Given the velocity distribution in a laminar boundary layer on a flat plate as $\frac{\upsilon}{U}=2(\frac{y}{\delta})-2(\frac{y}{\delta})^3+(\frac{y}{\delta})^4$

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 boundary layer thickness, shear stress, and force on one side of the plate in terms of Reynolds number.

Solution :

The velocity profile, $$\quad \frac{v}{U}=\frac{2 y}{\delta}-\frac{2 y^{3}}{\delta^{3}}+\frac{y^{4}}{\delta^{4}}$$

$$ \frac{\tau_{0}}{\rho U^{2}}=\frac{\partial}{\partial x}\left[\int_{0}^{\delta} \frac{v}{U}\left(1-\frac{v}{U}\right) d y\right] $$

Substituting the given velocity profile in the above equation

$$ \begin{aligned} \frac{\tau_{0}}{\rho U^{2}} & =\frac{\partial}{\partial x}\left[\int_{0}^{\delta}\left(\frac{2 y}{\delta}-\frac{2 y^{3}}{\delta^{3}}+\frac{y^{4}}{\delta^{4}}\right)\left(1-\left\{\frac{2 y}{\delta}-\frac{2 y^{3}}{\delta^{3}}+\frac{y^{4}}{\delta^{4}}\right\}\right) d y\right] \\ &=\frac{\partial}{\partial x}\left[\int_{0}^{\delta}\left(\frac{2 y}{\delta}-\frac{2 y^{3}}{\delta^{3}}+\frac{y^{4}}{8^{4}}\right)\left(1-\frac{2 y}{8}+\frac{2 y^{3}}{\delta^{3}}-\frac{y^{4}}{\delta^{4}}\right) d y\right] \\ &=\frac{\partial}{\partial …