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When one talks about the pressure of a flowing fluid, I get confused. For example, say I want to measure the pressure in a flowing river. I can think of 2 ways of measuring the pressure at a point in the flowing river.

(1) Stick a pressure probe in the river and measure the pressure while keeping the probe fixed relative to the ground.

(2) Stick a pressure probe in the river while traveling with the river on a boat and measure the pressure while moving with the river.

Surely, these two would give you different pressure readings. With that said, when one talks about the pressure field of a flowing fluid, are they talking about the pressure felt by a stationary probe or the pressure felt by one that is flowing with the fluid?

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  • $\begingroup$ What exactly is your question? Do you want to measure the pressure of fluid flowing in pipes? Do you doubt that the Bernoulli equation is correct? Do you need an explanation of why pressure drops when velocity increases? $\endgroup$ Oct 31, 2021 at 15:57
  • $\begingroup$ I apologize if my question was worded in a confusing way. I have edited to make it more clear. What I want to know is: when one is talking about the pressure field P(x,y,z,t) of a flowing fluid at some point P, are they talking about the pressure felt by a stationary probe at P or one that’s moving with the fluid that at P? $\endgroup$ Oct 31, 2021 at 17:09
  • $\begingroup$ In scenario 1, you need to place your pressure probe opening parallel to the streamlines. If you place the pressure probe opening normal to the streamlines, you will be measuring the "dynamic pressure," which includes a $\rho \frac{v^2}{2}$ term, over and above the static pressure. $\endgroup$ Oct 31, 2021 at 20:13

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Pressure of liquid is an unambiguous property of any spatial point inside the liquid: it is the magnitude of normal force acting on a liquid element flat surface per unit surface area, due to the surrounding liquid.

Method (1) is appropriate when measuring pressure at desired points of space relative to reference frame (river bank, laboratory). It has the problem that the measuring probe will modify the liquid flow and thus change the pressure that is being measured. The change depends on size, shape and orientation of the probe. So some correction is needed to get the desired value of pressure that would be there without the probe.

Method (2) is appropriate when measuring changes of pressure of a liquid element as it flows downstream. Since the (non-zero-sized) probe is moving much more slowly relative to the liquid around it, modification of the flow and pressure there is much smaller than with (1) and can be ignored.

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