Chapter 5, Problem 44P

A parabolic approximate velocity profile was used in Problem 5.11 to model flow in a laminar incompressible boundary layer on a flat plate. For this profile, find the x component of acceleration, a_x, of a fluid particle within the boundary layer. Plot a_x at location x = 0.8 m, where δ = 1.2 mm, for a flow with U = 6 m/s. Find the maximum value of a_x at this x location.

5.11 A useful approximation for the x component of velocity in an incompressible laminar boundary layer is a parabolic 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 = 2 (y/δ) − (y/δ)², where δ = cx^1/2 and c is a constant. Show that the simplest expression for the y component of velocity is $\frac{\upsilon }{U}=\frac{\delta }{x}\left[\frac{1}{2}{\left(\frac{y}{\delta }\right)}^2-\frac{1}{3}{\left(\frac{y}{\delta }\right)}^3\right]$

Plot υ/ U versus y/δ to find the location of the maximum value of the ratio u/U. Evaluate the ratio where δ = 5 mm and x = 0.5 m.