Dimensionless Numbers ,Formula derivation and there use

 Dimensionless Numbers,Formula derivation and their use

In 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)