Question

Reynold’s number R decides that the flow in any pipe is streamlined or not? This constant is combination of velocity (v), density (ρ) and coefficient of viscosity (η). It is given that value of R is proportional to diameter (D) of the pipe. Establish the relation for R using dimensional method.

Answer 1

Suppose the value of R depends upon v, ρ, η and D as under:

R ∝ D¹; R ∝ v^a

R ∝ ρ^b; R ∝ η^c

∴ R = k D¹v^aρ^bη^c .........(1)

Dimensional fromula of L.H.S. = [M⁰L⁰T⁰]

Dimensional formula of R.H.S. = [L¹]¹ [L¹T^-1]^a [M¹L^-3]^b [M¹L^-1T^-1]^c

= L¹.L^aT^-aM^bL^-3bM^cL^-cT^-c

or Dimensional formula of R.H.S. = [M^b+cL^a-3b-c+1T^-a-c]

For the validity of formula (1), the dimensions on both sides should be equal. Therefore on comparing the dimensions, we get

b + c = 0 ...........(2)

a - 3b - c + 1 = 0 .............(3)

-a - c = 0 ...........(4)

On solving equations (2), (3) and (4), we get

a = 1; b = 1 and c = -1

∴ From equation (1)

R = \(\frac{kDup}{\eta}\)

This is the required relation.