Restricted gradient constitutive dissipative model

Use the time-oriented Lorentzian four-manifold, signature $(-,+,+,+)$, Levi-Civita connection, smooth future unit observer congruence, and length-coordinate convention $x^0=ct$ of Rest fluid state, frame and geometric current definitions. Here a frame means this smooth observer field $u$ with $g(u,u)=-1$ and its pointwise rest spaces; it does not mean a coordinate chart or a tetrad. On a nonempty open region $O$, take a $C^2$ fixed-composition state map $(v,{\sigma }_p):O\to X$ with $X\subset (0,\mathrm{\infty })\times \mathbb{R}$, state function $e\in C^2(X)$, and $T={\mathrm{\partial }}_{{\sigma }_p}e>0$ as adopted by Restricted local equilibrium parcel with adopted material state function. The mathematical state maps are $n=v^{-1}$, $s=n{\sigma }_p$, $\epsilon =ne$, and $p=-{\mathrm{\partial }}_{v}e$; their meanings and units are those in Rest fluid state, frame and geometric current definitions and Thermodynamics-GR fields, currents, reference normalization and units.

Write $u_b=g_{bc}{u}^c$ and define the rest-space projector, derivative along the observer congruence, expansion, acceleration, and spatial symmetric trace-free shear by ${\mathrm{\Delta }}^a{}_b={\delta }^a{}_b+u^a{u}_b,{\mathrm{\Delta }}^{ab}=g^{ab}+u^a{u}^b,D=u^a{\mathrm{\nabla }}_a,\theta ={\mathrm{\nabla }}_a{u}^a,a_b=u^c{\mathrm{\nabla }}_c{u}_b,$ ${\sigma }_{ab}={\mathrm{\Delta }}_a{}^c{\mathrm{\Delta }}_b{}^d{\mathrm{\nabla }}_{(c}{u}_{d)}-\frac{\theta }{3}{\mathrm{\Delta }}_{ab},$ where parentheses denote the averaged symmetric part. Put $Y_b={\mathrm{\nabla }}_{b}T+T{a}_b$ and $Z_b={\mathrm{\nabla }}_b(\mu \mathrm{/}T)$, using the chemical potential $\mu$ from Rest fluid state, frame and geometric current definitions. The number current, stress tensor and entropy current in the selected branch are $J^a=n{u}^a+V^a,{\mathsf{T}}^{ab}=\epsilon u^a{u}^b+(p+\mathrm{\Pi }){\mathrm{\Delta }}^{ab}+u^a{Q}^b+Q^a{u}^b+{\pi }^{ab},S^a=s{u}^a+\frac{Q^a}{T}-\frac{\mu V^a}{T},$ with $u_a{V}^a=u_a{Q}^a=u_a{\pi }^{ab}=0$ and ${\pi }^{ab}$ symmetric and trace-free. Adopt the source-free balances ${\mathrm{\nabla }}_a{J}^a=0$ and ${\mathrm{\nabla }}_a{\mathsf{T}}^{ab}=0$ together with the first-order gradient closures $\mathrm{\Pi }=-\zeta c\theta ,{\pi }^{ab}=-2\eta c{\sigma }^{ab},Q^a=-\frac{\kappa }{c}{\mathrm{\Delta }}^{ab}{Y}_b,V^a=-L{\mathrm{\Delta }}^{ab}{Z}_b\mathrm{.}$

