## 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

```
```The following forces can be formed from these variables.

```
```Inertia force = mass x acceleration = ρL3 x(V/t)= ρL2V x(L/t)= ρL2V2

```
```Viscous force = area X shear stress = L2 x µ x (V/L) = µVL

```
`Gravity Force = mass X acceleration due to gravity = ρL3g`

Pressure Force = pressure x area = pL2

Elastic Force = bulk modulus of elasticity X area = E x L2 = ρC2L2, Since E = ρC2

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, NRe

It is the ratio of inertia force and viscous force.

Reynolds number formula,  NRe = Inertia force / Viscous force = ρL2V2 / 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 =  ρL2V2 / ρL3g = V2/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 = pL2 / ρL2V2 = p/ρV2 = F/ ρV2L2

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 =  (ρL2V2 / ρC2L2)1/2 = V/C

The ratio v2/c2 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 = ρL2V2/ σL = ρLV2/ σ

Weber number is important when surface tension is important. For example, capillary tube flow, droplet formation, human in blood flow etc.