Bernoulli Principle from a molecular scale

[tmox]

Good evening everyone.
Can you tell me if it is possible to mathematically derive the Bernoulli principle from a microscopic analysis?
In particular, in the hypothesis of an incompressible stationary flow, at a constant altitude, it states that:

P + 0.5ρV^2 = cost

Well, in textbooks this report is generally derived through considerations related to work and kinetic energy. The result is that, in order for the flow to remain stationary under the assumed conditions, it is necessary that as the speed increases the pressure decreases in a quadratic manner. How to demonstrate this fact on a molecular scale? How to convince us of the fact that at the microscopic level an increase in the average speed of the particles reduces the pressure in a quadratic way?

Thank you so much!

 

[sophiecentaur]

How to demonstrate this fact on a molecular scale?

Are you suggesting that the dynamics of a vast number of molecules should be calculated explicitly? That would be a tall order and would it be any better than treating it statistically? The introduction to the gas laws starts with individual molecules with mass and mean speed and then derives the pressure. Is that the sort of thing you are after?

 

[Lord Jestocost]

Something about "Molecular Scale Derivation" regarding Bernoulli's principle can be found on the NASA website https://www.grc.nasa.gov/www/k-12/airplane/bern.html

 

[russ_watters]

My understanding is that fluid dynamics isn't really fluid dynamics at a molecular level; Velocity and pressure are bulk properties. I'm sure it can be modeled, but I'm not sure it can be derived.

 

[boneh3ad]

The vast majority of fluid mechanics is performed under the continuum hypothesis; it assumes that the fluid forms a continuous medium rather than a collection of individual molecules. Bernoulli's equation is one of these concepts that arises from this paradigm.

You could derive it from molecular considerations, I suppose. I've never tried, but you can certainly relate molecular motion to energy contained in a volume containing said molecules, and that is ultimately what Bernoulli's equation is tracking.

 

[tmox]

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."

[Text from: https://www.grc.nasa.gov/www/k-12/airplane/bern.html ]

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?
Thank you all!

 

[jartsa]

Let's say a piston emits momentum at constant rate into a liquid in a pipe, the pipe is shaped so that a liquid molecule accelerates at constant rate, which means that said molecule absorbs momentum at constant rate.

If a molecule starts accelerating at time zero, then at time t the molecule sees behind it a line of molecules, the length of the line is $s=1/2 *at^2$

The speed of the molecule is proportional to time, the pressure drop that happens between the piston and the molecule is proportional to time squared. So the pressure drop is proportional to speed squared.Here's another example of pressure drop being caused by absorption of momentum:

The first train car of a train receives all the momentum that the engine emits, while the last car receives the momentum emitted by the engine minus the momentum absorbed by the other train cars. And that is why the negative pressure experienced by a train car is proportional to its distance from the engine.

 

[boneh3ad]

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?
Thank you all!

That work is being performed by the static pressure, in essence. That's a source entirely internal to the flow. If that work is being done to add kinetic energy to the flow, then that energy had to come from somewhere, and since it is static pressure "doing" the work, that energy is coming out of the pool of energy represented by static pressure.

 

[tmox]

That work is being performed by the static pressure, in essence. That's a source entirely internal to the flow. If that work is being done to add kinetic energy to the flow, then that energy had to come from somewhere, and since it is static pressure "doing" the work, that energy is coming out of the pool of energy represented by static pressure.

That's true. But the work is performed by the static pressure "around" the particle, while we are dealing with the drop of static pressure of the particle that "receive" work.

 

[boneh3ad]

That particle is also pushing on its adjacent particles. It's all a big interconnected system of "particles" in a continuous medium.