(Solved):   Using the momentum equation amd the continuity equation. Derive using the necessary assumpt ...

 

Using the momentum equation amd the continuity equation. Derive using the necessary assumptions the non-dimentionless momentum equation to flow stream situation in the preamble.

A uniform steady flow with velocity Uo approaches a horizontal flat plate of length \( L \) and a width assumed to be semi-infinite. Starting from the momentum and continuity equations obtained a nondimensionalized momentum equation for the domain. (Hint: the dimensionless number to look out for is the Reynolds number). Also derive Prandtl's boundary layer approximation for the flow problem described in the preamble. a. Starting from Prandtl's boundary layer equation and applying the similarity transformations \( \eta=y \sqrt{\frac{U_{o}}{2 v x}} ; \mathrm{u}=U_{o} \frac{d f}{d \eta} \) and \( \mathrm{v}=\left(\eta f^{\prime}-f\right) \sqrt{\frac{2 U_{o} v}{x}} \), show that \[ \frac{d^{3} f}{d \eta^{3}}+f \frac{d^{2} f}{d \eta^{2}}=0 ; f(0)=1, f^{\prime}(\infty)=0 \] b. By using an appropriate solver, Plot the solution of \( f \), \( f \) ' and \( f^{\prime \prime} \) against \( \eta \). c. Determine the wall shear stress