$3.(30$ pts$)$ In a laboratory experiment, the boundary layer is measured along a solid wall. The

velocity profile shape in the boundary layer fits almost exactly to a sine wave, i$.$e$.,$

u$($y$)=$ umax sin$($c$1$y$),$ where umax is the maximum velocity, measured at the edge of the boundary

layer. Let be the y location of the edge of the boundary layer, as sketched.

$($a$)$ Generate an expression for constant c$1$ such that the no slip condition is satisfied at the wall

$($y$=0),$ and u $=$ umax at the edge of the boundary layer $($y $=$ $).$ Then find an expression for the

shear stress, du

dy

$=,$ at any y location in the boundary layer.

$($b$)$ The fluid is air at $14\mathrm{.}83$ oC$,$ umax is $10\mathrm{.}0$ m$/$s$,$ and the boundary layer thickness is $6\mathrm{.}00$ mm$.$ Calculate the shear stress in

Pascals at y $=0$ mm$,$ i$.$e$.,$ right at the wall. Note: Use Sutherland$$s law for air viscosity.

$($c$)$ For these same conditions, calculate the shear stress in Pascals at y $=3\mathrm{.}0$ mm $($y $=$ $/2).$

$($d$)$ For these same conditions, calculate the shear stress in Pascals at y $=6\mathrm{.}0$ mm $($y $=$ $).$