Q$1$

For a laminar flow over a flat plate, we can assume the velocity profile as a polynomial in $y$ as: $u=a+by+c{y}^2.$

$($i$)$ If the incoming velocity is $U,$ state all physical boundary conditions for the velocity $u$ and $\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{y}}$ on the wall and at the upper edge of the boundary layer $$ as shown in Figure Q$1$

$[15\%]$

$($ii$)$ Based on the boundary conditions in $($i$),$ calculate the value of $a,b,c$ and determine the velocity profile $u=u(y)$;

$[20\%]$

$($iii$)$ From the calculated velocity profile, calculate the wall shear stress ${}_w={}_w(),$ and displacement boundary layer thickness ${}^{**}$;

$[25\%]$