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I have just been introduced to the Bernoulli Equation for fluid flow. However, I am unable to understand why the pressure and velocity are inversely proportional.

Because, as the fluid goes through a smaller cross-sectional area, the flow velocity increases, but due to the Bernoulli equation this increase in velocity means that there is a decrease in pressure. However, if say the fluid is not an ideal fluid, when it goes through a smaller cross section, it compresses, and hence is more "focused" therefore, it should surely mean that the pressure is increased. Furthermore, using the fundamental pressure equation: P = F/A, a smaller cross-section will mean that there is more pressure.

I have seen all of the mathematical proof (the derivation of the Bernoulli equation), observed it in practice (using a U-tube manometer) and read some analogies to GPE to KE, but I still don't underdstand, physically how this can happen.

If someone has a good explanation, I would love to hear it, thanks.

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As Chet Miller alluded in the comment, since a liquid element is accelerating as it approaches the bottleneck, there has to be force that is causing this acceleration. The only forces present are pressure forces from surrounding liquid, so pressure upstream has to be higher than downstream, to cause this acceleration.

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    $\begingroup$ Yes, this makes sense to me thank you, I understand the concept now $\endgroup$ – Barry B Benson Oct 14 '20 at 21:28
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If the pressure in the small diameter, where the flow has greater velocity, was equal or greater, we could take a derivation from that region and inject the fluid back in the region of big diameter.

In this case, the average velocity in the region of big diameter would increase, leading to increased velocity in the region of small diameter and so on. Energy would be created from nothing.

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