Question

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 = cx1/2 and c is a constant. (a) Derive the stream function for this flow field. Locate streamlines at (b) "one-quarter" and (c) one-half the total volume flow rate in the boundary layer.

Answer 1

Answer:

1) Stream function, Ψ = {U/c^(2)} [(cy^(2) / x^(1/2)) - (y^(3)/3x)]

2) At one quarter and one-half, The streamlines pass through the coordinates;

x1 = (3/(8cU))^2 , y1 = 3/(8U)

And x2 =(3/(4cU))^2 , y1 = 3/(4U)

Explanation:

Due to the cumbersome nature of the symbols and squares, i explained it all in a 3 pages file i have attached.