We can make an argument that the pressure across a cross-section must be constant, (or just increase linearly downward because of the gravitational hydrostatic force) otherwise there will be flow in the radial direction to relax the pressure.  Since we are requiring only one dimensional flow, there can be no pressure change in the r direction.  Now that the pressure is constant across a cross section, we realize that the pressure change in the axial (z) direction must be the same in each phase.