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Comments on Eötvös Number versus Bond Number

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Sanda-Carmen Georgescu at National University of Science and Technology POLITEHNICA Bucharest
Sanda-Carmen Georgescu
  • National University of Science and Technology POLITEHNICA Bucharest
Jean-Luc Achard
Jean-Luc Achard
Jean-Luc Achard
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Abstract

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?” Related to the paper: Quasi-steady formation of bubbles and drops viewed as processes that break bifurcation, by Jean-Luc Achard, Sanda-Carmen Georgescu, published in Journal of Engineering Mathematics, vol.51, no.2, pp 147-164, February 2005.
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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.
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'
... The dimensionless Eötvös number Eö is very similar to the Bo number. Depending on the researcher, there are different definitions of the Eötvös number [2,14,16,17]. Khandekar and Groll [11] define the the Eö number by Equation (3). Table 1 shows different thresholds which are standardized by the La number [2]. ...
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