Chapter 5, Problem 45P

A cubic approximate velocity profile was used in Problem 5.12 to model flow in a laminar incompressible boundary layer on a flat plate. For this profile, obtain an expression for the x and y components of acceleration of a fluid particle in the boundary layer. Plot a_x and a_y at location x = 3ft, where δ = 0.04 in., for a flow with U = 20 ft/s. Find the maxima of a_x at this x location.

5.12 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 u/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.