREST

POWER ENGINEERING FACULTY, HYDRAULICS DEPARTMENT

313 Spl. Independentei, S6, 060042, Bucharest, ROMANIA

Phone: +40 21 4029705; Email: carmen.georgescu@upb.ro

1

Note:

Extracted from the letter addressed by Dr. Sanda-Carmen Georgescu, to Prof. H. K. Kuiken,

the Editor-in-Chief of the Journal of Engineering Mathematics, on April 26, 2004, related to

the submitted paper: "Quasi-steady formation of bubbles and drops viewed as processes that

break bifurcation", by Jean-Luc Achard and Sanda-Carmen Georgescu.

Task:

Corroborating the Eötvös number and the Bond number by some historical evidence, e.g.

“When did Bond’s first paper on the subject appear and when Eötvös’s?”

Answer:

COMMENTS ON EÖTVÖS NUMBER VERSUS BOND NUMBER

Upon investigations, we found few data on that subject. It seems that Eötvös published his

results about 4 to 5 decades before Bond. The historical evidence would be as following.

The Hungarian physicist Lóránd Baron von Eötvös was born on July 27, 1848 in Pest (now

part of Budapest), and died on April 8, 1919 in Budapest. The Hungarian name of Lóránd,

Báró Eötvös appears in some biographies as Roland, Baron von Eötvös, e.g. in Encyclopaedia

Britannica (2004) [1].

Lóránd Baron von Eötvös introduced the concept of molecular surface tension. He

published on capillarity between 1876 and 1886, then he published on gravitation for the

rest of his life [2]. We don’t dispose of exact references for Eötvös’ work on capillarity. Some

historical curiosities about his work on gravitation and the controversy over the modern

reanalysis of Eötvös’ results are described by Nieto et al [3].

The Eötvös number, as ratio between the gravitational force and the surface tension

force appears in literature till the 8

th

decade of the last century, e.g. in the study of Davis

and Acrivos in 1966 [4], in the study of Marmur and Rubin in 1973 [5], or in the monograph

on bubbles and drops by Clift et al in 1978 [6]. Upon our modest investigations, we found

no Bond number before that.

We don’t dispose of W. N. Bond’s biography, but we found exact references for his work.

He published on the effect of surfactants on the motion of drops and bubbles in 1927 [7],

and in 1928 [8]. The dimensionless quantity that today is associated with Bond’s name

appeared within his paper from 1928 [8], without a specific name, even if that quantity

was equivalent to the already existent Eötvös number.

In 1950, Datta et al [9, pp.20] pointed out that Bond and Newton [8] have tested the equation

defining the true terminal velocity of bubbles, by using viscous liquids, and found that when

the viscosities ratio

is small, allowance should be made for the surface energy effect [7],

which is related to the dimensionless quantity

2

gR

, where the density

and the

UNIVERSITY “POLITEHNICA” of BUCHAREST

POWER ENGINEERING FACULTY, HYDRAULICS DEPARTMENT

313 Spl. Independentei, S6, 060042, Bucharest, ROMANIA

Phone: +40 21 4029705; Email: carmen.georgescu@upb.ro

2

dynamic viscosity

correspond to the liquid,

and

correspond to the gas, R is the

spherical bubble radius,

the surface tension and g the gravity.

In 1966, Davis and Acrivos [4] published their results for the drag force as a function of the

Eötvös number, noticing that the results agree qualitatively with the observations of Bond and

Newton [8]. So the results obtained by Bond in 1928 [8] were evaluated in 1966 with

respect to the Eötvös number.

During the last two decades of the last century, the Bond number was associated to the

Eötvös number, and replaced the former in many references around the World, e.g. in the

extensive publication of Hideki Tsuge in 1986 [10, pp.192].

In 1997, the monograph on bubbles and drops by Sadhal et al [11] presents both the

Eötvös number and the Bond number, as following:

- “The Eötvös number is a measure of the gravitational force (or body force) compared to the

surface tension force. Where gravity forces are dominant, the appropriate parameter to be

considered in a study of particle deformation is the Eötvös number.” [11, Chapter 2: Shape

and size of fluid particles, pp.20]. Here the length scale is the particle radius R.

- “For a drop on a plane solid surface, the appropriate dimensionless parameter is called the

Bond number, which is also known as the Eötvös number.” [11, Chapter 5: Wall

interactions, pp.215]. Here, the Bond number considers a length scale L corresponding to a

drop on a plane solid, inclined or level.

- The Bond number is defined by the same formula as the Eötvös number, with a length scale

related to the size of the orifice where bubbles or drops are attached [11, Chapter 7:

Formation and breakup of bubbles and drops, pp.314].

- The Bond number is defined by the same formula as the Eötvös number, with a length scale

related to the initial radius of the drop [11, Chapter 8: Compound drops and bubbles,

pp.431].

Facing the historical evidence, we conclude that the Bond number became predominant in

literature during the last say 25 years, while the Eötvös number, old from about 120

years and quite widespread 50 years ago, became now less familiar.

We would like to denote as Eötvös number, the ratio between the gravitational force and the

surface tension.

REFERENCES

1. Web page: http://www.britannica.com/eb/article?eu=33321

2. Web page: http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Eotvos.html

3. M. M. Nieto, R. Hugues and T. Goldman, Actually, Eötvös did publish his results in

1910, it’s just that no one knows about it..., American Journal of Physics, 57, no.5 (1989)

397-404.

4. R. E. Davis and A. Acrivos, The influence of surfactants on the creeping motion of

bubbles, Chem. Engng Sci., 21 (1966) 681-685.

5. A. Marmur and E. Rubin, Equilibrium shapes and quasi-static formation of bubbles at

submerged orifice. Chem. Engng Sci. 28 (1973) 1455-1464.

UNIVERSITY “POLITEHNICA” of BUCHAREST

POWER ENGINEERING FACULTY, HYDRAULICS DEPARTMENT

313 Spl. Independentei, S6, 060042, Bucharest, ROMANIA

Phone: +40 21 4029705; Email: carmen.georgescu@upb.ro

3

6. R. Clift, J. R. Grace and M. E. Weber, Bubbles, Drops, and Particles, New York:

Academic Press Inc. (1978).

7. W. N. Bond, Bubbles and drops and Stokes’ law, Phil. Mag., 4 (1927) 889-898.

8. W. N. Bond and D. A. Newton, Bubbles, drops and Stokes’ law (Paper 2), Phil. Mag., 5

(1928) 794-800.

9. R. L. Datta, D. H. Napier and D. M. Newitt, The properties and behaviour of gas bubbles

formed at a circular orifice, In: Proc. Conf. on Formation and Properties of Gas Bubbles,

Institution of Chemical Engrs, February 14, London, U.K. (1950) 14-26.

10. H. Tsuge, Hydrodynamic of bubble formation from submerged orifices. In: N. P.

Cheremisinoff (ed.), Encyclopedia of Fluid Mechanics. Houston, TX: Gulf Publishing

(1986) pp.191-232.

11. S. S. Sadhal, P. S. Ayyaswamy and J. N. Chung, Transport Phenomena with Drops and

Bubbles, New-York: Springer (1997) 520pp.

Sanda-

Carmen

GEORGESCU

Digitally signed by Sanda-Carmen

GEORGESCU

DN: cn=Sanda-Carmen GEORGESCU,

o=University POLITEHNICA of

Bucharest, ou=Hydraulics and

Hydraulic Machinery Department,

email=carmen.georgescu@upb.ro,

c=RO

Date: 2013.06.21 23:05:35 +03'00'