Consider a steady, incompressible $(=$ constant$),$ laminar boundary layer flow over a surface with free$-$stream velocity $U(x),$ which may vary with the distance from the leading edge, $x.$ Assuming negligible body forces and constant fluid viscosity:

$($a$)$ Show that the pressure in the boundary layer will be:

$P(x)=P_0+\frac{}{2}(U_0^2-U(x{)}^2)$

where $P_0$ and $U_0$ are the velocity and pressure at $x=0.$ List the assumptions you made in showing the result.

$($b$)$ How would the KMIE given below change if $P(x)=$ constant for the flow?

$\frac{{}_W}{}=\frac{d}{dx}\{U^2{\displaystyle \underset{0}{\overset{}{}}}\frac{u}{U}(1-\frac{u}{U})dy\}+U\frac{dU}{dx}\{{\displaystyle \underset{0}{\overset{}{}}}(1-\frac{u}{U})dy\}$

$($c$)$ Let's assume that for $P(x)=$ constant, I want to analytically find the shape of the velocity profile in the boundary layer using a quadratic equation:

$\frac{u}{U}=a+b(\frac{y}{})+c(\frac{y}{}{)}^2$

where $a,b$ and c are constants. I have $3$ unknowns in $a,b$ and c $.$ What boundary conditions for velocity would you use to solve this problem? Why?