Answer

Verified

53.7k+ views

**Hint**The Reynolds number can be said as the ratio of the inertial force to that of the viscous force in a fluid. It can be derived from the given parameters by writing its dimensions and equating it with the unit less Reynolds number and simplifying it.

**Complete step by step solution**

It is given that the Reynold number depends on the velocity, density, coefficient of viscosity and the diameter. Hence this relation is also written as

$R = K{v^a}{\rho ^b}{\eta ^c}{D^1}$

The constant $K$ is added because there may be some change in the value even though it is derived from the given relation. Let us write the dimension of the above parameters to derive the formula. We know that the dimension of the velocity is $m{s^{ - 1}}$ , the dimension of the density is $Kg{m^{ - 3}}$ , coefficient of viscosity is $sc{m^{ - 1}}$ and for the diameter is $m$.

${M^0}{L^0}{T^0}$= ${\left[ {L{T^{ - 1}}} \right]^a}{\left[ {M{L^{ - 3}}} \right]^b}{\left[ {M{L^{ - 1}}{T^{ - 1}}} \right]^c}{L^1}$

In the above step, the Reynold number is substituted with no unit.

${M^0}{L^0}{T^0}$= ${M^{b + c}}{L^{a - 3b - c + 1}}{T^{a - c}}$

By equating the corresponding dimensions and the values. First let us equate the values of the mass.

$b + c = 0$

$b = - c$ --------------(1)

Then the parameters of length are considered.

$a - 3b - c + 1 = 0$

Substitute the value of the (1) in the above equation,

$a + 3c - c = - 1$

$a + 2c = - 1$ ---------------(2)

Then the parameter of time is considered.

$ - a - c = 0$

$a = - c$

Substituting this value in the equation (2),

$ - c + 2c = - 1$

$c = - 1$

$a = b = 1$

Substituting these in the dimension equation, we get

$R = K{v^1}{\rho ^1}{\eta ^{ - 1}}{D^1}$

Hence the above equation may also written as

$R = K\dfrac{{v\rho D}}{\eta }$

The above formula holds for the Reynolds number.

**Note**The Reynolds number determines the characteristics of the fluid that either it is streamlined or not. If the Reynolds number is less than $2000$ , it is laminar flow and if the Reynolds number is between $2000$ to $4000$ , it is turbulent flow.

Recently Updated Pages

Why do we see inverted images in a spoon class 10 physics JEE_Main

A farsighted man who has lost his spectacles reads class 10 physics JEE_Main

State whether true or false The outermost layer of class 10 physics JEE_Main

Which is not the correct advantage of parallel combination class 10 physics JEE_Main

State two factors upon which the heat absorbed by a class 10 physics JEE_Main

What will be the halflife of a first order reaction class 12 chemistry JEE_Main

Other Pages

The nitride ion in lithium nitride is composed of A class 11 chemistry JEE_Main

The resultant of vec A and vec B is perpendicular to class 11 physics JEE_Main

A given ray of light suffers minimum deviation in an class 12 physics JEE_Main

Electric field due to uniformly charged sphere class 12 physics JEE_Main

Oxidation of succinate ion produces ethylene and carbon class 12 chemistry JEE_Main

when an object Is placed at a distance of 60 cm from class 12 physics JEE_Main