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The velocity profile in fully developed laminar flow in a pipe of diameter D is given by \(u = {u_0}\left( {1 - \frac{{4{r^2}}}{{{D^2}}}} \right)\), where r is the radial distance from the center. If the viscosity of the fluid is μ, the pressure drop across a length L of the pipe is:
1. \(\frac{{\mu {u_0}L}}{{{D^2}}}\)
2. \(\frac{{4\mu {u_0}L}}{{{D^2}}}\)
3. \(\frac{{8\mu {u_0}L}}{{{D^2}}}\)
4. \(\frac{{16\mu {u_0}L}}{{{D^2}}}\)

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Correct Answer - Option 4 : \(\frac{{16\mu {u_0}L}}{{{D^2}}}\)

Explanation:

Pressure drop across a length L of pipe is:

\({\rm{\Delta }}p = \frac{{32\mu\ {u_{av}}L}}{{{D^2}}}\)    ......(i)

where uavg =  average velocity.

Given velocity profile is

\(u = {u_0}\left( {1 - \frac{{4{r^2}}}{{{D^2}}}} \right)\)

For parabolic velocity profile, maximum velocity occurs at r = 0 i.e. umax =  u0.

Relation between average and maximum velocity is

\({u_{av}} = \frac{{{u_0}}}{2}\)    ......(ii)

From equation (i) and (ii):

\({\rm{\Delta }}P = \frac{{16\mu {\mu _0}L}}{{{D^2}}}\)

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