Q$4.$ Reduce the Navier Stokes equations for the laminar boundary layer with clearly stated assumptions as given by Prandtl for fluid flow over a flat plate where $U_{}$ : free stream velocity. Evaluate the ratio of displacement thickness $({}^{**})$ to the boundary layer thickness $()$ i$.$e$.,*\frac{*}{}$ for the following velocity profiles $($i$)($ii$)($iii$)$ and state for which case the shear stress acting on the flat plate is maximum.

Velocity profiles: $($i$)\frac{u}{U_{}}=(\frac{y}{})$

$($ii$)\frac{u}{U_{}}=2(\frac{y}{})-(\frac{y}{}{)}^2$

$($iii$)\frac{u}{U_{}}=sin(\frac{}{2})(\frac{y}{}).$

Sketch the boundary layer for the three cases $($i$)($ii$)($iii$)$ with the y $-$axis: $\frac{y}{}$ and x $-$axis: $\frac{u}{U_{}}.$

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