Boundary layer separation and stagnation point

Boundary layer separation

When flow moves into a region of rising pressure (an adverse pressure gradient, $dp/dx > 0$, as on the rear of a cylinder or in a diverging channel), the fluid near the wall loses momentum to friction and cannot go further. The velocity gradient at the wall becomes zero, $(du/dy)_{y=0} = 0$. Beyond this point of separation, the flow near the wall reverses, the boundary layer leaves the surface and a wake of eddies forms.

 free stream ->  ___________________
                 \  boundary layer   separated flow
 body surface ====\=======S==========  S = separation point
                    --> <-- back flow    (du/dy = 0 at wall)

Separation causes high pressure (form) drag and, on a wing, loss of lift (stall).

Stagnation point

It is the point on the surface of a body in a flowing fluid where the fluid velocity becomes zero. The streamline dividing the flow splits here, and the pressure reaches its maximum value (stagnation pressure):

$$p_0 = p + \tfrac12\rho V^2$$

   ---->  ---->  ___
   ---->  ---->  /  |  stagnation point
   ---->  ----> (*  |  at the nose, v = 0