Understanding the Bernoulli Effect

[freefall]

I do not completely understand it.

 

I see how, if fluid is running though a pipe, the pressure decreases as the radius decreases because the acceleration of the fluid requires the unbalanced force of the pressure differential.

 

What I don't see is why the pressure must decrease absolutely (instead of relatively) as the radius decreases. For example, if you have a pipe with water flowing though it, and you add a narrower pipe to the end, it seems that the pressure in the larger pipe would increase. I'm pretty sure it would, because you can do the experiment with your mouth. Hold your mouth wide open and blow air out, then close your mouth almost entirely, and your cheeks will bulge outwards. The pressure in the larger-radiused area increased because it had to accelerate the air though the small gap.

 

So I guess what I'm confused about is: to me, the bernoulli equation proves that there must be a relative pressure decrease as radius of a pipe decreases, but that does not justify the fact that faster moving fluids have absolutely lower pressures, which everybody always says.