I know that I'm thinking about this problem wrong, but I've been pondering it for a while and can't understand something.
On my fluid mech. exam I had a problem that asked to determine the radius of the top half of an hourglass as a function with respect to vertical coordinate (z), such that the level of the hourglass (z) lowered constantly at some specific rate.
So based on continuity, A1V1=A2V2 where A1V1 is the area at the current height of fluid in the hourglass * its downward velocity, and A2V2 is the area* downward velocity at the small hole where it would exit to the bottom half of the hourglass. Both sides of the fluid are exposed to atmospheric pressure.
So obviously the velocity must be faster at the smaller cross section, and relations can be set up to solve for the radius at height z. My answer included finding an increase in fluid velocity due to elevation difference (z) through Bernoulli's equation, then relating that with continuity. This got me thinking though, what would happen if the cross section does not change, and you have vertical flow of fluid in some pipe with an obvious change in elevation, but then had both the entrance and exit at atmospheric pressure. The energy equation, which is supposed to remain constant for a continuous fluid without losses, would have a differing height term without differing pressure or velocity terms. Is it because it's not a continuous fluid, or because friction from the wall eventually accounts for head loss of the elevation differential after the fluid speeds up enough?
Sorry if this is poorly worded, it just isn't making sense in my head. Thanks a lot.