Thermomechanical coupling requires additional assumptions

The periodic mechanical model has a velocity field and a kinetic-energy inequality with viscous dissipation, but it has no temperature variable. The separate thermal model uses a fixed rectangle $\mathrm{\Omega }={\prod }_{i=1}^3(0,L_i)$ with lengths $L_i>0$ in metres, an open time interval $I\subseteq \mathbb{R}$ measured in seconds, and a temperature field $T:\bar{\mathrm{\Omega }}\times I\to \mathcal{T}\subset (0,\mathrm{\infty })$, where $\mathcal{T}$ is an open interval, measured in kelvins; its time derivative through order one and spatial derivatives through order two extend continuously to $\bar{\mathrm{\Omega }}\times I$. In that thermal supplier, $u\in C^1(\mathcal{T},\mathbb{R})$ is the internal-energy density in $\mathrm{J}{\mathrm{m}}^{-3}$ and $c(T):=u^{\mathrm{\prime }}(T)>0$ is the volumetric heat capacity in $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$; write $e_{\mathrm{t}\mathrm{h}}:=u$ to distinguish this caloric function from the mechanical velocity notation. The thermal flux is $j_Q:\bar{\mathrm{\Omega }}\times I\to {\mathbb{R}}^3$ in $\mathrm{W}{\mathrm{m}}^{-2}$; in the adopted local-energy postulate it and its first spatial derivatives extend continuously to $\bar{\mathrm{\Omega }}\times [t_0,t_1]$ for every compact $[t_0,t_1]\subset I$, and $j_Q\cdot n=0$ on every face, where $n$ is the outward unit normal. Its local balance is ${\mathrm{\partial }}_t{e}_{\mathrm{t}\mathrm{h}}(T)+\mathrm{div}{j}_Q=0$, with $\mathrm{div}{j}_Q={\sum }_{i=1}^3{\mathrm{\partial }}_{x_i}(j_Q{)}_i$. The Fourier constitutive assumption is $j_Q=-K(x,T)\mathrm{\nabla }T$, where $\mathrm{\nabla }T=({\mathrm{\partial }}_{x_i}T{)}_{i=1}^3$, $K:\bar{\mathrm{\Omega }}\times \mathcal{T}\to {\mathbb{R}}^{3\times 3}$ has entries in $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$, and $K_s:=(K+K^{\mathsf{T}})\mathrm{/}2$ satisfies $z^{\mathsf{T}}{K}_s(x,T)z\ge 0$ for every $z\in {\mathbb{R}}^3$. The separate entropy-production theorem and constant-conductivity Neumann heat-evolution theorem retain their own regularity and initial-data conditions.

Neither branch identifies any part of the mechanical viscous dissipation with a source in the thermal balance, and the two branches use different domains. A model relating the mechanical and thermal fields requires additional assumptions specifying that relation and a common domain/field interface. In particular, identifying mechanical viscous dissipation with a source in the thermal balance requires an additional assumption specifying that conversion. A coupled temperature evolution does not follow from the weak periodic continuation or from the thermal entropy and heat-evolution results.

Weak continuation of the periodic mechanical model and its kinetic-energy inequality

Insulated local thermal energy balance with no volumetric source

Fourier heat-flux constitutive assumption

Conditional local thermal entropy production

Conditional constant-conductivity Neumann heat evolution and relaxation

Lenya Ryzhik, Lecture Notes for Math 256B, Version 2024, §1.3, printed pp. 9–11

Berselli and Spirito, On the Existence of Leray–Hopf Weak Solutions to the Navier–Stokes Equations, §1, printed pp. 1–3