Governing equations and problem set up

From Denn the flow past a wedge (figure 15-1) is given by
U(x) = A xm  , m=beta/(2 pi-beta)

Recall that the pressure gradient, , will change as -V(x) V'(x). This gives

(vx + vy ) = + ν


******  Thus, the flow past a wedge is a model problem for telling how the boundary layer will change as the pressure gradient changes. *****

Denn tells us that the equation can be reduced to a nonlinear ode again taking advantage of similarity of solution profiles.  This gives

        f''' + (m+1)/2 f f''/2 + m (1-f'2)  = 0.
        
        If m=0, the equation is the same as for the flat plate.  


We again need to make a system of first order ODE's .  These are

y1' = y2
y2' = y3
y3' = - (m+1)/2 y1 y3 + m*(1-y22)   

This time I use a real shooting method, with a pseudo Newton-Raphson iteration.  It works for most cases.  If the angle is too negative, the stress will go through 0, this means that there is no layer and thus it is not surprising that the answer blows up.  Of course if we look at the physical situation, we might be surprised that we can get any solution for negative β.  The trend is correct and it is interesting to examine the case of a diverging flow.