 Energy fluxes of a strong adiabatic shock

In the rest frame of a strong planar shock wave, a perfect gas has ${\rho }_2/{\rho }_1=(\gamma +1)/(\gamma -1)$, $u_2/u_1=(\gamma -1)/(\gamma +1)$ and $p_2=2{\rho }_1{u}_1^2/(\gamma +1)$. Its downstream specific internal energy is ${ϵ}_2=p_2/[(\gamma -1){\rho }_2]=2u_1^2/(\gamma +1{)}^2$. Multiplying by mass flux $j={\rho }_1{u}_1$ gives the internal-energy flux in the title. The lost bulk kinetic energy flux instead equals $j(u_1^2-u_2^2)/2=\gamma F_{\mathrm{int}}$: this includes downstream pressure work and is the increase of specific enthalpy flux in the cold-upstream limit. Neither quantity is a temperature change per unit time of a fixed target mass without further information about that target.