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I have encountered two different explanations of why the pressure term appears in the equation for Bernoulli's principle. The first one is:

(1) "The fluid has to speed up as it enters the narrower region. That means that the bit of fluid just entering the region has to be being pushed from behind. So the pressure behind it must be larger than in front." (from this post)

Here, the pressure we're talking about is the pressure in the direction of the fluid flow.

The second explanation is:

(2) The fluid molecules that enter the narrow region of a pipe are the ones that have a large velocity component parallel to the pipe. Because their velocities are aligned with the direction of the pipe, this means that only a small component of their velocity points perpendicular to the direction of the walls of the pipe. Hence, fluid in narrower parts of the pipe exert a smaller pressure on the walls of the pipe. This explanation was given by this video.

Here, the pressure we're talking about is the pressure on the walls of the pipe.

After hearing these two explanations (which both make intuitive sense), I am confused as to which pressure that the pressure term in Bernoulli's principle is referring to. Is it referring to the pressure in the direction of fluid flow or is it talking about the pressure on the pipe walls?

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  • $\begingroup$ Pressure in fluids is always the same in all directions, and is connected to the random motion of particles, so when most of the kinetic energy is in one direction, the pressure is smaller than the case where all of the kinetic energy is random. $\endgroup$
    – Ofek Gillon
    Commented Sep 11, 2021 at 19:06
  • $\begingroup$ Isn't pressure same in all directions only for a stationary fluid (which only has random motion) and not for a moving fluid (which has ordered motion in a certain direction)? $\endgroup$
    – silverbackgorilla
    Commented Sep 11, 2021 at 19:10
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    $\begingroup$ No, like internal energy, pressure is measured in the rest frame of the fluid-particle (fluid-particles here has a different meaning than the particles of the fluid) $\endgroup$
    – Ofek Gillon
    Commented Sep 11, 2021 at 19:14
  • $\begingroup$ So the pressure in Bernoulli's equation is the pressure measured in the frame of a moving parcel of fluid? $\endgroup$
    – silverbackgorilla
    Commented Sep 11, 2021 at 19:31
  • $\begingroup$ There is no difference between the pressure in our frame of reference to the parcel's frame of reference $\endgroup$
    – Ofek Gillon
    Commented Sep 11, 2021 at 20:00

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