Bernoulli's Principle On Atomic Scale

The spheres represent the atoms of a fluid or gas. Although Bernoulli’s principle is often discussed, what is almost never discussed is what the individual atoms are doing in order to make this principle true.

Why should an individual atom exert less pressure if it moves with a higher velocity?

Should not the opposite be the case, since we would expect an atom moving at a higher velocity to exert a greater and not a smaller force on the walls of the pipe?
Before we answer these questions, let us first review what Bernoulli’s principle says about this example, and the highly unsatisfying explanation typically given for Bernoulli’s principle. The fluid has approximately the same density everywhere throughout the pipe. Therefore, the total number of atoms per second flowing through each section of pipe will be the same.

 Since fewer atoms can fit in the narrow section of the pipe, the atoms must flow through the narrow section of the pipe faster, to keep the total flow constant. The explanation typically given is that in order for the velocity of a fluid to increase as it enters the narrow section of the pipe, the pressure inside the narrow section has to be smaller.

The larger pressure before the narrow section allegedly creates a net force pushing the fluid into the narrow section as it enters, causing the velocity of the fluid to increase.


The problem with these explanations is that there is no reason to expect an atom with a higher velocity to exert a lower pressure on the walls of the pipe.

We should expect just the opposite of the Bernoulli principle, since an atom with a higher velocity would have a higher kinetic energy, and therefore exert a larger force on the pipe walls, rather than a smaller one.

So what is actually happening?

An atom is more likely to enter the narrow section of the pipe if it has a larger component of its velocity parallel to the pipe walls.

If the atom has a larger component of its velocity perpendicular to the pipe walls, the atom is more likely to bounce back. The atoms which make it into the narrow section are more likely to have more of their velocity parallel to the pipe walls.

As atoms leave the narrow section, their collisions with other atoms cause them to again have a large component of their velocity perpendicular to the pipe walls. It is the component of velocity perpendicular to the walls that is responsible for the pressure against the pipe. A larger component of velocity perpendicular to the walls creates a greater pressure in the pipe.
A larger component of velocity parallel to the walls increases the velocity of the flow through the pipe.

The atoms inside the narrow section are more likely to have a smaller component of their velocity perpendicular to the pipe and a larger component of their velocity parallel to the pipe. Hence, the atoms in the narrow section will exert a smaller pressure on the walls, but will have a larger flow velocity.