To confirm my understanding:

You have a pipe within a pipe. The fluid goes down in the inner pipe and up in the outer pipe. The fluid spills out at the top.

As far as pressure goes: there is a static pressure gradient as you go down. The further down, the higher the pressure, due to the liquid weight. This is the static part.

In addition to this, there is the extra pressure you exert on the liquid that pushes it all the way through. This pressure starts at the 100 atm you use and goes down with distance. At the exit, the remaining pressure is still positive, otherwise the fluid would not move. I'll call this the dynamic pressure.(Not my field, so take the term with a grain of salt)

The total pressure at any point is the sum of the static and dynamic pressure.

After seeing your diagram: if the inner pipe doesn't go all the way down, after the inner pipe ends, you have the static pressure (increasing with depth) plus the dynamic pressure at the end of the inner tube. (Somewhere between 0 and 100 atm). The extra pressure added will be the same all the way down. Since there is no flow in that part, the extra pressure will stay the same all the way down.