# How does a diffuser increase fluid pressure?

In diffusers the velocity of a flow slows and the pressure generally increases. Nozzles do the opposite. My question is, "What about the geometry of a diffuser causes the pressure to increase?" I understand the conservation equations that net that result and I understand why velocity changes to keep flow steady.

What I don't have is an intuitive understanding of why a slower flow pushes harder per unit area, or if the the expaning geometry somehow causes increased pressure.

1. The velocity needs to decrease due to mass conservation (this has nothing to do with unsteadiness, by the way): The product of cross section times mean flow velocity in the cross section must remain constant.
2. Thus, each of the fluid particles must decelerate while going through the diffuser.
3. For them to do so, a net force acting in the direction opposite the flow must be present (Newton's Second Law).
4. That net force must be provided by a positive pressure gradient, meaning increasing pressure in the streamwise direction.

Not in every case a diffusor increases the pressure in a fluid. Take a water reservoir and make a lot of small holes into the ground. The water will go through the holes, the water level inside the reservoir and by this the potential energy and the the pressure decreases.

Now take a long pipe which is connected to the ground of the reservoir and let water going out through this pipe. The water will have some velocity and by this some kinetic energy. Now decries the cross section of the valve and for a moment the pressure inside the pipe increases. For a long pipe it is somehow forbidden to use fast closing valves because a hydraulic knock appears and could destroy the pipe.

Now imagine that your water reservoir is below the level you want to use the water and between your reservoir and the valve is a pump with an electric device. Open the valve, start the pump and water is flowing. Now make the cross section of the valve smaller and the constant power of the pump will increase the pressure in the pipe.

In detail, why in the mentioned examples the decrease of a cross section increases the pressure? Take a meat grinder and remove the diffusor. The meat will go through faster (but not very well shredded). So the decrease of the cross section changes the possible flow of a fluid at this place and if some kinetic energy applied at the starting side stays unchanged the pressure increases.

Fluid going through the diffuser experiences viscous drag (a velocity-squared retarding force) when the fluid exceeds some threshold velocity. That threshold (roughly Reynolds number about 1000) is much lower for the small-passages diffuser than it is for the pipe which presumably is in the streamline flow regime. The diffuser has higher Reynolds number than the pipe.

So, the diffuser exerts a back-pressure onto the fluid on the input side, and that means the input side of the diffuser has higher pressure fluid than the output side of the diffuser.