Past exam of the theoretical mechanics course of the University of Oxford / 2010 / B / 4 / a / ii / Solution

Continuity gives $D\log \rho /Dt=-\nabla \cdot \mathbf{u}$. Using $p=\rho RT$, the temperature equation gives $D\log T/Dt=-(R/c_v)\nabla \cdot \mathbf{u}$. ThereforeConsequentlyEach smooth parcel evolves isentropically under these adiabatic inviscid equations; the argument does not cover entropy-producing shocks.