Since the equations of RET have the structure of balance laws, they can be recast into the following form:
$$\begin{aligned} \partial _{t} {\mathbf {F}}^{0}({\mathbf {u}}) + \partial _{i} {\mathbf {F}}^{i}({\mathbf {u}}) = {\mathbf {f}}({\mathbf {u}}), \quad i = 1,2,3. \end{aligned}$$
(32)
In (
32),
\({\mathbf {u}} \in \mathbb {R}^{N}\) denotes the vector of state variables,
\({\mathbf {F}}^{0}({\mathbf {u}}) \in \mathbb {R}^{N}\) is the vector of densities,
2\({\mathbf {F}}^{i}({\mathbf {u}}) \in \mathbb {R}^{N}\) are the components of fluxes, and
\({\mathbf {f}}({\mathbf {u}}) \in \mathbb {R}^{N}\) is the vector of source (production) terms. Certain components of
\({\mathbf {F}}^{0}({\mathbf {u}})\) and
\({\mathbf {F}}^{i}({\mathbf {u}})\) are determined by the physical laws, but some of the fluxes are constitutive quantities, as well as the source terms
\({\mathbf {f}}({\mathbf {u}})\). Since the dependence of the densities and fluxes on
\({\mathbf {u}}\) is of local type (in accordance with assumption (b)), the system turns into a quasi-linear system of PDE’s:
$$\begin{aligned} {\mathbf {A}}^{0}({\mathbf {u}}) \partial _{t} {\mathbf {u}} + {\mathbf {A}}^{i}({\mathbf {u}}) \partial _{i} {\mathbf {u}} = {\mathbf {f}}({\mathbf {u}}), \end{aligned}$$
(33)
where
\({\mathbf {A}}^{0} = \partial {\mathbf {F}}^{0}/\partial {\mathbf {u}}\) and
\({\mathbf {A}}^{i} = \partial {\mathbf {F}}^{i}/\partial {\mathbf {u}}\). If all the eigenvalues
\(\lambda ({\mathbf {u}})\) of the eigenvalue problem
\((- \lambda {\mathbf {A}}^{0} + n_{i} {\mathbf {A}}^{i}) {\mathbf {d}} = {\mathbf {0}}\) are real, where
\(n_{i}\) are the components of the unit-normal vector, the system (
33) is said to be hyperbolic in
\(t-\)direction.
For any thermodynamic process described by (
32), in accordance with assumption (d), the state variables
\({\mathbf {u}}\) must satisfy the entropy inequality:
$$\begin{aligned} \partial _{t} h^{0}({\mathbf {u}}) + \partial _{i} h^{i}({\mathbf {u}}) = \Sigma ({\mathbf {u}}) \ge 0, \end{aligned}$$
(34)
where
\(h^{0}\) is the entropy density,
\(h^{i}\) are the components of the entropy flux and
\(\Sigma \) is the entropy production rate. Due to local dependence, the entropy inequality can be put into quasi-linear form:
$$\begin{aligned} \frac{\partial h^{0}}{\partial {\mathbf {u}}} \partial _{t} {\mathbf {u}} + \frac{\partial h^{i}}{\partial {\mathbf {u}}} \partial _{i} {\mathbf {u}} = \Sigma ({\mathbf {u}}). \end{aligned}$$
(35)
The entropy principle states [
40] that for any thermodynamic process the state variables
\({\mathbf {u}}\) must obey the entropy law (
34), provided they satisfy the balance laws (
32). In other words, the balance laws are regarded as
constraints. Since both the balance laws (
33) and the entropy law (
35) are quasi-linear with respect to
\({\mathbf {u}}\), the problem with constraints can be transformed into a problem without constraints at the expense of introduction of Lagrange multipliers
\({\mathbf {u}}^{\prime } \in \mathbb {R}^{N}\):
$$\begin{aligned} \partial _{t} h^{0} + \partial _{i} h^{i} - \Sigma = {\mathbf {u}}^{\prime } \cdot \left( \partial _{t} {\mathbf {F}}^{0} + \partial _{i} {\mathbf {F}}^{i} - {\mathbf {f}} \right) . \end{aligned}$$
(36)
As a consequence, the following relations must hold:
$$\begin{aligned} \mathrm {d}h^{0} = {\mathbf {u}}^{\prime } \cdot \mathrm {d}{\mathbf {F}}^{0}, \quad \mathrm {d}h^{i} = {\mathbf {u}}^{\prime } \cdot \mathrm {d}{\mathbf {F}}^{i}, \end{aligned}$$
(37)
or equivalently:
$$\begin{aligned} \frac{\partial h^{0}}{\partial {\mathbf {u}}} = {\mathbf {u}}^{\prime } \cdot \frac{\partial {\mathbf {F}}^{0}}{\partial {\mathbf {u}}}, \quad \frac{\partial h^{i}}{\partial {\mathbf {u}}} = {\mathbf {u}}^{\prime } \cdot \frac{\partial {\mathbf {F}}^{i}}{\partial {\mathbf {u}}}. \end{aligned}$$
(38)
Moreover, the residual inequality has to be satisfied as well:
$$\begin{aligned} \Sigma = {\mathbf {u}}^{\prime } \cdot {\mathbf {f}} \ge 0. \end{aligned}$$
(39)
Equations (
36)–(
39) require some comments. Since certain components of the fluxes are constitutive quantities, Eq. (
38) can serve for the determination of the multipliers
\({\mathbf {u}}^{\prime }\), as well as constitutive components of the fluxes. Moreover, one of the legacies of RET is the assumption that entropy flux
\(h^{i}\) is not necessarily proportional to the heat flux, i.e., it is not defined as
\(h^{i} = q_{i}/T\). Instead, it is treated as a constitutive quantity, which brings certain freedom in its derivation. Finally, once the multipliers
\({\mathbf {u}}^{\prime }\) are determined, one may choose the source terms
\({\mathbf {f}}({\mathbf {u}})\) in such a way that the residual inequality (
39) is satisfied for all thermodynamic processes. Although this statement introduces the flavor of arbitrariness, it is usually reduced to similar arguments as those used in classical TIP, or as in NET-IV in the previous section. Namely, the components of the source terms
\({\mathbf {f}}\) are chosen as linear forms of
\({\mathbf {u}}^{\prime }\) (in the appropriate way which takes into account their tensorial order), so that
\(\Sigma \) becomes quadratic form in
\({\mathbf {u}}^{\prime }\).