For the Velocity profile for Laminar Boundary Layer : $\frac{u}{U}=\frac{3}{2}(\frac{y}{\delta})-\frac{1}{2}(\frac{y}{\delta})^2$ Determine Boundary layer thickness, Shear stress, Drag force and coefficient of Drag in terms of Reynold’s number.

Solution :

Given :

Velocity distribution,

$$\quad \frac{u}{U}=\frac{3}{2}\left(\frac{y}{\delta}\right)-\frac{1}{2}\left(\frac{y}{\delta}\right)^{3}$$

We have

$$\frac{{t}_{0}}{\rho U^{2}}=\frac{\partial}{\partial x}\left[\int_{0}^{5} \frac{u}{U}\left({I}-\frac{\mu}{U}\right) d y\right]$$

Substituting the value of $\frac{u}{U}=\frac{3}{2}\left(\frac{y}{\mu}\right)-\frac{1}{2}\left(\frac{y}{\mu}\right)^{3}$ in the above equation

$\begin {aligned} \frac{\tau_{0}}{\rho U^{2}} &=\frac{\partial}{\partial x}\left[\int_{0}^{\delta}\left[\frac{3}{2}\left(\frac{y}{\delta}\right)-\frac{1}{2} \left(\frac{y}{\delta}\right)^{3}\right]\left[1-\left\{\frac{3}{2}\left(\frac{y}{\delta}\right)-\frac{1}{2}\left(\frac{y}{\delta} \right)^{3} \right\} \right] d y\right]\\ &=\frac{\partial}{\partial x} \left[\int_{0}^{\delta} \left(\frac{3 y}{2 \delta}-\frac{y^{3}}{2 \delta^{3}} \right)\left(1-\frac{3 y}{2 \delta}+\frac{y^{3}}{2 \delta^{3}}\right) d y\right]\\ …