Here $c>0$ has units $\mathrm{m}{\mathrm{s}}^{-1}$; $\theta$, $a_b$ and ${\sigma }_{ab}$ have inverse-length components; $\mathrm{\Pi }$, ${\pi }^{ab}$ and $Q^a$ have units Pa; $\zeta ,\eta \in C^1(O)$ have units Pa s; $\kappa \in C^1(O)$ has units $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$; and $L\in C^1(O)$ has formal units $\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}{\mathrm{r}}^2\mathrm{K}{\mathrm{J}}^{-1}{\mathrm{m}}^{-2}$. The temperature and chemical-potential gradients have units K m${}^{-1}$ and J marker${}^{-1}$ K${}^{-1}$ m${}^{-1}$, respectively. The coefficient fields satisfy $\zeta ,\eta ,\kappa ,L\ge 0$. On $O$, require $\mathrm{\Pi }$, ${\pi }^{ab}$, $Q^a$, and $V^a$, together with the resulting $J^a$, ${\mathsf{T}}^{ab}$, and $S^a$, to be $C^1$. The state map and state function are $C^2$ and $u$ is smooth, as specified above. These regularities make the displayed balances and entropy-current divergence pointwise classical expressions; require the balances to hold pointwise.

Each application declares a nonempty open regime of the pointwise state values and first covariant derivatives of $u,T,\mu \mathrm{/}T$ (with the listed algebraic constraints) on which the constitutive model is used. The regime separately requires $J$ to be future timelike. Since $u_a{V}^a=0$, this condition is $n>0$ and $g_{ab}{V}^a{V}^b<n^2$, as follows from $g(J,J)=-n^2+g(V,V)$ and $-u_a{J}^a=n$. It is an additional applicability restriction, not a consequence of the gradient closures.

This is an adopted constitutive model for one declared material regime. The closures are not derived from geometry, measurements, or the second law; they do not assert that a solution exists, that the model applies to every material, or that the branch is causal, stable, or globally entropy-increasing. No empirical observation is supplied.

Use the fixed-composition local-equilibrium state, smooth time-oriented Lorentzian geometry of signature $(-,+,+,+)$, smooth future unit observer congruence $u$, and length coordinates $x^0=ct$ from Restricted local equilibrium parcel with adopted material state function and Rest fluid state, frame and geometric current definitions. In particular, $n>0$, $T>0$, ${\mathrm{\Delta }}^{ab}=g^{ab}+u^a{u}^b$, $D=u^a{\mathrm{\nabla }}_a$, and $\theta ={\mathrm{\nabla }}_a{u}^a$ have those suppliers' meanings. In a selected bulk-only branch, adopt the stress, number current, and source-free balances ${\mathsf{T}}^{ab}=\epsilon u^a{u}^b+(p+\mathrm{\Pi }){\mathrm{\Delta }}^{ab},J^a=n{u}^a,{\mathrm{\nabla }}_a{\mathsf{T}}^{ab}=0,{\mathrm{\nabla }}_a{J}^a=0,$ where $\mathrm{\Pi }$ is a smooth scalar bulk pressure. Let $\zeta ,\tau \in C^1(O)$ be positive coefficient fields with SI units Pa s and s, respectively, and define $\beta =\tau \mathrm{/}\zeta$, with units Pa${}^{-1}$. Adopt the extended entropy current and derivative-corrected relaxation law $S^a=(s-\frac{\beta {\mathrm{\Pi }}^2}{2T})u^a,\beta D\mathrm{\Pi }+\frac{\mathrm{\Pi }}{\zeta c}=-\theta -\frac{T\mathrm{\Pi }}{2}{\mathrm{\nabla }}_a\text{\hspace{0.17em}⁣}\left(\frac{\beta u^a}{T}\right)\mathrm{.}$ Here $\mathrm{\Pi }$ has units Pa, $D$ and $\theta$ have inverse-length units, and each side of the relaxation law has units m${}^{-1}$; $\beta {\mathrm{\Pi }}^2\mathrm{/}(2T)$ has entropy-density units J K${}^{-1}$ m${}^{-3}$.

