Exploitation of the second law: Coleman-Noll and Liu procedure in comparison
The authors consider the balance of internal energy \(\overset{.}{\varepsilon }+\text{div}\,\mathbf{q}-r=0\) for a material at rest. Here \(\varepsilon \) is the internal energy density, \(\mathbf{q}\) is the heat flux and \(r\) is the heat supply density. The second law is expressed as \(\sigma _{S}=\overset{.}{ s}+\text{div}\,\mathbf{J}_{s}-r_{s}\geq 0\), where \(s\) means the entropy density, \(\mathbf{J}_{s}\) is the entropy flux and \(r_{s}\) is the entropy supply density. The authors then specialize to a rigid heat conductor whose constitutive state space is spanned by the internal energy \(\varepsilon \) and vector fields~\(\mathbf{z}\), which do not contain the derivative of this internal energy. The authors derive the Coleman-Noll inequality through some substitution of values and Liu's inequality when introducing a Lagrange-Farkas multiplier \(-\lambda (\varepsilon ,\mathbf{z})\). The first main result proves the equivalence between these two inequalities. The authors then repeat the study, but now assuming that the thermodynamic state space is spanned by the set of variables \(\varepsilon \), \(\mathbf{z}\), \( \nabla \varepsilon \) and \(\nabla \mathbf{z}\). An equivalence result also holds true. As a last example, they consider the case where the thermodynamic state space is spanned by the set of variables \(\varepsilon \), \(\nabla \varepsilon \), \(\alpha \) and \(\nabla \alpha \), for some internal variable \(\alpha \) satisfying a partial differential equation \(\overset{.}{ \alpha }=f(\alpha ,\nabla \alpha )\). Here the Lagrange-Farkas multipliers must be linked through linear relations in order to satisfy the equivalence between the two inequalities. In the conclusion, the authors briefly discuss the advantages of the Coleman-Noll technique and of Liu's procedure.
scientific article; zbMATH DE number 5309510