Is the pressure of an incompressible fluid discontinuous or continuous across an aperture, like the nozzle of a hose?

Question

Is the pressure of an incompressible fluid discontinuous or continuous across an aperture? E.g. inside the fire hose water is at high pressure, outside of it it's at 1atm, but what does the transition look like at the aperture of the nozzle?
Since water is only mostly incompressible I'm sure there's at least continuity to the extent that it was very slightly compressed in the hose and returned to a very slightly greater volume once outside. But what about an idealized incompressible fluid?
If it's not discontinuous, are there formulae for how it looks for different flow types?

D. Halsey · Answer

The flow in and around a fire hose can be modelled in a similar way to a jet in the vicinity of a wing - a configuration studied by Carl Shollenberger and Peter Lissamon in 1971.
Shollenberger_PhDThesis
In this idealization, there is a jump in total pressure across a thin actuator disk at the inlet, and there are velocity discontinuities (modelled by vortex sheets) across the wake boundaries, but the static pressure is continuous everywhere.

Is the pressure of an incompressible fluid discontinuous or continuous across an aperture, like the nozzle of a hose?

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