Important Dimensionless Groups in Fluid Mechanics–Reynolds Number

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 = ρL³x(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, N_Re

It is the ratio of inertia force and viscous force.

Reynolds number formula, N_{Re =}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.

Dimensionless groups in fluid mechanics reynolds number

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.

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