= Pressure work
{title2=$\delta W=p\,dV$}
<Pressure work> is mechanical <work> transferred by <pressure forces> acting through a moving boundary. For a piston of <area> $A$ moving through $dx$, $dV=A\,dx$ and the opposing applied force is $p_{\rm ext}A$. The <work> done by the <fluid> is therefore
$$
\delta W_{\rm by}=p_{\rm ext}\,dV.
$$
The familiar $\delta W=p\,dV$ uses the system <pressure> when it equals the applied boundary <pressure>, as in a reversible mechanical step. In a <free expansion> into vacuum, $p_{\rm ext}=0$ and the boundary <work> vanishes even if the gas initially has nonzero internal <pressure>.
In steady <incompressible flow>, the <Bernoulli equation> contains $p/\rho$ as flow <work> per unit <mass>. This accounts for <pressure forces> at a <control volume>'s entrances and exits; it is not an independently conserved <internal energy> of an isolated parcel. The <first law of thermodynamics> keeps the complete work and <energy> balance.
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