 Isobaric conductive cooling in mass coordinates

With fixed pressure $p$, let $A=\mu p/\mathcal{R}$, so $\rho =A/T$. A material mass coordinate has ${\partial }_x=\rho {\partial }_m$ and the material time derivative becomes ${\partial }_t{|}_m$. The energy and continuity equations then give the displayed equation with $\tau =Ct$ and $C=(\gamma -1)(\mu /\mathcal{R}{)}^{2}p/\gamma$. A mass integral from a fixed origin is material only when the mass flux there vanishes; otherwise subtract its time-integrated flux to label parcels. Uniform pressure is a low-Mach approximation when sound crossing is faster than thermal evolution.