Chapter 5, Problem 12P

A useful approximation for the x component of velocity in an incompressible laminar boundary layer is a cubic variation from u = 0 at the surface (y = 0) to the freestream velocity, U, at the edge of the boundary layer (y = δ). The equation for the profile is $u/U=\frac{3}{2}(y/\delta )-\frac{1}{2}{(y/\delta )}^3$, where δ = cx^1/2 and c is a constant. Derive the simplest expression for υ/U, the y component of velocity ratio. Plot u/U and υ/U versus y/δ, and find the location of the maximum value of the ratio υ/U. Evaluate the ratio where δ = 5 mm and x = 0.5 m.