For the velocity profile for laminar boundary flow $\frac{\upsilon}{U}=sin(\frac{\pi y}{2\delta})$

where u is the velocity at the distance y from the surface of the flat plate and U be the free stream velocity at the boundary layer thickness $\delta$. Obtain an expression for the

boundary layer thickness and the average drag coefficient in terms of the Reynolds Number.

Solution :

The velocity profile is $\dfrac v U=sin\left(\dfrac \pi 2 \dfrac y \delta\right)$

$$ \begin{aligned} \frac{\tau_{0}}{\rho U^{2}} &=\frac{\partial}{\partial x}\left[\int_{0}^{\delta} \frac{v}{U}\left(1-\frac{v}{U}\right) d y\right]\\ &=\frac{\partial}{\partial x}\left[\int_{0}^{\delta} \sin \left(\frac{\pi}{2} \frac{y}{\delta}\right)\left[1-\sin \left(\frac{\pi}{2} \frac{y}{\delta}\right)\right] d y\right] \\ &=\frac{\partial}{\partial x}\left[\int_{0}^{\delta}\left[\sin \left(\frac{\pi}{2} \frac{y}{\delta}\right)-\sin ^{2}\left(\frac{\pi}{2} \frac{y}{\delta}\right)\right] d y\right] \\ &=\left[\frac{\partial}{\partial x}\left[\frac{-\cos \frac{\pi y}{2 \delta}}{\frac{\pi}{2 \delta}}\right]-\left[\frac{\frac{\pi y}{2 \delta} …