Problems #$1$ & #$2(2x$ credit$)$

Consider two$-$dimensional, steady flow along a flat plate of length $L$ and width $b$ at zero inclination

to incoming flow of constant speed $U.$ Assume the velocity profile is given by the $4$th order

polynomial:

$u=U[2(\frac{y}{})-2(\frac{y}{}{)}^3+(\frac{y}{}{)}^4].$

Determine the following:

$($a$)$ Boundary layer thickness $$ as a function of $x$ and downstream distance$-$based Reynolds

number $(R{e}_x=U\frac{x}{v}).$

$($b$)$ Momentum thickness $$ as a function of $x$ and $R{e}_x.$

$($c$)$ Displacement thickness ${}^{**}$ as a function of $x$ and $R{e}_x.$

$($d$)$ Skin friction coefficient $c_f$ as a function of $R{e}_x.$

$($e$)$ Drag force acting on both sides of the plate, evaluated using wall shear stress $({}_w)$ :

$D_{1-\text{s}\text{i}\text{d}\text{e}}=b{\displaystyle \underset{0}{\overset{L}{}}}{}_{w}dx$