consider a hypodermic needle like structure with a piston in the larger tube and a piston in the smaller tube as well. let the cavity be filled with an ideal incompressible fluid. if i understand the Bernoulli theorem correctly, if a pressure is applied to the larger piston, causing the fluid to move with respect to the structure, then the pressure on the smaller piston will be less than that pressure, in accordance with Bernoulli's equation. (i'm ignoring gravity in this. ) notice that the cavity shape is changed, but the volume is unchanged. if i replace the liquid with an ideal gas, and keep the volume unchanged as the larger piston is pushed, does the same equation apply? if so, is there a reference where Bernoulli's theorem is proved using the kinetic theory of gasses?