Bond number

Physical key figure
Surname	Bond number, Eötvös number
Formula symbol	${\displaystyle {\mathit {Bo}},{\mathit {Eo}}}$
dimension	dimensionless
definition	${\displaystyle {\mathit {Bo}}={\frac {f\,L^{2}}{\sigma }}}$
${\displaystyle f}$ Force density of the volume force ${\displaystyle L}$ characteristic length ${\displaystyle \sigma }$ Surface tension
Named after	Wilfrid Noel Bond , Loránd Eötvös
scope of application	Phase interfaces of fluids

The bond number ( Symbol : after the English physicist Wilfrid Noel Bond (1897-1937)), or Eötvös number ( according to the Hungarian mathematician and geophysicists Lorand Eötvös ) is a dimensionless measure of fluid mechanics . It can be interpreted physically as the ratio of the volume force that acts on the liquid to the force due to surface tension : ${\displaystyle {\mathit {Bo}}}$ ${\displaystyle {\mathit {Eo}}}$

${\displaystyle {\mathit {Bo}}={\frac {F_{\text{Volumen}}}{F_{\text{Oberfläche}}}}}$

The term Eötvös number can be used

• as a synonym for bond number
• as a special case of the bond number in the case of buoyancy or
• as a generalization of the bond number for any characteristic parameter . ${\displaystyle L}$

Similar to the Reynolds number , the bond number is suitable for comparing systems that differ in individual parameters such as density , size or surface tension. In contrast to the Reynolds number, which is used in flows , the Bond number characterizes static systems. A small value means that the system is determined by the surface tension, while a large value means that the surface tension can be neglected for estimating the behavior. Together with the Morton number , the Bond number describes, for example, the shape of a fluid particle ( air bubble , water droplets, etc.) under the influence of gravity .

Special case: gravitation as volume force

If the volume force is given by gravity, the bond number is formed as follows:

${\displaystyle {\mathit {Bo}}={\frac {\text{Gravitationskraft}}{\mathrm {Oberfl{\ddot {a}}chenkraft} }}={\frac {\text{hydrostatischer Druck}}{\text{Kapillardruck}}}={\frac {\rho \cdot g\cdot H\cdot R}{2\sigma }}}$

It describes

• ${\displaystyle H}$ the vertical height
• ${\displaystyle R}$the radius responsible for the capillary pressure z. B. a drop . Both do not have to be identical, so that two length scales are often included in the bond number (e.g. vertical capillary : fill level , radius ). ${\displaystyle H}$ ${\displaystyle R}$

Furthermore is

• ${\displaystyle \rho }$the density
• ${\displaystyle g}$the acceleration due to gravity
• ${\displaystyle \sigma }$ the surface tension.

In the event that the buoyancy cannot be neglected or predominates, for example an air bubble in the water, the volume force must be calculated from the difference in the densities of the two phases , here water and air: ${\displaystyle \Delta \rho }$

${\displaystyle {\mathit {Bo}}={\frac {\Delta \rho \cdot g\cdot R^{2}}{\sigma }}}$

Example: a drop

Shape of raindrops depending on their size

With a drop of liquid on a flat, horizontal surface, the bond number allows a prediction of the shape it will assume. In this case, the bond number with the characteristic radius is determined as follows: ${\displaystyle R}$

${\displaystyle {\mathit {Bo}}={\frac {\rho gR^{2}}{\sigma }}}$

In this case, the radius is a maximum of twice the weight force ( ) and is responsible for the capillary pressure. In contrast to the Morton number, which only depends on the properties of the fluid, the Bond number changes with the radius of the drop. ${\displaystyle H=2R}$

If is very much smaller than one, gravity doesn't matter and the drop is spherical to a good approximation . With larger values ​​of it is elliptical and with a low Morton number (mostly with liquids of low viscosity , for example water) it is rather wobbly. With even larger Bond numbers, the drop takes on the shape of a round cap, which in the case of raindrops ultimately splits into two smaller drops. ${\displaystyle {\mathit {Bo}}}$ ${\displaystyle {\mathit {Bo}}}$

Individual evidence

1. ^ ^{A b} Josef Kunes: Dimensionless Physical Quantities in Science and Engineering . Elsevier, 2012, ISBN 0-12-391458-2 , pp. 95 ( limited preview in Google Book search).
2. ^ Willi H. Hager: Wilfrid Noel Bond and the Bond number . In: Journal of Hydraulic Research . tape 50 , no. 1 , p. 3–9 , doi : 10.1080 / 00221686.2011.649839 .
3. ↑ R. Schmel: Dissertation: Drop deformation and post-disintegration in technical mixture preparation . In: Research report of the ITS . tape 23 . LOGOS-Verlag, 2004, ISBN 3-8325-0707-8 , p. 53 ( kit.edu ).
4. ↑ Satish Kandlikar, Srinivas Garimella, Dongqing Li, Stephane Colin, Michael R. King: Heat Transfer and Fluid Flow in Minichannels and Microchannels . Butterworth-Heinemann, 2013, ISBN 0-08-098351-0 , pp. 229 ( limited preview in Google Book search).
5. ↑ CB Jenssen et al .: Parallel Computational Fluid Dynamics 2000: Trends and Applications . Gulf Professional Publishing, 2001, ISBN 0-08-053840-1 , pp. 80 ( limited preview in Google Book search).