Bernoulli's equation :
Do you mean : $P+\tfrac{1}{2}\rho v^2+\rho g h=$constant

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question 1: can this be simulated with an ideal gas between the pistons?
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Could you describe your model in more detail? I can not catch your ideas.

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question 2: is Bernoulli's equation  true for an ideal gas?
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The Bernoulli's principle is valid for incompressible flows (e.g. most liquid flows) and also for compressible flows (e.g. gases) moving at low Mach numbers