Consider a boundary layer with no external pressure gradient (a flat plate)

Consider a boundary layer with no external pressure gradient (a flat plate). Then, from the Prandtl boundary-layer equation, deduce that μ d2vx/ dy2 = 0 at y = 0 Now derive an expression for the second derivative of velocity at the surface for the case where there is a finite pressure gradient. In particular, show that μ d2vx /dy2 y=0 = −ρve dve dx where ve is the x-component of the velocity at the edge of the boundary layer. Using the above result, show that the following velocity profile can be used in the presence of a finite pressure gradient: vx/ ve = 3/ 2 η − η3/ 2 + δ2 /4ν dve/ dx [η − 2η2 + η3 ]where η is defined as y/δ. Also ν = μ/ρ as usual. This can then be used in conjunction with the von Kármán integral momentum balance to find the boundary-layer thickness and the velocity profiles for non-flat surfaces.