Please show step by step works Engineering mechanic of Flui

Please show step by step works ( Engineering mechanic of Fluids)

At what distance r from the center of a tube of radius r_0 does the average velocity occur in a laminar flow?

Solution

Radial distribution of velocity in tube with laminar flow is given by U=Umax(1-(r/r₀)²).

Also average veleocity U_av = U_max /2.

U_av =Umax(1-(r_a/r₀)²) = U_max/2

(1-(r_a/r₀)²) = 1/2

(r_a/r₀)² = 1-1/2=1/2

r_a/r_o =Sq root of (1/2)

r_{a =} 0.707 r_o