a$)$ For a laminar boundary layer, the steamwise distribution of the edge velocity is described as:

$\backslash [$

U$\_\{$e$\}($x$)=$K$\backslash$left$[1-\backslash$frac$\{2$ x$\}\{$L$\}\backslash$right$]$

$\backslash ]$

where $\backslash ($ K $\backslash )$ is a constant and $\backslash ($ L $\backslash )$ is the length of the surface.

i$)$ Does this flow have an adverse or a favourable streamwise pressure gradient? Explain.

ii$)$ Assuming that the boundary layer starts to develop at $\backslash ($ x$=0\backslash ),$ where along the surface would you expect the wall shear stress to have the highest value? Explain without performing calculations.

iii$)$ Using Thwaites' method, determine the streamwise variation of momentum thickness.

iv$)$ Determine whether the boundary layer separates at some streamwise location along the surface, and if so$,$ the coordinate $(\backslash ($ x $/$ L $\backslash ))$ of this location.

b$)$

i$)$ What is the advantage of using the momentum integral equation to solve for the streamwise development of a boundary layer compared to solving the boundary$-$layer equations as derived by Prandtl?

ii$)$ Why does Thwaites' method rely on experimental data to integrate the momentum integral equation?

iii$)$ How is reliance on experimental data avoided in the method of Karman and Pohlhausen for integrating the momentum integral equation?