Perfect stress and particle balances with entropy identity

Given: A nonempty open set $O$ in a smooth time-oriented Lorentzian four-manifold $(M,g,N)$ of signature $(-,+,+,+)$; a smooth future unit field $u$ on $O$; a $C^2$ state map into a nonempty open material-state domain; and the fields and balances specified in the Statement.

The geometric and tensor-bundle construction carries Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)); no stronger choice principle is assumed.

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model: $M$ is a smooth four-manifold without boundary, $g$ is a smooth symmetric nondegenerate metric of signature $(-,+,+,+)$, and $N$ selects the future cone. Its stated tangent-bundle construction assumes only ${\mathrm{A}\mathrm{C}}_{\omega }$.

Rest fluid state, frame and geometric current definitions: on a nonempty open set $O$, the observer field is smooth and future unit timelike; a local state has a nonempty open domain $X\subset (0,\mathrm{\infty })\times \mathbb{R}$, a specified $e\in C^2(X)$, and a $C^2$ state map $(v,{\sigma }_p):O\to X$. It defines $n=v^{-1}$, $s=n{\sigma }_p$, $ϵ=ne$, $T=e_{{\sigma }_p}$, $p=-e_v$, $H=e+pv$, $h=ϵ+p=nH$, and $\mu =H-T{\sigma }_p$. The convention is signature $(-,+,+,+)$ with $x^0=ct$; $c$ has units $\mathrm{m}{\mathrm{s}}^{-1}$, $u$ is dimensionless, $n$ has units $\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}\mathrm{r}{\mathrm{m}}^{-3}$, $s$ has units $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$, $T$ has units K, $\mu$ has units $\mathrm{J}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}{\mathrm{r}}^{-1}$, and $ϵ,p,h$ have units Pa.

Local Gibbs, Euler, parcel first law and response identities (current batch-53 draft): for this state, $dϵ=Tds+\mu dn$ and $ϵ+p=Ts+\mu n$ as identities of differentials and state functions. The identities are used conditionally at the current carrier's stated $C^2$ hypotheses.

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields: for a smooth nondegenerate metric, the metric-selected connection is torsion-free and metric-compatible, and its induced tensor derivative obeys the product and contraction rules. This is the published mathematical supplier actually inspected here.

Define ${\mathrm{\Delta }}^{ab}=g^{ab}+u^a{u}^b$, $D=u^a{\mathrm{\nabla }}_a$, $\theta ={\mathrm{\nabla }}_a{u}^a$, and $a^b=u^a{\mathrm{\nabla }}_a{u}^b$. The perfect model data are $T^{ab}=ϵu^a{u}^b+p{\mathrm{\Delta }}^{ab}$, $J^a=n{u}^a$, and ${\mathrm{\nabla }}_a{T}^{ab}=f^b$, ${\mathrm{\nabla }}_a{J}^a=0$. In SI, $f^a$ has units $\mathrm{N}{\mathrm{m}}^{-3}$, $ϵ,p,h$ have units Pa, $n$ has units $\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}\mathrm{r}{\mathrm{m}}^{-3}$, and $s$ has units $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$.