Local energy balance for an insulated continuum body

For a field record on $\mathrm{\Omega }\times I$ from Rectangular local-equilibrium field data, require the supplied heat-flux field $j_Q$ and each of its first spatial partial derivatives to extend continuously to $\bar{\mathrm{\Omega }}\times [t_0,t_1]$ for every compact interval $[t_0,t_1]\subset I$. Thus $\mathrm{div}{j}_Q$ is the classical coordinate divergence in $\mathrm{\Omega }\times I$, and $j_Q\cdot n$ denotes its continuous trace on each closed face. For these records, adopt the local balance ${\mathrm{\partial }}_{t}u(T(x,t))+\mathrm{div}{j}_Q(x,t)=0$ and the insulated-boundary condition $j_Q(x,t)\cdot n(x)=0$ on every face of $\mathrm{\partial }\mathrm{\Omega }$. Both terms in the local equation have units $\mathrm{W}{\mathrm{m}}^{-3}$. The boundary condition says that no heat energy crosses the body's outer faces in this model.

This local continuum balance is an additional adopted postulate, with the sign and energy convention of First-law energy balance for closed process records. It is not deduced from that global process-record balance. Matter transfer, volume work, internal sources, moving boundaries, and external-field energy require separately specified balance models and are outside this item.