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We are talking about a Venturi pipe for an ideal liquid,
not viscous and not compressible.
1) Suppose we have an ideal horizontal tube of constant cross
section. No work is needed to maintain its liquid in motion at
a constant speed.
2) The same tube is bent upward near the exit. This time we
must have a force to push water at constant speed. Suppose
the volume ejected during a short time interval (at very
small kinetic energy) is dV and the output is at an elevation
h, with respect to the axis of the horizontal pipe. The work
done must be dW=rho*g*dV*h, where rho is the density. It
is equal to PE gained by the dV parcel of water. It is
also equal to P*dV, where P is the pushing pressure applied
by the piston at the pushing end. This is not the same as
P*dV for a gas whose volume changes.