$3.$ Boundary layer grows as air $($kinematic viscosity $\backslash (=1\mathrm{.}5\backslash$times $10^\{-5\}\backslash$mathrm$\{$~m$\}^\{2\}/\backslash$mathrm$\{$s$\}\backslash )$ and density $\backslash (=1\mathrm{.}23\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ))$ moves over a flat plate with a freestream velocity of $\backslash (10\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$

$($a$)$ Consider two parallel streamlines entering the boundary layer. Sketch them within the boundary layer showing their position with respect to the flat plate and the edge of the boundary layer.

$($b$)$ Considering a parabolic velocity distribution within the laminar boundary layer, find out its expression.

$($c$)$ For the above velocity profile, find out the expression of wall shear stress.

$($d$)$ Using momentum integral equation find out the variation of boundary layer thickness along the flat plate.

$($e$)$ Find out the distance from the leading edge of the plate where the thickness of the boundary layer is $2\mathrm{.}5$ mm $.$