# Bernoulli Principle at a Microscopic Level

I'm trying to understand the microscopic mechanisms that justify the Bernoulli principle. I found an interesting discussion on the site of NASA that tries to describe, in a general way, the mechanism underlying the pressure reduction in favor of the increase in speed. I propose the text in question, and then I explain to you my doubt.

"We can make another interpretation of the Bernoulli equation by considering the motion of the gas molecules. The molecules within a fluid are in constant random motion and collide with each other and with the walls of an object in the fluid. The motion of the molecules gives the molecules a linear momentum and the fluid pressure is a measure of this momentum. If a gas is at rest, all of the motion of the molecules is random and the pressure that we detect is the total pressure of the gas. If the gas is set in motion or flows, some of the random components of velocity are changed in favor of the directed motion. We call the directed motion "ordered," as opposed to the disordered random motion. We can associate a "pressure" with the momentum of the ordered motion of the gas. We call this pressure the dynamic pressure. The remaining random motion of the molecules still produces a pressure called the static pressure. From a conservation of energy and momentum, the static pressure plus the dynamic pressure is equal to the original total pressure in a flow (assuming we do not add or subtract energy in the flow). The form of the dynamic pressure is the density times the square of the velocity divided by two."

From the text we learn that the pressure reduction with speed can be explained by thinking that part of the "chaotic" kinetic energy translates into "ordered" kinetic energy. The remaining chaotic kinetic energy gives rise to static pressure, while the ordered kinetic energy is the so-called "dynamic pressure". If a particle of fluid initially at rest is set in motion, then a force must have acted. This force comes from the pressure difference on the sides of the fluid particle. The text suggests that the motion of the particle is due to the conversion of a part of the chaotic kinetic energy into ordered kinetic energy, thanks to the pressure unbalance on the sides of the fluid particle. Therefore the speed increases and the static pressure decreases."

I'm trying to understand this better. The force acting on the fluid particle, performing work, should add kinetic energy to the system, not converting an energy already present in the particle into another. Why do we have a conversion of potential energy (static pressure) into ordered kinetic energy to the action of a force on our system (the fluid particle)? Should we not maintain the same chaotic kinetic energy and, if anything, have that extra due to the work of force?

The quoted explanation is misleading, almost incorrect. The connection between pressure and chaotic motion is valid only in a limited way and only for gases; it is not valid for incompressible liquid flows such as water in pipes.

The force acting on the fluid particle, performing work, should add kinetic energy to the system, not converting an energy already present in the particle into another.

This is exactly right and this work of external forces on the fluid element is the origin of the pressure term in the Bernoulli equation. This equation is a convenient rewrite of the work-energy theorem applied to non-viscous fluid flows: work done by external forces equals change of kinetic energy of a body. If the work is done only by forces due to pressure gradients (no gravity or other kinds of external forces), then this work is entirely captured by the pressure term in the Bernoulli equation.

In this derivation of the Bernoulli equation from the work-energy theorem there is no chaotic motion involved; such motion is a microscopic detail that has no appearance or role in the theory behind the Bernoulli equation. For adiabatic gas flows, it is true that lowering pressure means comparable lowering of average kinetic energy of chaotic motion of molecules (indicated by lowered temperature in an adiabatic process of gas expansion), but for liquid flows such as water in pipes there is no universal basis for such effect (water expansion with pressure decrease is significantly smaller and decrease of the average kinetic energy of chaotic motion is negligible).

Bernoulli equation in its standard (simplest) form is most accurate model of incompressible flow of liquids, where there is no simple relation between adiabatic decrease of pressure and change of temperature. So it is not a good idea to explain the Bernoulli equation based on behaviour of ideal gases.

• Thank you! Then how can we explain the reduction in pressure with speed at a microscopic level? I mean considering a smlal cube containing gas molecules. How it work the convertion of Chaotic Kinect Energy in Ordered Kinect Energy while a gradient of pressure is acting on the particle? Commented Apr 3, 2019 at 8:01
• It is explained in molecular statistical physics: the expanding gas works on its environment and therefore loses internal energy. This manifests as decreased temperature. Commented Apr 3, 2019 at 11:10
• Ok. Would you like to describe this aspect in more detail? It is precisely the part that I want to understand. I imagine a fluid particle initially at rest, which is set in motion due to a pressure imbalance in its surroundings. How do you justify its conversion of energy from chaotic to ordinate, also taking into account the energy introduced by the work of the force acting as a pressure force? Commented Apr 3, 2019 at 14:42
• Perhaps it would be better if you asked that as a new question, without mentioning the misleading justification for the Bernoulli equation. Commented Apr 3, 2019 at 20:33

This paragraph is based on a misconception:

The force acting on the fluid particle, performing work, should add kinetic energy to the system, not converting an energy already present in the particle into another. Why do we have a conversion of potential energy (static pressure) into ordered kinetic energy to the action of a force on our system (the fluid particle)? Should we not maintain the same chaotic kinetic energy and, if anything, have that extra due to the work of force?

The walls exert a force on the gas, but they do no work. Work requires a force acting through a distance, and the walls don’t move.

Since the walls do no work, they can’t change the energy of the flow.

But they can change the direction of the momentum by bouncing particles back into the flow at an angle. And that’s how the perpendicular motion that gives rise to pressure is (partially) converted to longitudinal motion.

Let’s imagine what is happening to molecules inside a pipe, where flow goes through a large diameter and then into an orifice and then a tiny diameter. In the large diameter, the molecules are bouncing around randomly, with a fixed average velocity, but more or less evenly in all directions. Now in order for the molecules to enter the orifice, they need to be traveling in the direction of the orifice. Molecules moving up or down or backwards won’t enter the orifice. So the molecules that enter have their velocities aligned with the direction of flow. And because more of their velocity is in the flow direction, it leaves less available velocity in the perpendicular direction. This reduction in the perpendicular velocity component is equivalently reduced pressure on the pipe wall, and also between the molecules inside the fluid. To help visualize it further, imagine all the molecule’s available velocity moving exactly parallel with the pipe, all at the same velocity. So there is no perpendicular motion. And now place yourself in the frame of reference of the fluid. There is no relative motion between the fluid molecules! Pressure has dropped to zero! This doesn’t happen in reality, but the thought experiment may help understand.

Thanks for posting your explanation. To rephrase it according to my understanding, the reduction in diameter is resulting in a selection of particles that move in "a forward direction"; the rejected ones have the directions of their motion further randomized, eventually acquiring a forward component, and then they try again to make it through the orfice.

About energy: if two particles are moving in opposite directions in the large volume, then their combined center of gravity is at rest and has no kinetic energy. On the other hand, when the two particles are moving in the same direction in the small volume after passing through the orfice, they have net kinetic energy, since the average speed is greater zero.

Further, if we assume a piston closes off a portion of the volume with the large diameter and the pressure is maintained by applying a (constant) force onto the piston, then, as particles are escaping through the orfice, the piston is moving in. Since force times distance equates to work, this is the kinetic energy that the escaping particles are carrying away: it is a process, where "pV" is converted into "1/2 m v2", so to speak.

The microscopic explanation of Bernoulli's equation has been bugging me for decades, basically, and I think now it is starting to make sense. Thank you!