$($a$)$ For fully developed turbulent flow in a channel between two parallel plates, the velocity profile is: overline u u$^*=1/$k ln$($ u$^*$ y nu $)+$D where u is the frictional velocity, k $=0\mathrm{.}41$ and D $=5$

$($i$)$ Find an expression for the average velocity.

$[5]$

$($ii$)$ Defining the Reynolds number in terms of the hydraulic diameter D h $(=4$h h is the channel half width$),$ show that the Darcy's friction factor is approximately given by $1/($sqrt$($f$))=2*$ log$\_10($R$*$e$\_\{10\}*$ sqrt$($f$))-1\mathrm{.}19[2]$

$($b$)$ Find the friction drag on the second half of a flat plate of length L if the boundary layer is