Pressure and velocity in fluid dynamcis

Question

I want to ask whether someone could help me solving a problem in understanding the relationship between pressure and velocity in fluid dynamics, please?

Why do I know that particles are not by accident equally moving, i.e. equal magnitude and equal direction? If this would happen, I end up with a problem in the relationship between pressure and fluid velocity. Say I would think about pressure being the transported momentum across the boundary of a fluid element. Then I have a problem, since I know that for equal pressure the fluid velocity should be zero. But if I think about a cloud of particles all moving with equal velocity the pressure in the given sense would be zero, i.e. constant, but obviously the velocity would not be zero.

Answer 1

The Bernoulli equation addresses your question in the macroscopic sense; it is an energy balance for fluid in steady flow. See a good thermodynamics book, such as one by Sonntag and Van Wylen, for an explanation of this equation; for example, how pressure can be reduced as velocity increases.

I defer to others for a microscopic level answer to your question.