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Bernoulli (horizontal)



JohnM showed a better derivation of the Bernoulli formula.
P+0.5*rho*v^2+rho*g*y=const. My question, however,
was primarily about the interpretation, not the derivation.

Presumably the system is water between the two pistons
The naive explanation of the observed increase of KE is
"because the volume of the liquid must be conserved."
A more sophisticated explanation is the work-energy
theorem applied to the CM of the system. The net work
is equal to deltaKE.

Suppose the setup is horizontal; the tube becomes narrower
but the average PEgrv of "parcels of water" does not change
along the tube. In that simple case we have

P1+0.5*rho*v1^2=P2+0.5*rho*v2^2.

where P is pressure. Is it OK to say that P is also the energy
per unit volume (because N/m^2=J/m^3)? We know that P
decreases along the tube as the cross sectional area becomes
smaller. The situation looks like an energy transformation;
process. Bernoulli tells us that P decreases by the same amount
by which the KE (also per unit volume) increases. What is
wrong with saying that P is the "mechanical energy of
pressure," per unit volume? Is this energy kinetic or potential?
Ludwik Kowalski