$u/U= 2(y/δ) - (y/δ)^{2}$Where U is the mean stream velocity and δ is the boundary layer thickness . Determine the displacement thickness and momentum thickness.

1)Determine momentum thickness:

θ=$∫_0^δ\frac{u}{U}(1-\frac{u}{U})dy$

Substitute value of $\frac{u}{U}$ in above equation we get,

$θ=[∫_0^δ{\frac{2y}{δ}-\frac{y^2}{δ^2}}{1-(\frac{2y}{δ}-\frac{y^2}{δ^2})} dy]$

Making substitution

Let $\frac{y}{δ}=$t

∴$y=δt$

$dy=δ$dt

Changing limit at $y=0, t=0$

At $y=δ,t=0$

∴$θ=∫_0^1[2t-t^2 ][1-2t+t^2 ]δdt$

=$δ[∫_0^1[2t-t^2 ][1-2t+t^2 ]$

$θ=0.1333δ ……… (ans)(1)$

Boundary layer thickness:

$\frac{u}{U}=\frac{2y}{δ}-(\frac{y}{δ})^2$

We know, $\frac{\tau_0}{ρU^2}=\frac{d}{dx} [∫_0^δ \frac{u}{U}(1-\frac{u}{U})dy]$

Substituting the value of …