Incompressible Fluid Flow in horizontal pipe segment which encounters tilted pipe segment - Continuity Equation
Say that we have water flowing horizontally in a pipe of constant diameter. The continuity equation for incompressible fluids, which is a statement of mass conservation, guarantees that the flow rate will be the same throughout the pipe. Given that the diameter is constant it also guarantees that the velocity will be the same throughout.
Now say the pipe reaches a point where it is tilted. The tilted segment leads downhill and reaches another horizontal segment.
Water is now going downhill and so it is accelerating. Let us assume an ideal situation with negligible friction and hydraulic resistance. 1018
If the water is accelerating under gravity then velocity must change so I assume the diameter of the water flow itself must shrink and the water will "stretch" in order to maintain continuity. Is this actually the case?
What happens at the bottom when the water flow becomes horizontal again? Does it gradually "bulge up" at the bottom in order to reestablish the horizontal flow of constant diameter in the lower horizontal pipe? Assuming that the conservation of mass holds and continuity is preserved, the horizontal flow and speeds must be the same in both horizontal pipes.
One additional remark on this situation. What if the water were being pumped uphill in a pipe in the same configuration? Is it possible to pump water uphill and maintain a constant velocity in a constant diameter pipe? How can you demonstrate whether it is or isn't from first principles? It seems like water in the pipe would have to slow down when it encounters and uphill segment, but it can't pile up anywhere in the pipe either, and certainly doesn't expand, so it must also preserve continuity somehow I would assume? I feel like I am really missing something.
I apologize if this question seems trivial or silly. My backgrounds are in chemistry and in geology and I could not find any explanations on the internet regarding this. My fluid mechanics, physical hydrology, and hydrogeology textbooks also were of no help. 1018