Problem $4(20$ pts$.)$: An excellent approximation for the two$-$dimensional incompressible laminar boundary layer on the flat surface as shown in the figure

$u$~~$U(2\frac{y}{}-2\frac{y^3}{{}^3}+\frac{y^4}{{}^4})$ for $y$

where $=C{x}^{\frac{1}{2}},C=$ constant.

a$)$ Assuming a no$-$slip condition at the wall, find an expression for the velocity component $v(x,y)$ for $y.$

b$)$ Then find the maximum value of $v$ at the station $x=1m,$ for the particular case of airflow, when $U=3\frac{m}{s}$ and $=1\mathrm{.}1cm.$

c$)$ Find the equation describing the stream functions and plot them by using your favourite software.