Question about streamline in boundary layers I was wondering what the streamlines look like in a boundary layer. I found a picture that shows this: 

I would like to know if this presentation of the streamline is correct. I don't think that it is correct in the turbulent region, but I'm more interested in the laminar region. 
The streamline seems horizontal to me in that region, so it looks like there are no velocities in the y-direction, so $v_y=0$. I wonder if this is correct, because this seems to contradict the boundary-layer equations in which they use a $v_y$. I feel like that the picture isn't correct, but that the streamlines tend to follow the angle of the dotted line. So I wonder: is that feeling correct and do the streamlines have more or less the same slope as the dotted line? I also feel that the more we go in the y-direction the more the tendency to follow the dotted line becomes, so close to the plate they are horizontal, can anyone confirm or contradict this?
 A: Short Answer: The fluid velocity in the y-direction in the boundary layer is approximately zero; close enough that it is commonly (and safely) assumed to be zero in mathematical calculations. 
Why?:
Well, it simplifies the Navier-Stokes equations! But lazy reasons aside, here's the reason why it's just set to zero:
Knowing that,


*

*the boundary layer is thin (when Reynolds number is large),

*boundary conditions indicate that $u=0$ and $v = 0$ ($u, v$ being the
horizontal and vertical velocity of the fluid correspondingly), and that

*fluid velocity outside the boundary layer would be $u=v_∞$ and $v = 0$


we can conclude that,


*

*$\frac{\partial u}{\partial y}$ will be large by comparison with changes along the length of the plate, i.e., $\frac{\partial u}{\partial y} >> \frac{\partial u}{\partial x}$


And so from the continuity equation it follows that $\frac{\partial u}{\partial y} >> \frac{\partial v}{\partial y}$. In other words, the change in vertical velocity is relatively small. Since we know that the velocity at the boundary is $v = 0$, we can safely and correctly assume that it remains very close to this!
This answer was done consulting:
Marine Hydrodynamics by J.N. Newman
