# Why do flows gain pressure with decreasing velocity?

I know it isn't always the case, but in many conservation equations velocity and pressure of a flow are inversly related, or sometimes velocity and enthalpy. My question is, "What about slowing molecules down makes then push harder?"

I understand the math but not intuitively why a flow that is moving slower can push harder, in fact I would have guessed that a faster flow pushes harder. Specifically I am looking at nozzles and diffusers.

• Faster flow 'pushes' objects the flow impinges upon because of momentum exchange. Static gas has no (bulk) momentum. – docscience Sep 13 '16 at 0:45

It's just Bernoulli's principle which is nothing more than $F=ma$ in a fluid.

Force in a fluid is called pressure. The only way a fluid can accelerate is by going from high pressure to low (i.e. feeling a force pushing it forward). Similarly, the only way it can slow down is by going from low pressure to high (feeling a force pushing it backward).

@docscience:

Do the whole experiment horizontally, so you can ignore potential energy. Energy and momentum are linked by velocity. So $F=ma$ is both conservation of momentum and conservation of kinetic energy. Excuse me while I do a little math to show that $F=ma$ conserves energy:

An increment of work is $Fdx$.
An increment of distance is $dx = vdt$.
$F=ma$ says $F=mdv/dt$.
So the increment of work is $Fdx = ma \times dx = mdv/dt \times vdt = mvdv$.
The increment in kinetic energy is $dKE = 1/2 m(v^2 + 2vdv + (dv)^2 - v^2) = mvdv$,
(because you can ignore $(dv)^2$).
So $Fdx = dKE$.

• But what I recall from all of my fluids courses is that the Bernoulli equation deals with energy, not forces. The name Energy Equation is very often used interchangeably with Bernoulli's equation. Energy, momentum and continuity equations were the basic building blocks for fluid motion. So shouldn't the pressure rather be explained in terms of energy? Perhaps kinetic vs. potential? – docscience Sep 13 '16 at 3:22
• Ok so correct me If I am misunderstanding. Flow pressure in such equations is actually defined as pressure pushing in on the flow, since you said a force pushing it forward corresponds to low pressure and backwards to high. So if you have flow passing through a nozzle the flow accelerates and the pressure pushing in on the control volume decreases. And so outward pressure actually increases at the nozzle exit along with velocity, as intuition would suggest. – BoddTaxter Sep 13 '16 at 4:32
• Because then it would make alot more sense to me. – BoddTaxter Sep 13 '16 at 4:38
• @BoddTaxter: Fluid is just a bunch of little billiard balls bouncing around with heat. Put them in a cylinder, with a piston at one end, and an opening at the other. Now push on the piston. That applies force (pressure) at the piston end, and there is no force (pressure) pushing back at the outlet. So where do the little balls go? What accelerates and why? That's the basic idea. – Mike Dunlavey Sep 13 '16 at 12:37