# 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 dynamicsDimensionless 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)