Question

EXAMPLE 58. Reynold number Ne (a dimensionless
quantity) determines the condition of laminar flow of a viscous
liquid through a pipe. NR is a function of the density of the
liquid 'p', its average speed 'v' and coefficient of viscosity 'm'.
Given that NR is also directly proportional to 'D' (the
diameter of the pipe), show from dimensional considerations
that NR
POD
n​

Answer 1

Answer:

Let

NR = K × (p^a) × (v^b) × (n^c) × D^1

...... (i)

, where K is dimensionless constant. Writing dimensions in (i) we get,

[M° × L° × T°] = (ML ^ (-3))^a × (LT ^(-1))^b × (ML^(-1))^c

= M^(a + b) × L^(-3a + b - c + 1) × T^(-b - c)

Equating the power of M, L, T, we get , a+c =0,

=> -3a +b -c +1 = 0,

=> -b-c = 0

:. a = -c, b = -c

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

= 3c -c-c+1 = 0, c = -1,

:. a = 1, b = 1.

Since,

Nr = K × p^1 × v^1 × n^(-1) × D = KpvD / n

Thus,

Nr ∝ KpvD / n

Hence, proved.

Answer 2

Answer:

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