Derivation of Reynolds Number

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What is Reynolds Number?

When liquid flows through the pipe, it hits the pipe. The engineers make sure that the liquid flow through a pipe all over the city should be as steady as possible.

So, for this, a number called Reynolds number predicts if the flow of the liquid will be steady or turbulent.

Sir George Stoke introduced this concept for the first time. Later on, it was popularized by Osborne Reynolds, then the name of this number was given as Reynolds number.

Reynold number is a pure number that determines the flow of liquid through a pipe.

According to Reynold, the critical velocity vₙ of a liquid flowing through a tube of diameter D is given by  

                                             vₙ  = Nᵣη/ρD 

                  Or                       Nᵣ = ρDvₙ/η

Where η is the coefficient of viscosity of the liquid flowing through the tube

            ρ = density of the liquid

             Nᵣ = It is a constant called as a Reynold number

Here, vₙ is the critical velocity.


Point To Be Noted

The average speed of the fluid is not the same at all the places in the pipe.

It means in between the pipe; the speed is maximum, while at the surfaces, the speed is lesser, you can say close to zero, not exactly zero because of the friction introduced by the walls of the pipe.


Critical Velocity

The critical velocity is the velocity of the liquid flow, up to which the flow of the liquid is streamlined or laminar, and above which the liquid flow becomes turbulent. It is given by,

                                      vₙ  = Kη / ρr      

vₙ depends upon η, ρ, and a radius of the tube (r).

For the flow of liquid to be streamlined, the value of vₙ should be larger, while for η, the value should be as small as possible.


Derivation of Reynolds Number

Reynold’s number is defined as the ratio of the inertial forces divided to the viscous force per unit area for a flowing fluid. 

Consider a tube of a small area of cross-section A, through which a fluid of density ρ is flowing with velocity v.

The mass of the fluid through tube per second, 

∆m = volume of fluid flowing per second x density                     

     =  A v x ρ

∴ Inertial force per unit area = rate of change of momentum/area

              =  (∆m)v/A = (A v x ρ)v/A = v2ρ…(1)

Since viscous force, F = ηAv/r

Here, r is the radius of the tube, v/r is the velocity gradient between the layers of the liquid flow.

∵ Viscous force per unit area = F/A =   ηv/r….(2)

Therefore, Reynolds number = inertial force per unit area/viscous force per unit area 

 = eq(1)/eq(2), we get,                

Nᵣ = v2ρ/ ηv/r = vρr/η


The value of  Nᵣ is independent of the system of units it is being measured.


What is the significance of Reynolds number?

Let’s understand this through an example:

When you try to take out honey from the jar, the honey being viscous takes time to come out because of an adhesive force between the honey and the walls of the jar. As we give a force, the honey starts experiencing inertial force, and it starts accelerating, but at a very slow rate. 

We know that the average speed of honey would vary, however, to determine if the flow of the honey from the jar would be laminar or turbulent will depend on a constant value called the Reynolds number.

Here, mu =  dynamic viscosity of honey, as it is in motion.

So, 

Reynolds number (Nᵣ) = inertial force per unit area divided by viscous force per unit area 

Quantitatively, the value of  Nᵣ for honey is in the order of 10-4.     


What is the value of Reynolds Number?

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If the value of Nᵣ lies between 0 to 2000, the flow of the liquid is streamlined or laminar.

For values above 4000, the flow is turbulent, and between 2000 to 3000, the flow of the liquid is unstable, i.e., changing between the laminar and turbulent flow.


Reynolds Number Calculation Example

Let’s take an example to calculate the Reynolds number

Suppose the water is flowing through a pipe with a diameter of 3.5 cm. The velocity with which the water is flowing is 1.5 m/s. If the density of water is 1000 kg/m3 and the coefficient of viscosity of water is 9 x 10-4 Pa.s. Find the Reynolds number to determine if the flow of water is streamlined or turbulent.

Solution: Here we are given with,

          D = 3.5 mm = 3.5/1000 m

           vₙ  = 2 m/s

           η  = 9 x 10-4

           ρ  =  1000 kg/m3

We know the formula, i.e.,

                             Nᵣ = ρDvₙ/η

Putting the values given we are provided with, in this formula, we get,

                    = 1000 x 3.5 x 2/9 x 10-4 x 1000

On calculating, we get the value of Reynolds number:

                                 Nᵣ = 7,7777

Here, we can see the value of  Nᵣ > 4000. It indicates that the flow of the liquid is turbulent.

FAQ (Frequently Asked Questions)

Q1: What does Reynolds Number Mean?

Ans: The Reynolds number (Nᵣ) of the liquid flowing through the pipe is defined as the product of density times and velocity times length divided by the coefficient of viscosity. 

The value of  Nᵣ is proportional to the ratio of inertial forces and viscous forces per unit area in a fluid flow.

Q2: What affects Reynolds Number?

Ans: The Reynolds determines several factors that affect the flow of liquid, as the type of liquid, factors are:

  1. The roughness of the surface of the pipe.

  2. Heat transfer

  3. Vibrations

  4. Noise and other disturbances

Q3: What is the critical Reynolds Number?

Ans: A Reynold number at which the flow of liquid changes from laminar to turbulent flow is called the critical Reynolds number.

Q4: What are low and high critical points in Reynolds Number?

Ans: The low critical points in Reynolds number indicate that below a lower critical value of Reynolds number, the flow of liquid is streamlined while above a higher critical value, flow is turbulent. Between these values, the flow is in transition.