Lagrangian entropy inequality (${\rho }_0\dot{\eta }\ge {\rho }_{0}r/\theta -\mathrm{Div}(q/\theta )$)

The local entropy balance has nonnegative production, with heat supply ${\rho }_{0}r/\theta$ and thermodynamic entropy flux $q/\theta$ at positive temperature $\theta$. Its integral version holds on every material reference subvolume.