Problem $5(25$ pts$.)$: In this problem, we will study the boundary layer developed near a sharp flat plate with lenght $L$ and width $b$ that is parallel to a stream of velocity $U_{}.$ Assume the following velocity profile

$u$~~$U_{}sin(\frac{y}{2})$ for $y.$

a$)$ Explain which constraints are satisfied by this velocity profile. Can we use $u$~~$U_{}sin(\frac{y}{})$ or $u$~~$U_{}cos(\frac{y}{2})?$

b$)$ Calculate the drag force $D$ from the control volume analysis.

c$)$ Calculate the shear stress ${}_w$ on the plate at a distance $x$ from the leading edge by using your result at part a$.$

d$)$ Calculate ${}_w$ from Newton's law of viscosity.

e$)$ Comparing the two expressions from parts b and c $,$ find $\frac{}{x}$ as a function of Re$.$

f$)$ Explain the concept of displacement thickness ${}^{**}$ and calculate it for the given approximate velocity profile.

g$)$ Explain the concept of momentum thickness $$ and calculate it for the given approximate velocity profile.