Isn't Bernoulli's principle only applicable when talking about the same flow? I've always found the "above path is longer than the lower path" explanation to be unintuitive because we're not talking about the same flow. They're separate flows.
- points 1 and 2 lie on a streamline,
- the fluid has constant density (note effects of height difference > gravitational potential energy between point 1 and 2),
- the flow is steady, and
- there is no friction.