For each application, specify a nonempty open near-equilibrium regime in the pointwise state and first-derivative data around $\mathrm{\Pi }=0$, based at constant zero-gradient reference data, and use the current and law only while the fields remain in that declared regime. The state map and state function are $C^2$, $u$ is smooth, the coefficient fields are $C^1$, and $\mathrm{\Pi }$ is smooth; thus $J^a$, ${\mathsf{T}}^{ab}$, and $S^a$ are $C^1$. Require the balances and corrected law to hold pointwise. The mathematical result Corrected bulk relaxation and exact extended entropy supplies the exact matching conditional entropy identity and one homogeneous witness; its actual statement and proof are conditional on the listed mathematical inputs and assume exactly countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). It motivates this selection and is used by the downstream deductions, but it does not derive the adopted physical postulate.

This is a restricted constitutive assumption for the selected material. It does not derive the equation of state, assert universal validity, or adopt existence of a general nonlinear solution. It makes no causal or stability claim; positive relaxation time alone supplies neither. No empirical observation is supplied.

Let $O$ be a nonempty open region of a smooth time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$, with its metric-compatible torsion-free connection. Use the $C^2$ state map $(v,{\sigma }_p):O\to X$ and the associated $n=v^{-1}$, $s=n{\sigma }_p$, $T=e_{{\sigma }_p}$, $p=-e_v$, and $\mu =e+pv-T{\sigma }_p$ of Rest fluid state, frame and geometric current definitions, with $n,T>0$ and the fixed-composition branch adopted by Restricted gradient constitutive dissipative model. The frame is the smooth future unit timelike observer congruence $u$ of that definition, not a chart or tetrad. Define ${\mathrm{\Delta }}^{ab}=g^{ab}+u^a{u}^b$, $\theta ={\mathrm{\nabla }}_a{u}^a$, and $a_b=u^c{\mathrm{\nabla }}_c{u}_b$. Let ${\sigma }_{ab}$ be the spatial symmetric trace-free part of ${\mathrm{\nabla }}_{(a}{u}_{b)}$, namely ${\mathrm{\Delta }}_a{}^c{\mathrm{\Delta }}_b{}^d{\mathrm{\nabla }}_{(c}{u}_{d)}-(\theta \mathrm{/}3){\mathrm{\Delta }}_{ab}$, where parentheses denote averaged symmetrization. Let the fields in Restricted gradient constitutive dissipative model satisfy its source-free stress and particle balances, closures, and declared applicability regime, including the separately required future-timelike condition on $J$. The observer congruence is smooth and the state function and map are $C^2$ as in Rest fluid state, frame and geometric current definitions. To match the exact hypotheses of Frame-qualified dissipative entropy cancellation, assume $\mathrm{\Pi }$, $Q^a$, $V^a$, and ${\pi }^{ab}$ are smooth on $O$; the transport coefficients remain $C^1$ as in the postulate. The balances and entropy-current divergence are then classical pointwise expressions. Put $Y_a={\mathrm{\nabla }}_{a}T+T{a}_a,Z_a={\mathrm{\nabla }}_a\text{\hspace{0.17em}⁣}\left(\frac{\mu }{T}\right),a_b=u^c{\mathrm{\nabla }}_c{u}_b\mathrm{.}$ Then the entropy current $S^a=s{u}^a+Q^a\mathrm{/}T-\mu V^a\mathrm{/}T$ has the exact pointwise divergence ${\mathrm{\nabla }}_a{S}^a=\frac{\zeta c{\theta }^2+2\eta c{\sigma }^{ab}{\sigma }_{ab}}{T}+\frac{\kappa }{c{T}^2}{\mathrm{\Delta }}^{ab}{Y}_a{Y}_b+L{\mathrm{\Delta }}^{ab}{Z}_a{Z}_b\ge 0,$ where $\theta ={\mathrm{\nabla }}_a{u}^a$, ${\mathrm{\Delta }}^{ab}=g^{ab}+u^a{u}^b$, and ${\sigma }_{ab}$ is the spatial symmetric trace-free shear defined in the postulate. Thus this selected branch satisfies the adopted local entropy-current admissibility condition of Local entropy-current admissibility in a separately defined model. With length coordinates $x^0=ct$, the divergence has units J K${}^{-1}$ m${}^{-4}$ and $c{\mathrm{\nabla }}_a{S}^a$ is the physical production rate per volume in J K${}^{-1}$ m${}^{-3}$ s${}^{-1}$.

