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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?

Qmechanic
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silverbackgorilla
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2 Answers2

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If the pipe has a convergence section and the flow rate inside the pipe is constant, then the velocity of the convergence section must be high. How can the speed be high? Only the pressure upstream of the convergence section is higher. High pressure is necessary to increase fluid velocity.

If you flatten the outlet of a water pipe, the flow of water in the upstream pipe will be obstructed, resulting in a decrease in water velocity. Once the velocity decreases, the pressure will increase, just like the crowding at the exit of a movie theater or the increase in pressure at the stagnation point of a fluid. Due to the increase in pressure, the speed of convergence of the cross-section will increase.

Assuming that the energy provided by the upstream is constant, the pressure and velocity in the pipe will stabilize. So it will result in a state where the upstream pressure is high but the velocity is low, while the downstream convergence section has a high velocity but low pressure.

enbin
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The pressure term in Bernoulli's principle is referring to static pressure, not dynamic pressure. From my understanding, static pressure is specifically the part of pressure that is isotropic (equal in all directions). This is also why it's treated as a scalar, not a vector.

The first explanation you mentioned implies the pressure gradient (change in pressure) is in the direction of the pipe, not the pressure itself.

Ishaan Jain
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