Thermodynamic framework for a generalized heat transport equation (Q325609)

The authors propose an extended irreversible thermodynamic framework for a generalized heat transport equation taken as \(\tau \frac{\partial \mathbf{q} }{\partial t}+\mathbf{q}=-\lambda \nabla T+m_{1}\mathbf{q}\nabla \cdot \mathbf{q}+m_{2}\mathbf{q}\cdot \nabla \mathbf{q}+m_{3}\nabla \mathbf{q} ^{2}+m_{4}\nabla ^{2}\mathbf{q}+m_{5}\nabla (\nabla \cdot \mathbf{q})+m_{6} \mathbf{q}(\mathbf{q}\cdot \nabla T)+m_{7}\mathbf{q}^{2}\nabla T\), where \( \mathbf{q}\) is the heat flux, \(T\) the temperature and \(m_{i}\), \(i=1,\dots ,7 \), are constants which allow covering different known and already studied heat transport equations. The authors consider the internal energy \(u\), the heat flux \(\mathbf{q}\) and the flux of the heat flux \(\mathbf{Q}\) as basic state variables. They first write the entropy production \(\sigma ^{s}\) as a linear relation \(\sigma ^{s}=\mathbf{q}\cdot X_{\mathbf{q}}+\mathbf{Q}:X_{ \mathbf{Q}}\) which involves two conjugate thermodynamic forces \(X_{\mathbf{q} }\) and \(X_{\mathbf{Q}}\) whose expressions are given in terms of \(\mathbf{q}\), \(\mathbf{Q}\), \(\mathbf{Q}\cdot \mathbf{q}\) and \(\mathbf{qq}\). They here use a generalized specific entropy \(s(u,\mathbf{q},\mathbf{Q})\), its derivative and the entropy flux \(\mathbf{J}^{s}\). They finally describe the coupled evolution equations for \(\mathbf{q}\) and \(\mathbf{Q}\). Then, simplifying the evolution equation for \(\mathbf{Q}\), the authors obtain an expression which gives an approximation of \(\mathbf{Q}\) in terms of \( \mathbf{q}\) and \(\mathbf{qq}\). This refers to the Guyer-Krumhansl model described by \textit{D. Jou} et al. [Extended irreversible thermodynamics. 4th ed. Dordrecht: Springer (2010; Zbl 1185.74002)]. This leads to a simplified evolution equation for \(\mathbf{q}\). The paper ends with an approximate expression of the thermal conductivity \(\lambda \) in terms of the unit tensor \(\mathbf{I}\) and of \(\mathbf{q}^{2}\) and \(\mathbf{qq}\).