This is a conditional model deduction for a sufficiently regular solution already supplied in the declared regime. It does not assert that the model has a solution, that the constitutive law describes nature, or that the branch is causal, linearly stable, or globally entropy-increasing. It uses no empirical premise.

On a nonempty open region of the smooth $(-,+,+,+)$ Lorentzian model of Specified bulk extended entropy and relaxation law, assume an actual solution of its source-free stress and particle balances and corrected relaxation law. Use the state maps and units of Rest fluid state, frame and geometric current definitions and Thermodynamics-GR fields, currents, reference normalization and units, with smooth future unit observer congruence $u$ (the frame, not a chart or tetrad), ${\mathrm{\Delta }}^{ab}=g^{ab}+u^a{u}^b$, $D=u^a{\mathrm{\nabla }}_a$, $\theta ={\mathrm{\nabla }}_a{u}^a$, $n,T,c>0$, positive $C^1$ coefficient fields $\zeta ,\tau$, and $\beta =\tau \mathrm{/}\zeta$. The state function and map are $C^2$ and $u$ is smooth as in Rest fluid state, frame and geometric current definitions; take $\mathrm{\Pi }$ smooth as required by Corrected bulk relaxation and exact extended entropy, with the positive coefficients $C^1$ as in the postulate. Thus the displayed stress and current and $S^a$ are $C^1$; require the balances and corrected law to hold pointwise. The solution remains in the application's declared open near-equilibrium regime. With ${\mathsf{T}}^{ab}=\epsilon u^a{u}^b+(p+\mathrm{\Pi }){\mathrm{\Delta }}^{ab},J^a=n{u}^a,S^a=(s-\frac{\beta {\mathrm{\Pi }}^2}{2T})u^a,$ the exact pointwise identity is ${\mathrm{\nabla }}_a{S}^a=\frac{{\mathrm{\Pi }}^2}{\zeta cT}\ge 0.$ This is conditional on an actual solution and does not assert general solution existence.

There is also an explicit homogeneous solution under the additional application condition that the declared open regime contains constant zero-gradient reference data with $\mathrm{\Pi }=0$. Let the selected material's state domain contain $(v_0,{\sigma }_0)$ with $n_0=1\mathrm{/}{v}_0>0$ and $T_0=e_{{\sigma }_p}(v_0,{\sigma }_0)>0$, and let the regime contain those constant state data. On $M={\mathbb{R}}^4$ with global length coordinates $(x^0,x^1,x^2,x^3)=(ct,x^1,x^2,x^3)$, Minkowski metric $g=-(d{x}^0{)}^2+{\sum }_{i=1}^3(d{x}^i{)}^2$, and future direction ${\mathrm{\partial }}_{x^0}$, choose constant positive ${\zeta }_0,{\tau }_0$ and set $u={\mathrm{\partial }}_{x^0}$. For some $\delta >0$ seconds take the open region $O=(-c\delta ,\mathrm{\infty })\times {\mathbb{R}}^3$, put $t=x^0\mathrm{/}c$, and define $\mathrm{\Pi }(t)={\mathrm{\Pi }}_0{e}^{-t\mathrm{/}{\tau }_0},t>-\delta \mathrm{.}$ Choose ${\mathrm{\Pi }}_0$ sufficiently small that the full field and first-derivative data stay in the declared regime throughout $O$. Then the source-free balances and corrected law hold, and the entropy production is strictly positive at every finite $t\ge 0$ when ${\mathrm{\Pi }}_0\mathrm{\ne }0$, and zero when ${\mathrm{\Pi }}_0=0$.

The argument is a conditional model deduction, not an observation or a general nonlinear existence, causality, or stability theorem. No empirical premise is used.