# 9.4.2: Relationship Between Dimensionless Numbers

The Dimensionless numbers since many of them have formulated in a certain field tend to be duplicated. For example, the Bond number is referred in Europe as Eotvos number. In addition to the above confusion, many dimensional numbers expressed the same things under certain conditions. For example, Mach number and Eckert Number under certain circumstances are same.

Example 9.16

Galileo Number is a dimensionless number which represents the ratio of

\[

\label{GalileoNumber:def}

Ga = \dfrac{\rho^2\,g\,\ell^3}{\mu^2} \tag{64}

\]

The definition of Reynolds number has viscous forces and the definition of Froude number has gravitational forces. What are the relation between these numbers?

Example 9.17

Laplace Number is another dimensionless number that appears in fluid mechanics which related to Capillary number. The Laplace number definition is

\[

\label{Laplace:def}

La = \dfrac{\rho \, \sigma \, \ell }{\mu^2} \tag{65}

\]

Show what are the relationships between Reynolds number, Weber number and Laplace number.

Example 9.18

The Rotating Froude Number is a somewhat a similar number to the regular Froude number. This number is defined as

\[

\label{RotatingFr:def}

Fr_R = \dfrac{\omega^2\,\ell}{g} \tag{66}

\]

What is the relationship between two Froude numbers?

Example 9.19

Ohnesorge Number is another dimensionless parameter that deals with surface tension and is similar to Capillary number and it is defined as

\[

\label{ohnesorge:def}

Oh = \dfrac{\mu}{\sqrt{\rho\,\sigma\,\ell} } \tag{67}

\]

Defined \(Oh\) in term of \(We\) and \(Re\) numbers.

### Contributors

Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.