Problem $3($Outcomes $4$&$1)$

The velocity profile $u(y)$ inside of a turbulent boundary layer is a function of dynamic viscosity $,$ density $,$ and shear stress ${}_w.$

$($a$)$ Determine the primary dimensions of the quantities $u_{}=\sqrt[\phantom{2}]{\frac{{}_w}{}}$ and $l=\frac{}{\sqrt[\phantom{2}]{{}_w}}.$

$($b$)$ Applying the Buckingham$-$Pi theorem with $,,$ and ${}_w$ as your repeating variables, show that there are two $-$groups and express them in terms of $\frac{u}{u_{}}$ and $\frac{y}{l}.$

The experimentally determined functional relationship, famously known as the "Law of the Wall", is

$\frac{u}{u_{}}=\frac{1}{0\mathrm{.}41}ln(\frac{y}{l})+5\mathrm{.}0$

In an experiment with a flat plate, the local air speed at distance $y=0\mathrm{.}81$ in from the plate is found to be $u=16f\frac{t}{s}.$

Note ${}_{\text{a}\text{i}\text{r}}=2\mathrm{.}33{10}^{-3}$ slugs $\frac{?}{f{t}^3}$ and ${}_{\text{a}\text{i}\text{r}}=0\mathrm{.}38{10}^{-6}lb.\frac{s}{f}{t}^2.$

$($c$)$ Using the relationship above, determine the wall shear stress ${}_w.$

pls can u send it handwritten