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.