Problem $1(10$ pts$.)$: A sharp plate with lenght $L$ and width $b$ is parallel to a stream of velocity $U_{}.$ The expected functional relations for the boundary layer thickness $$ and the shear stress on the plate ${}_w$ is as follows

$=f(x,,U_{}),{}_w=f(x,,U_{}),$

where $x$ is the distance to the leading edge, $$ and $$ are the density and the viscosity of the fluid respectively.

a$)$ By using Buckingham $$ theorem, show that the dimensionless form of these functional relationships are given by

$\frac{}{x}=f(Re),C_F=f(Re),$

where $Re=\frac{U_{}x}{}$ is the Reynolds number and $C_F=\frac{{}_w}{\frac{1}{2}{U}_{}^2}$ is the skin friction coefficient.

b$)$ Can you justify our use of drag coefficient as $C_D=\frac{D}{\frac{1}{2}{U}_{}^{2}Lb}$ where $D$ is the total drag force.