## Variables needed for getting dimensionless groups

In most of the fluid phenomena the following variables may be important
- Length. L

- Acceleration due to gravity, g

- Mass density, ρ

- Velocity, V

- Pressure, p

- Viscosity, µ

- Surface tension, σ

` `

-Velocity of sound C## Forces needed to get the Dimensionless Numbers in Fluid Mechanics

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The following forces can be formed from these variables.
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`Inertia force = mass x acceleration = ρL`^{3 }x(V/t)= ρL^{2}V x(L/t)= ρL^{2}V^{2}

`Viscous force = area X shear stress = L`^{2} x µ x (V/L) = µVL

`Gravity Force = mass X acceleration due to gravity = ρL`^{3}g

Pressure Force = pressure x area = pL

^{2}

Elastic Force = bulk modulus of elasticity X
area = E x L

^{2}= ρC^{2}L^{2}, Since E = ρC^{2}
Surface Tension Force = σL

The following dimensionless groups can be formed
by combining inertia force with each of the independent forces.

##
Reynolds Number in Fluid Mechanics, N_{Re }

It is the ratio of inertia force and viscous
force.

Reynolds number formula, N

_{Re = }Inertia force / Viscous force = ρL^{2}V^{2 }/ µVL = ρVL/µ
So , Reynolds number equation = ρVL/µ . For getting

**Reynold Number**using kinematic viscosity we have to divide the dynamics viscosity µ by the density ρ.
Reynolds number similarity is used when viscous
force is predominant. For example, flow through pipes completely submerged
flow, flow through venturimeter and orificemeter etc.

### Froude Number, Fr

It is the ratio of inertia force to gravity
force.

Froude Number , Fr = Inertia Force / Gravity
Force = ρL

^{2}V^{2 }/ ρL^{3}g = V^{2}/Lg
Froude
number is used when gravitational force is predominant in the fluid motion. For
example open channel flow, wave motion in the ocean, forces on bridge piers and
off shore structures.

### Euler Number , E

It is the ration of pressure force and inertia
force.

Euler Number = Pressure force / Inertia Force = pL

^{2}/ ρL^{2}V^{2}= p/ρV^{2}= F/ ρV^{2}L^{2 }
Euler number is important when pressure force is
predominant. For example, flow through pipes, flow over submerged bodies, flow
of water through orifices and mouthpieces etc.

### Mach number, M

`Square root of the ratio of inertia force to the elastic force `

**Mach Number**= (Inertia force / elastic force)

^{1/2 }= (ρL

^{2}V

^{2 }/ ρC

^{2}L

^{2})

^{1/2 }= V/C

The ratio v

^{2}/c^{2}is known as Cauchy’s number. Mach number is important in compressible fluid flow at high velocities. For example high velocity flow in pipes, motion of high velocity projectiles and missiles.**Weber Number, W**

It is defined as the ratio of inertia force to
the surface tension force.

Weber Number = Inertia force / Surface tension
force = ρL

^{2}V^{2}/ σL = ρLV^{2}/ σ
Weber number is important when surface tension
is important. For example, capillary tube flow, droplet formation, human in
blood flow etc.

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