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Comments on Eötvös Number versus Bond Number



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.
313 Spl. Independentei, S6, 060042, Bucharest, ROMANIA
Phone: +40 21 4029705; Email:
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.
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?”
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
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
, where the density
and the
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Phone: +40 21 4029705; Email:
dynamic viscosity
correspond to the liquid,
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,
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.
1. Web page:
2. Web page:
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)
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.
313 Spl. Independentei, S6, 060042, Bucharest, ROMANIA
Phone: +40 21 4029705; Email:
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
DN: cn=Sanda-Carmen GEORGESCU,
o=University POLITEHNICA of
Bucharest, ou=Hydraulics and
Hydraulic Machinery Department,,
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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The operating limits of oscillating heat pipes (OHP) are crucial for the optimal design of cooling systems. In particular, the dryout limit is a key factor in optimizing the functionality of an OHP. As shown in previous studies, experimental approaches to determine the dryout limit lead to contradictory results. This work proposes a compact theory to predict a dryout threshold that unifies the experimental and analytical data. The theory is based on the influence of vapor quality on the flow pattern. When the vapor quality exceeds a certain limit (x = 0.006), the flow pattern changes from slug flow to annular flow and the heat transfer decreases abruptly. The results indicate a uniform threshold value, which has been validated experimentally and by the literature. With that approach, it becomes possible to design an OHP with an optimized filling ratio and, hence, substantially improve its cooling abilities.
In a previous paper by one of the authors it was shown that spherical drops or bubbles surrounded by a more viscous fluid might have a terminal velocity as great as one and a half times that of a solid sphere of equal size and mass. The present paper shows experimentally and theoretically that the surface-tension of the surface of the drop or bubble decreases the terminal velocity. For any radii appreciably less than a certain critical value the drop or bubble behaves almost like a rigid sphere. After a fairly rapid transition, for all radii appreciably larger than the critical the effect of surface-tension is small. Experiments on the terminal velocity for air in water-glass, air in syrup, mercury in syrup, and water in castor-oil, all give critical radii of the order predicted; but the different media show appreciably mutual disagreement, the cause of which is not vet certain.
A simple model is proposed for the creeping motion of a bubble contaminated with an “insoluble” surface-active agent. The existence of a stagnant cap over the rear of the bubble is taken for granted, and the cap size and associated bubble drag are then computed theoretically, as functions of the surface pressure at which the contaminant film collapses. The model, which employs no adjustable parameters, is consistent with experimental observations and, in particular, explains the success of the empirical correlation relating the bubble drag to the Bond number.RésuméLes auteurs proposent pour cette étude, l'exemple simple, de l'ascension d'une bulle contaminée par un agent insoluble à surface active. L'existence sur la partie arrière de la bulle d'une calotte permanente, est admise, et la taille de cette calotte, ainsi que celle de la bulle entraînée, à laquelle elle est attachée, sont calculées théoriquement en les considérant comme des fonctions de la pression de surface qui règne à la limite de disparition du film contaminant.Cet exemple qui n'utilise que des paramètres fixes, se limite à des observations, expérimentales, et ces dernières expliquent la justesse de la relation empirique qui lie, les caractéristiques de la bulle entraînée, au nombre de liaisons chimiques qui lui sont attachées.ZusammenfassungDie Bewegung einer Blase, deren Oberfläche mit einer inerten Oberflächenaktiven Substanz belegt ist, kann durch ein einfaches Modell interpretiert werden. Geht man von der Existenz einer stagnierenden Schicht auf der Rückseite der Blase aus, dann faβt man die Ausdehnung dieser Schicht mit der Blasenreibung theoretisch zusammen, und zwar in Abhängigkeit jenes Oberflächendruckes, bei dem der Inertfilm zerstört wird. Das Modell enthält keine unabhängigen Variablen und stimmt mit experimentellen Beobachtungen überein; darüber hinaus erklärt es jene empirischen Korrelationen, die den Blasenwiderstand in Beziehung zur Bond-Zahl setzen.
Actually Eötvös did publish his results in 1910, but no one knows about it because it was added at the end of a geophysics paper published in the proceedings of an international conference. After a short review of Eötvös' life, this article reports how the first publication of the results of his famous experiment was ``lost.'' This and other historical curiosities about his work are related to the current interest in his results and modern theories of quantum gravity. The appendices explain how the Eötvös balance works and also describe the controversy over the modern reanalysis of Eötvös' results.
This paper extends Stokes' calculations for the slow rectilinear motion of a solid sphere through viscous fluid, in the way he outlined, to the case where the sphere is composed of fluid. Experiments on the rate of rise of air bubbles in water-glass and in golden syrup, and on the velocity of the fluid in the neighbourhood of the bubbles in the former liquid, are in substantial agreement with the prediction that the bubble should rise one and a half times as fast as it would if it were solid. A smaller number of experiments on air bubbles and drops of syrup in castor oil also show reasonable agreement with theory. But those on water bubbles in castor oil indicate an effect due to surface contamination.
A detailed analysis of the effect of chamber volume, orifice radius, orifice submergence and contact angle on quasi-static formation of bubbles is presented. It is shown, that many aspects of slow bubble formation, involving phenomena leading to various modes of the bubble release, as well as the maximum orifice diameter which sustains a bubble at equilibrium, can be explained on the basis of information on equilibrium shapes and conditions. Scaling rules enabling adoption of results for water to other liquids are also presented.