Question $2$: $(12$ Marks$)$

Consider the following laminar boundary layer velocity profile,

dary layer velocity profile, $Pa**S\frac{m}{s}**\frac{}{m}$

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

Obtain an exact expression for the shear stress ${10}^{-3}$ Pa$.$s$).$

Show that a second order finite difference approximation of $d\frac{u}{d}y$ using forward differencing at a general point $y_i$ and step $y$ is given by$,$

$($Use Taylor series expansion to evaluate $u(y_i+y)$ and $u(y_i+2y)$ to obtain the required expression$)$

$3.$ Evaluate the shear stress using expression derived in part $(2)$ at $y_i=0$ using $y={10}^{-3},{10}^{-4}$ and ${10}^{-5}.$

$4.$ Evaluate the error in calculating the shear stress. Show that the truncation error is proportional to $y^2.$