# What happens to flow in a constant area duct when there is a pressure gradient?

In nozzles and diffusers the cross-section area follows the one defined by the pressure gradient. In a nozzle it decreases to keep mass flow rate constant. In a diffuser the opposite happens.

But say you had a regular pipe with air, constant area, and the pressure was greater on one end. In my mind the velocity would have to increase because of the pressure gradient and this would mean one of two things happens:

1. The density decreases a lot to keep mass flow rate constant. This seems unlikely because density doesn't change much for subsonic flow.

2. The density drops a little but mainly the flow pulls away from the walls, forcing an area constriction to keep mass flow rate constant. This also seems unlikely because there would be a vacuum where it pulls away.

So, my question is what really happens in this scenario?

• For a given pressure difference, there is definite mean flow speed. Flow speed does not increase indefinitely.
– Deep
Dec 29 '16 at 4:15
• The question's starting assumptions are probably incorrect. May 26 '20 at 19:08

We actually have three factors if the fluid is compressible, like air.

1. Mass flow rate

2. Pressure (and hence density of the air)

3. Velocity.

If we force the air to flow and there were some friction losses along the length, then yes a pressure gradient would be present to keep the air flowing. The mass flow rate is constant at any point: Area times density times velocity, that product stays the same the whole length. Density is proportional to pressure, so the density in the beginning is higher than the density at the end of the pipe (because pressure is higher at the beginning). The area is constant.

All that means is the the velocity of the air at the outlet is higher to maintain a constant mass flow rate at a lower pressure.

I mean in general the pressure difference drives a flow across the pipe which increases until the "head loss" (energy lost due to friction with the sides) is equal to the pressure difference. The flow speed at the boundary is 0 and it rapidly increases to a maximum at the center; for example with laminar flow in a circular pipe it is I believe parabolic, $u(r) = u_\text{max}~(1 - (r/R)^2).$ Then if it gets turbulent I think it flattens out in the center.

For air flow in a uniform pipe, continuity requires that the total mass flow rate must be constant under equilibrium conditions. It should be roughly proportional to the pressure gradient. As the pressure drops, the density decreases and the velocity must increase.