# Dimensionless Numbers,Formula derivation and their useIn this blog we will learn what are dimensionless Numbers , their formula derivation and what are there applications in real life. These dimensionless Number play very important role in fluid dynamics

Dimensionless Number : A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1), which is not explicitly shown

Example :

- Reylond's Number
- Froude's Number
- Euler's Number
- Weber's Number
- Mach's Number

**:**

__Reylond's Number__It is defined as the ratio of the inertia of flowing fluid to the viscous force of the flowing fluid.

Inertia Force (`F_{i}`) = mass X acceleration of the flowing Fluid

= `\rho AV^{2}`

Viscous Force(`F_{v}`) = Shear stress X Area

= `\tau A` = `\mu\frac{\partial u}{\partial y}A`

= `\mu\frac{V}{L}A`

`R_{e} = \frac{F_{i}}{F_{v}} = \frac{\rho AV^{2}}{\mu\frac{V}{L}A} =\frac{\rho VL}{\mu}`

we know that kinematic viscosity `\upsilon = \frac{\mu}{\rho}` put this value in above equation we can get the reylond's equation in more shorter form

`R_{e}`= `\frac{VL}{\upsilon}`

Application

- Pipe flow calculation
- Can use to find resistance experienced by sub-marines, airplanes, fully immersed etc.

*Froude's Number :* It is defined as the square root ratio of the inertia force of flowing fluid to the gravity force

`F_{g}` = force due to gravity

= Mass X acceleration due to gravity

=`\rho ALg`

`F_{e} = \sqrt{\frac{F_{i}}{F_{g}}} = \sqrt{\frac{\rho AV^{2}}{\rho ALg}} = \sqrt{\frac{V^{2}}{Lg}} = \frac{V}{\sqrt{Lg}}`

Application of Froud's Number

- Analysis of flow of jet from nozzle and orifice
- free surface flow such as spillways and feirs
- Where different density of fluid flow one over another

*Euler's Number:*It is defined as the square root ratio of inertia force to the pressure force

`E_{u} = \sqrt{\frac{F_{i}}{F_{p}}} `

Fp = pressure X area

`E_{u} = \sqrt{\frac{\rho A V^{2}}{pA}} = \sqrt{\frac{V^{2}}{\frac{p}{\rho}}} = \frac{V}{\sqrt{\frac{p}{\rho}}}`

Application of Euler Number

This law is used where cavitation takes place

__Weber's Number :__It is defined as the square root ratio of the inertia force of the flowing fluid to the surface tension force.

`W_{e} = \sqrt{\frac{F_{i}}{F_{s}}} `

`F_{s}` = surface tension force

= surface tension per unit length = `\sigma L`

Area is A

A = `L^{2}`

`W_{e} = \sqrt{\frac{\rho AV^{2}}{\sigma L}} = \sqrt{\frac{\rho L^{2} V^{2}}{\sigma L}} = \sqrt{\frac{\rho L V^{2}}{\sigma}} = \sqrt{\frac{V^{2}}{\frac{\sigma}{\rho L}}} = \frac{V}{\sqrt{\frac{\sigma}{\rho L}}}`

Use of Weber's Law

- Capillary rise in narrow passage
- capillary waves in channel

__Mach's Number :__It is defined as the square root ratio of inertia force to the elastic force

`M = \sqrt{\frac{F_{i}}{F_{e}}} `

`F_{e}` = Elastic force = elastic stress X Area

= KA = `KL^{2}`

`M= \sqrt{\frac{\rho A V^{2}}{K L^{2}}} = \sqrt{\frac{\rho L^{2}V^{2}}{K L^{2}}} = \sqrt{\frac{V^{2}}{\frac{K}{\rho}}} `

`\sqrt{\frac{K}{\rho }} = C `= Velocity of sound in fluid

`M = \frac{V}{C}`

Use of Mach's Number

- Aerodynamic Testing
- torpedos testing
- flying object projectile (velocity greater than speed of sound)

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