Heat and viscous entropy specialization

Let $(M,g,N)$ be a smooth time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$, let $O\subseteq M$ be nonempty and open, and let $u$ be a smooth future unit timelike vector field on $O$. Use the metric-compatible torsion-free connection $\mathrm{\nabla }$ and a smooth local state map $(v,{\sigma }_p):O\to X$ with $X\subset (0,\mathrm{\infty })\times \mathbb{R}$ open and $e\in C^{\mathrm{\infty }}(X)$, as in Rest fluid state, frame and geometric current definitions. Thus $n=v^{-1}>0$, $s=n{\sigma }_p$, $ϵ=ne$, $T=e_{{\sigma }_p}$, $p=-e_v$, and $\mu =e+pv-T{\sigma }_p$; assume $T>0$ on $O$ and the Gibbs and Euler identities of Local Gibbs, Euler, parcel first law and response identities. Use the convention $x^0=ct$ with $c>0$ in $\mathrm{m}{\mathrm{s}}^{-1}$.

Define ${\mathrm{\Delta }}^a{}_b={\delta }^a{}_b+u^a{u}_b$, ${\mathrm{\Delta }}^{ab}=g^{ac}{\mathrm{\Delta }}^b{}_c$, ${\mathrm{\Delta }}_{ab}=g_{ab}+u_a{u}_b$, $D=u^a{\mathrm{\nabla }}_a$, $\theta ={\mathrm{\nabla }}_a{u}^a$, and $a^b=D{u}^b$. Let ${\sigma }_{ab}$ be the shear defined in Spatial shear properties and nonnegative norm: the spatially projected symmetrized derivative of $u$, minus $\theta {\mathrm{\Delta }}_{ab}\mathrm{/}3$, with averaged symmetrization. That lemma proves $\sigma$ symmetric, spatial, trace-free, and of nonnegative squared norm. Let $\kappa ,\eta ,{\zeta }_{\mathrm{b}}\in C^{\mathrm{\infty }}(O)$ be pointwise nonnegative, with SI units $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$, $\mathrm{P}\mathrm{a}\mathrm{s}$ and $\mathrm{P}\mathrm{a}\mathrm{s}$, respectively. Put

$$X^a={\mathrm{\Delta }}^{ab}({\mathrm{\nabla }}_{b}T+T{a}_b),q^a=-\kappa X^a,Q^a=q^a\mathrm{/}c,$$

so $q$ is the physical heat flux and $Q$ is its geometric stress component. Set the particle-diffusion current $V^a=0$, the bulk correction $\mathrm{\Pi }=-{\zeta }_{\mathrm{b}}c\theta$, and the symmetric spatial trace-free stress ${\pi }^{ab}=-2\eta c{\sigma }^{ab}$. Suppose ${\mathrm{\nabla }}_a{J}^a=0$ for $J^a=n{u}^a$, and suppose the symmetric stress

$${\mathsf{T}}^{ab}=ϵu^a{u}^b+(p+\mathrm{\Pi }){\mathrm{\Delta }}^{ab}+u^a{Q}^b+Q^a{u}^b+{\pi }^{ab}$$

satisfies ${\mathrm{\nabla }}_a{\mathsf{T}}^{ab}=f^b$ with zero local power $u_b{f}^b=0$. Then for $S^a=s{u}^a+Q^a\mathrm{/}T$ the exact entropy-production identity is

$${\mathrm{\nabla }}_a{S}^a=\frac{\kappa }{c{T}^2}{X}^a{X}_a+\frac{2\eta c}{T}{\sigma }^{ab}{\sigma }_{ab}+\frac{{\zeta }_{\mathrm{b}}c}{T}{\theta }^2\ge 0.$$

The conclusion is a conditional identity for the stated smooth fields and closure; it asserts neither existence nor causality, stability, or empirical applicability of an evolution.