Question $2$

a$)$ A laminar boundary layer profile may be assumed to be approximately of the form

$\frac{u}{U_e}=f()=f(\frac{y}{})$

i$)$ Use an integral analysis with the following two$-$segment velocity profile,

$f()=(\frac{}{6})(10-3-{}^3),$ for $00\mathrm{.}293$ and $f()=sin(\frac{}{2})$ for $0\mathrm{.}2931,$

to find expressions for the displacement thickness $,$ the momentum thickness $,$

the shape factor $H,$ the skin$-$friction coefficient $C_t,$ and the drag coefficient $c_D.$

ii$)$ Derive an expression for the velocity normal to the stream wise direction given that,

$\frac{u}{U_e}=f().$

iii$)$ Hence obtain the velocity normal to the stream wise direction for the above two$-$

segment velocity profile.