The motion of a bubble is described by three dimensionless parameters, usually the combination Reynolds, Evotos and Morton. The most famous graph in the field is proposed by Grace, and  is in attachment.

To learn how to get the relations between the different numbers, you can use the exponent matching method, knowing that every dimensionless number can be described as a combination of the other three.

Take paper and pen and follow me with an example, on which I will surely make some arithmetic mistakes: we want to write the Reynolds number in terms of We, Eo and Mo.

By definition, the Reynolds is:

Re= d^1 U^1 nu^-1

While the We, Eo and Mo are:

We=rho^1 U^2 d ^1 sigma^-1;  Eo= d^2 rho g sigma^-1; Mo= g nu^4 rho^3 sigma^3

Thus we will write:

Re=We^a Eo^b Mo^c

On the left side we will have:

 rho ^(a+b+3c)* U^(2a)* d^(a+2b)*sigma^(-a-b-3c)*g^(c+b)*nu^(4c)

Now we can match the exponents, and we do it from the easy ones: those that have only one coefficient. From the exponent of U it is evident that a=1/2 and the exponent of nu gives us c=-1/4. We can now get the last from the sigma, for instance, noticing that b=1/4, and then double check that the rho will also hold: 1/2+1/4-3/4=0.

Provided that I made no mistake (please tell me in case :D), we are done:

Re=We^1/2*Eo^1/4*Mo^-1/4

Now that you've learned how it works, you can go for any other combination!

In order to perform my numerical simulation, I need its theoretical aspect.

Hi Jai,

for the drag-coefficient--Reynolds-relationship for spherical bubbles you could look at

https://www.researchgate.net/publication/258217033_Drag_with_External_and_Pressure_Drop_with_Internal_Flows_A_New_and_Unifying_Look_at_Losses_in_the_Flow_Field_Based_on_the_Second_Law_of_Thermodynamics

and the sources cited within this publication. The CD(Re) relationship can be transformed to the Re(Ar)-behaviour, which is a more common description of the bubble-motion. Ar is the Archimedes-number. High Ar-values are the cause for high velocities expressed as the Reynolds-number in a non-dimensional description. Together with your Eotvos-number (and presumably further dimensionless numbers) you should get the following dimensionless relationship for non-spherical bubbles:

Re=Re(Ar,Eo,...)

see e.g.:

Peters, Gaertner:

Scaling parameters of bubbles and drops; Interpretation and case study with air in silicon oil,

Acta Mechanica

July 2011, Volume 219, Issue 3, pp 189-202

Best regards

Bastian

Article Drag with External and Pressure Drop with Internal Flows: A ...

The motion of a bubble is described by three dimensionless parameters, usually the combination Reynolds, Evotos and Morton. The most famous graph in the field is proposed by Grace, and  is in attachment.

To learn how to get the relations between the different numbers, you can use the exponent matching method, knowing that every dimensionless number can be described as a combination of the other three.

Take paper and pen and follow me with an example, on which I will surely make some arithmetic mistakes: we want to write the Reynolds number in terms of We, Eo and Mo.

By definition, the Reynolds is:

Re= d^1 U^1 nu^-1

While the We, Eo and Mo are:

We=rho^1 U^2 d ^1 sigma^-1;  Eo= d^2 rho g sigma^-1; Mo= g nu^4 rho^3 sigma^3

Thus we will write:

Re=We^a Eo^b Mo^c

On the left side we will have:

 rho ^(a+b+3c)* U^(2a)* d^(a+2b)*sigma^(-a-b-3c)*g^(c+b)*nu^(4c)

Now we can match the exponents, and we do it from the easy ones: those that have only one coefficient. From the exponent of U it is evident that a=1/2 and the exponent of nu gives us c=-1/4. We can now get the last from the sigma, for instance, noticing that b=1/4, and then double check that the rho will also hold: 1/2+1/4-3/4=0.

Provided that I made no mistake (please tell me in case :D), we are done:

Re=We^1/2*Eo^1/4*Mo^-1/4

Now that you've learned how it works, you can go for any other combination!

Thank you so much for taking interest in my problem. I am indebted to you all.

Actually I am trying to validate a paper regarding oblique coalescence of two gas

bubbles:

Numerical simulation of gas bubbles behavior using a three-dimensional

volume of fluid method.

Authors: M. van Sint Annaland, N.G. Deen, J.A.M. Kuipers

Chemical Engineering Science(2005).

They have performed their simulations by giving Morton, Weber and Reynolds number. However for defining Reynolds number their velocity scale is bubble terminal velocity, but in my case it:

U=g^1/2 D^1/2

and time scale:

t=D/U

length scale:

L=D

g= acceleration due to gravity,D=bubble diameter.

So after non dimensionalising the N-S equation, I get Reynolds and Weber number. Although the Froude number will be 1, if we follow these scales.

Now I am unable to convert their non-dimensional numbers into what I required.

Sincere thanks

Jai

===

Jai, you can find the relations for cited numbers as the following: 

We=Re^2(Mo/Eo)-0.5

We=(FrMo(Re)^4)1/3

for more detail go to ( INTECH site -Two-Phase Flow-Chapter11-- 2012 Awad, Licence InTech) ,this is an open access

Please see Section 2.1. Dimensionless parameters in my chapter "Two-Phase Flow"

http://www.intechopen.com/articles/show/title/two-phase-flow

You can download my chapter "Two-Phase Flow" for free using this link

http://www.intechopen.com/articles/show/title/two-phase-flow

Thank you all for your kind support..

I will definitely look into these study materials, and will try to figure it out ASAP.

Best Regards

Jai

Alsarkhi Abdelsalam, Sarica Cem, Pereyra Eduardo, New Dimensionless Number for Gas-Liquid Flow in Pipes, International Journal of Multiphase Flow, In Press, Accepted Manuscript, Available online 7 January 2016.

http://www.sciencedirect.com/science/article/pii/S0301932215300835