Say we have a closed loop system, where the flow $Q$ is constant through a pipe.
Then, at one segment of the pipe, we make the radius smaller. This causes the resistance $R$ to be greater, so the pressure $P$ must be higher as well. So, at this stretch of pipe, the fluid is under greater pressure.
Meanwhile, we've made the radius smaller, so that cross-sectional area $A$ is lower. Then, to compensate, $v$ has to be greater, to keep $Q$ equal.
So, in the same stretch of pipe, the velocity $v$ is greater, but so is the pressure.
Is this at odds with the Bernoulli principle, which says that when pressure and velocity of a fluid are inversely proportional?
I have a very limited understanding of the Bernoulli equation, so it's very possible that I'm misinterpreting it. But my thought was that if a fluid flows faster through a pipe, the collapsing pressure on that pipe would be greater. However, this seems at odds with the fact that the fluid appears to be under more pressure when it flows faster.