A parabolic velocity profile was used to model flow in a laminar incompressible boundary layer in Problem 5.11. Derive the stream function for this flow field. Locate streamlines at one-quarter and one-half the total volume flow rate in the boundary layer.

Data From Problem 5.11

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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 = 8). The equation for the profile is u/U= 2(y/6)-(y/6), where 8 = cx/2 and c is a constant. Show that the simplest expression for the y component of velocity is v U = x 2 2 1 -- Plot v/U versus y/8 to find the location of the maximum value of the ratio v/U. Evaluate the ratio where 8=5 mm and x = 0.5 m.