# How to explain Bernoulli's principle with Brownian motion?

Air pressure is generated by Brownian motion pushing against solid objects. The integration of all molecule collisions with the boundary is then the air pressure pushing against that object.

But can the Bernoulli principle be explained with the same model? How exactly? Ideally, I would also be interested in the math behind this question, therefore how can one derive Bernoulli's principle from this Brownian motion?

• Brownian Motion is not the right term here, even though they are both caused by the random motion of small particles. Brownian motion specifically refers to the erratic, random motion of larger particles due to interactions with smaller particles Commented Jul 9, 2022 at 21:14

Energy is conserved. If average molecular velocity increases without an external application of work, the only place for the energy to come from is the average speed of the molecules. For a compressible fluid like air, we must also account for the mechanical work done to compress the fluid $$W_c = \int_{V_1}^{V_2} P dV$$ (for pressure P, volume V). The difference in the energy-over-time sum of all the microscopic molecule collisions with the barrier between the slow region and the fast region must be equal and opposite the mechanical work done in the same time on the same mass of macroscopic fluid, $$W = W_c + \Delta K$$ where K is the kinetic energy. Hence the pressure caused by the aggregate of the particle impacts must decrease.