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According to Bernouli's equation, the static pressure inside a pipe with streamline and steady flow decreases with fluid's velocity. One explanation to rationalise this phenomenon is because velocity pressure increases and the sum of static and velocity pressure is a constant; thus static pressure decreases.

This inverse relationship is contradictory to our intuition that a fluid e.g. water from a hose with greater velocity exerts a greater pressure on us. I believe this contradiction to our intuition is because what we feel as 'higher pressure' is actually referring to 'higher velocity pressure'? (please correct me if i'm wrong)

And if so (or regardless if that logic is right/wrong), what is the physical meaning or intuitive meaning of static pressure? How does it differ with velocity pressure?

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Your reasoning is correct but it misses one step: when the water coming from a high velocity stream hits you, the velocity is 'converted' back to pressure again. When the water hits your body its velocity drops to zero because it can't move through you. The water starts piling up, increasing the pressure near your skin (this happens really fast). The pressure increases until it is large enough to divert the stream somewhere else.

So in conclusion, the water that's in the stream has high velocity and low pressure while the water that's near your body has low velocity and high pressure because it is being diverted by your body. This is all in agreement with Bernoulli's principle.

Note: after the stream hits your body the flow becomes highly turbulent and you can't apply Bernoulli's principle. When energy is lost you can't really apply Bernoulli's principle.

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