# Intuitive explanation of the causality behind a pressure gradient forming

The Bernoulli equation as a statement on the conservation of energy as I understand it is the observation in an idealized flow that the bulk fluid velocity relates to the kinetic energy of the fluid and must therefore increase/decrease to agree with the conservation of energy in the system. The internal energy of static pressure must be always "converted" into the kinetic energy of bulk fluid motion or vice versa. This is often used to explain why pressure gradients form to drive fluid flow.

I don't find that description of pressure gradient particularly satisfying because it is simply a relationship between the "types" of pressure in a fluid. That description only observes after the fact that when fluids have been accelerated there "must" be a pressure drop that formed due to a static pressure change since velocity changed. This does nothing to address the causality behind a pressure gradient even forming. The fluid seems to magically go from high to low pressure and accelerate because it needs to.

Obviously, to accelerate/decelerate a fluid, a pressure gradient must form to create an unbalanced force. Imagine the classical idealized example of a fluid flowing through a narrowing tube. It's velocity must have increased to agree with continuity/mass conservation. But according to Newton's 2nd law an unbalanced force must have changed the velocity. But the standard agreement is that the unbalanced force exists due to a pressure drop that the fluid moves through. But why does the pressure drop just automatically exist? How does the fluid actually know a pressure drop needs to be formed. Certainly, the higher velocity cannot exist in the narrowing first, before a gradient even formed. So how does the pressure start lowering in the first place, if at first the velocity and kinetic energy couldn't have increased without the existence of a pressure gradient? The causality of this makes no sense to me.

The fluid assumptions you describe assume speeds that are sufficiently lower than the speed of sound that it is valid to speak of a fluid particle "knowing" that it must speed up.

At a more exacting level, the particles are colliding a great many times every second. If there is a substantial pressure difference, such as when one suddenly opens a valve at the bottom of a tank, the fluid particles at the leading edge of this suddenly find more collisions from behind than in front, and that imparts a net movement. This information is "relayed" back to the molecules further into the tank in the form of the average of a bunch of these collisions.

What we end up seeing is that there is a relationship between pressure and velocity, based on continuity It isn't inherently causal in one direction or the other. The causality stems from parts of the problem which provide a limiting factor. For example, in a long enough tube, drag limits velocity, so velocity drives pressure. In an air cannon, the amount of pressurized gas provides a major limitation, so we often talk of the pressure driving the velocity.

In reality, both are just the result of lots of little collisions, but it can be useful to think of one causing the other in any particular problem in order to unwind them. In problems where you can't unwind them like this, you have to use more advanced equations like Navier Stokes to figure out how the interplay works out.

Narrowing the tube by itself is not the cause for increasing the fluid velocity. But if there is a fluid tank above for example, gravity potential is the source of the flow.

Once there is a flow, if part of the tube has a narrow section, the fluid velocity must be greater there to keep the continuity.

The pressure decreases as a consequence. If the section is horizontal, the potential energy of the molecules doesn't change with the tube diameter. But if the velocity increases in the flow direction (say $$x$$), and $$E_k = \frac{1}{2}\mu v^2 = \frac{1}{2}\mu (v_x^2 + v_y^2 + v_z^2)$$, then $$v_y$$ and $$v_z$$ must decrease, otherwise energy would be created from nothing when entering in the narrow section.

If $$v_y$$ and $$v_z$$ decrease, the momentum components $$p_y$$ and $$p_z$$ also decrease. As the pressure is a consequence of the transversal force on the tube walls: $$F = \frac{dp}{dt}$$, a decrease of the transverse momentum means a decrease of the pressure.

The reason why physics is interesting is that it can control complexity with simplicity. One inspiration or a keen observation can completely fix the whole theory.
[Landau] Landau, L. D. & Lifshitz, E. M. Fluid Mechanics, 2nd ed., translated from the Russian by J. B. Sykes & W. H. Reid, New York: Pergamon, 1987.
[WiKi1] [https://en.wikipedia.org/wiki/Acoustic_wave_equation][1]
[Wiki2] [https://en.wikipedia.org/wiki/Acoustic_wave_equation#/media/File:Derivation_of_acoustic_wave_equation.png][2]
[Falkovich] = Falkovich, G: Fluid mechanics, 2nd ed., Cambridge: Cambridge University Press, 2018.

The proof of [Landau, p.252, (64.7)] is vague, while the proof given in [Wiki2] provides specific details of Landau's method. For example, $$\rho ' \frac{\partial^2 \phi}{\partial t^2}$$ is a term of the third order, so the term should be discarded. The former describes the wave in velocity potential $$\phi$$, the latter describes the waves in pressure $$P$$. Both proofs use the same method. Because there are many smart people in the world, someone gives Landau's proof a complete analysis and applies the method to the right object: pressure, and then summarizes it in [Wiki2]. The latter proof shows that the irrotationality of the velocity field is a unnecessary assumption and that pressure gradients are firmly identified as the main cause of sound waves [see the first paragraph of [WiKi1]]. The idea of converting the application of Landau's method from velocity potentials to pressures was a major step forward in the progress of acoustic wave equation. [Landau, p.252, (64.7)] treated the velocity field as the familiar electric field and made a great contribution during the development of acoustic wave equation in its time, but by today's standard of physics, to regard [Landau, p.252, (64.7)] as the real acoustic wave equation should be considered incorrect because the useful method is applied to an improper object. Thus, the journey to the final form of acoustic wave equation was meandering and arduous.
Now go back to your question. The intuitive explanation of the causality behind pressure gradient forming must come from concrete examples like the above case: acoustic wave equation. In this case, sound causes pressure gradient forming. [Landau, p.252, (64.8)] indicates the speed of sound. [Landau, p.252,l.$$-$$19--l.$$-$$11] describes the sound wave. This answer is excerpted from Example 6.18 in [https://sites.google.com/view/lcwangpress/%E9%A6%96%E9%A0%81/papers/mathematical-methods][3].
You may be interested in reading [Falkovich, 1.1.2 Equations of motion for an ideal fluid: The Euler equation]: Before Euler's time, scholars focused on establishing relationships between external forces and fluid velocities. Euler thought the relationships might be indirect, causality might be different case by case; that approach might easily lead to misjudgments, so he studied the more direct and closer relationship between velocity changes and the pressure field (already existed there) inside the fluid. Euler's approach guides us to have a deeper understanding of fluid mechanics.

[1]: https://en.wikipedia.org/wiki/Acoustic_wave_equation
[2]: https://en.wikipedia.org/wiki/Acoustic_wave_equation#/media/File:Derivation_of_acoustic_wave_equation.png