Entropy production by Fourier heat conduction ($\rho \frac{DS}{Dt}=\nabla \cdot \frac{k\nabla T}{T}+\frac{k|\nabla T{|}^2}{T^2}$)

For an ideal gas with specific internal energy $c_{v}T$ and Fourier's law heat flux $\mathbf{q}=-k\nabla T$, subtraction of the kinetic-energy balance from total energy gives $\rho c_{v}DT/Dt+p\nabla \cdot \mathbf{u}=\nabla \cdot (k\nabla T)$. The thermodynamic entropy satisfies $TDS/Dt=c_{v}DT/Dt+p\nabla \cdot \mathbf{u}/\rho$, soThe divergence describes entropy transport by heat; the final term is local irreversible production. For $T>0$ and $k\ge 0$ it is nonnegative. In a fixed impermeable thermally insulated container, boundary entropy flux vanishes and mass conservation gives $d\int \rho SdV/dt=\int k|\nabla T{|}^2/T^{2}dV$. Insulation conserves total energy while permitting entropy to increase as temperature gradients relax.