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Neglecting any special cases and generally speaking, when the velocity of a fluid increases the pressure decreases (Bernoulli's principle). This can be observed in experiments like "air over paper" and in things like venturimeter.

What I struggle with is making sense of what could physically happen that decreases the pressure of the fluid. We believe that the pressure of a fluid (without gravity) is due to the collisions of molecules with the walls of the container. So the pressure depends on the frequency and the velocity of collisions perpendicular to the walls.

So what really happens that decreases either of the two when fluid velocity increases.

Thank you.

P.S:  I've seen many other answers on stackexchange related to the topic but I didn't find an answer anywhere.

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    $\begingroup$ Turn it around. Ask what causes the velocity of a fluid to increase when there's a negative pressure differential? Obviously F=ma. Now, ask this: Is there any other way for the fluid to speed up? $\endgroup$ – Mike Dunlavey Jul 3 '17 at 18:08
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Bernouillis principle is just F=ma. In a one-dimensional flow the force on a fluid parcel is NOT the pressure behind it, but the difference in pressure before and behind it.If the parcel has length dx, and cross section dA we have $F=-(dp/dx)dA dx$ and $ma=(\rho \;dA dx)(u \;du/dx)$ hence $dp/dx+\rho u du/dx=0$ and Bernouilli follows.

You CAN get the same result from conservation of energy, but you have to allow for energy inside the molecules. The real energy argument is harder, but the handwaving version is very simple, which accounts for its popularity.

Honesty compels me to admit that I fudged the momentum argument a bit, but the esence of it is that the fluid accelerates only if the pressure is decreasing along its path. (Or because something else is going on, in which case Bernouilli does not apply)

ADDED NOTE In your question you ask why does pressure fall when fluid speeds up? This is not the proper way to think about it. Bernoulli applies to steady flows that do not change with time. That means that whatever caused the flow happened in the past. It does not matter if the speeding up or the falling pressure happened first, but they have to have reached equilibrium by now.

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Mass conservation

A flowing liquid speeds up at a pipe contraction because of mass conservation:

$$\dot m_1=\dot m_2\qquad\Leftrightarrow\\ \dot V_1\rho=\dot V_2\rho\qquad\Leftrightarrow\\ \dot A_1v_1\rho=\dot A_2v_2\rho\qquad\Leftrightarrow$$

$\rho$ is constant if the liquid is incompressible. So speed increases for decreasing $A$.

Energy conservation

When something speeds up, that means a gain in kinetic energy. Energy conservation states that such energy must come from somewhere:

$$U_1+K_1+W=U_2+K_2$$

If there is no pump along the way, there is no external work done, $W=0$. Then the fain in the liquid kinetic energy must be taken from some other kind of energy, most likely a potential energy in some other form.

Pressure as energy

Pressure can be described as an energy form - we could call it mechanical potential energy - because:

$$p=\frac{F}{A}=\frac{Fx}{Ax}=\frac{W}{V}$$

$x$ is how far the liquid particles move while they accelerate to speed up. Such acceleration must mean that a force pushes them forward, and this force $F$ (which is essentially the push from the liquid particles behind) is doing the work $W$.

This "pressure energy", the mechanical potential energy due to pressure, is basically the combination of all the molecular kinetic energy of the liquid particles.

(We can call this mechanical potential energy because it is a driving factor for the motion, where particles always move towards lower potential (just like things fall towards lower gravitational potential energy and charges move towards lower electrical potential energy etc.)

Pressure as molecular collisions

All in all the gain in kinetic energy is thus taken from this "pressure energy". To increase the net flow speed, the individual random motion of the liquid particles decreases.

Less random motion means smaller pressure, because pressure - as you said - comes from the particles colliding with the pipe walls.

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    $\begingroup$ Thanks for your answer, I appreciate your time. But this is the kind of an answer I find everywhere I search. Consider an air pump as an example, the increased kinetic energy comes from the work done by the pump yet the pressure decreases. Even after trying to simulate it I've got no better seeing that in action. Certainly I'm missing something. $\endgroup$ – Knight Jun 25 '17 at 17:21

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