Consider laminar flow in a pipe of diameter D of a fluid of viscosity mu and den

Question

Consider laminar flow in a pipe of diameter D of a fluid of viscosity mu and density rho with average velocity experiences a shear force at the wall tau_w. The dimensionless friction factor f = tau_w/rho^2 depends on the inverse of the Reynolds number Re = rho D/mu. Show this by dimensional analysis using [tau_w] = []^a[rho]^b[mu]^c[D]^d which indicates that [M/Lt^2] = [L/t]^a[M/L^3]^b[M/Lt]^c [L]^d. Recall that M, L, t represent the fundamental dimensions of mass, length, time. You will have 3 equations and 4 unknowns, choose exponent c to solve for, i.e. retain c.

Explanation / Answer

[M/Lt2] = [L/t]a[M/L3]b[M/Lt]c[L}d

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

-1 = a-3b-c+d----------(2)

-2 = -a-c------(3)

According to dimensional formula for viscosity of a fluid c=1

Hence equations (1), (2) and (3) can be rewritten as

1 = b-1---------------(1) so b =2

-1 = a-3b-1+d----------(2)

-2 = -a-1------(3) so a =1

Substituting the values a, b in equation (2), we get

-1 = 1-6-1+d

So d=5

Thus, a =1, b=2, c=1, d=5

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