Dimensional Formula of Reynolds Number (Re)

The physical quantity which helps determine the flow pattern of the liquid categorising it into a turbulent or laminar flow while it flows through a pipe. The ratio of inertial forces to viscous results in Reynolds number. The quantity is a dimensionless quantity. The laminar flow is termed as the type of flow in which the fluid moves in a particular pattern of straight lines whereas the turbulent flow, on the other hand, is an irregular flow of the fluid with an undefined pattern. The flow under 1100 Reynolds number is laminar and above 2200 is turbulent.

Reynolds Number (Re):

In 1883 Osborne Reynolds, a British architect and physicist, showed that the progress from laminar to fierce stream in a line relies on the worth of a numerical amount equivalent to the average speed of stream times the measurement of the cylinder times the mass thickness of the liquid isolated by its outright consistency. This numerical amount, an unadulterated number without aspects, became known as the Reynolds number and was consequently applied to different kinds of the stream encased or including a moving item inundated in a liquid.

The Reynolds number is the proportion of inertial powers to gooey powers. The Reynolds number is a dimensionless number used to arrange the liquids frameworks. The thickness impact is significant in controlling the speeds or the stream example of a liquid. The Reynolds number is given by Re. 

The Reynolds number is used to decide if a liquid is in the laminar or violent stream. Reynolds number is not precisely or equivalent to 2100 to demonstrate a laminar stream, and a Reynolds number more prominent than 2100 to show a violent stream. 

Reynolds Number Formula

Re = ρVD / μ 

Where,

ρ = Inscribes the Fluid density

V = Inscribes the Fluid velocity

μ = Inscribes the Fluid viscosity

D = Inscribes the length or diameter of the fluid.

The Reynolds Number units

The Reynolds number is the proportion of a liquid’s inertial power to its gooey power. Inertial power includes power because of the mass of streaming liquid. Consider it a proportion of how troublesome it is to change the speed of a streaming liquid. Gooey powers manage the erosion of a streaming liquid. Consider pouring some tea as opposed to pouring cooking oil. The cooking oil has a higher thickness since it’s more impervious to stream.

You are right if you think inertia and thick power are the same. They are comparative in that they have similar units! This implies the Reynolds number is united. We can decide if a liquid stream is a laminar or violent dependent on the Reynolds number.

Assuming the Reynolds number is under 2300, the stream is laminar. Any Reynolds number more than 4000 shows a violent stream. The middle of these qualities demonstrates a transient stream, which implies the liquid stream is progressing among laminar and tempestuous streams. This happens typically for a brief timeframe toward the start or end of the liquid stream when a valve or fixture is turned on or off. How about we check out the situation for the Reynolds number.

Dimensional formula of Reynolds Number:

The dimensional formula of Reynolds Number can be written as:

[M0L0T0]

Where,

M = Mass

L = Length

T = Time

Derivation of Dimensional Formula of Reynolds Number (Re):

Reynolds Number = (Density). (Velocity). (Length). (dynamic viscosity)-1 

•  Density (ρ) = (Mass) / (Volume)

The dimensional formula of density equals to

[M1L-3T0] …(i)

• Velocity = (Displacement)/ (Time)

Therefore, the dimensions of velocity can be written as 

[M0L1T-1] …(ii)

Viscosity = (Distance between layers). (Force) / (Area × Velocity)

• The dimensional formula for the Force is [M1L1T-2]

 Therefore, the dimensional formula of viscosity can be written as

η =[L].[M1L1T-2].[L2 × L1T-1]-1 = [M1L-1T-1] …(iii)

• Reynolds Number = (Density).(Velocity). (Length).(dynamic viscosity)-1 

From substituting values from equations (i), (ii) and (iii) 

• The dimensional formula expressed as

Re = [M1L-3T0].[M0 L1 T-1].[L].[M1L-1T-1]-1 

Hence, the Reynolds Number can be represented dimensionally as [M0L0T0].

Conclusion:

The fluids possess the nature of laminar and turbulence while flowing which is dependent on various factors like shear force, viscosity, the density of the fluid, the movement of layers of fluid (parallel or unparallel). The term that determines the movement of the fluids and categorises them into laminar and turbulent is termed as Reynolds number, which is a range of numbers, it is a dimensionless physical quantity. The Reynolds number has the range below or 1100 for laminar flow of the fluids and 2200 or above for turbulent flow of the liquid. The critical Reynolds number determines the laminar-turbulent transition.