Question $2$

a$)($a$)$ The velocity profile within a flat$-$plate boundary layer may be approximated by $u=$

Asin$(By).$ Use the boundary conditions $u()=U_{},$ and del$\frac{u}{d}el{y}_{}=0,$ to determine the

constants A and B$.$

b$)$ Show that the momentum thickness for the profile of part $($a$)$ is given by $\frac{}{}=\frac{2}{}-\frac{1}{2}.$

You will need to make use of the relation ${\displaystyle \underset{0}{\overset{}{}}}sin2d=\frac{1}{2}(-sincos).$

Show also that

$\frac{c_f}{2}=\frac{}{2}\frac{v}{U_{}}$

c$)$ Use the results of part $($b$)$ and the momentum$-$integral equation for a flat$-$plate boundary layer

$\frac{d}{dx}=\frac{c_f}{2}$

to find a differential equation for $.$ Solve the equation and then show that

$\frac{c_f}{2}\sqrt[\phantom{2}]{R{e}_x}=0\mathrm{.}328$