Temp-Dependent Volumetric Heat Generation

• Temperature-dependent volumetric heat generation is an internal energy source that varies with local temperature and governs dissipative mechanisms in diverse materials.
• Mathematical models employ linear, nonlinear, and tensorial formulations to capture heat production in processes such as plastic deformation, chemical reactions, and nuclear burning.
• Coupled thermomechanical simulations demand precise numerical methods to manage feedback effects, ignition thresholds, and interfacial jumps in layered and composite systems.

Temperature-dependent volumetric heat generation refers to the class of internal energy source terms in the thermal energy equation whose magnitude depends explicitly on the local temperature field. This phenomenon is central to the modeling of thermomechanical processes in solids, fluids, and composite materials where dissipative physical mechanisms—such as mechanical loss, plastic deformation, chemical reactions, or nuclear processes—produce heat at rates that vary strongly with temperature. Mathematical formulations of these source terms span linear, nonlinear (often exponential), and tensorial forms, depending on the physics and the coupling between temperature and material constitutive response. The following sections review foundational formulations, provide comparisons across modeling disciplines, and summarize key consequences and applications.

1. Mathematical Formulation Across Physical Contexts

Temperature-dependent volumetric heat generation appears as a source term $Q(T)$ in the heat equation:

$$\rho\,C_p\,\frac{\partial T}{\partial t} = \text{diffusive/advective terms} + Q(T, \ldots)$$

Specific forms of $Q(T)$ include:

• Linear in $T$: $Q = \nu T$ (bilayer CDRS; exothermic first-order reactions or Joule heating) (Umbricht et al., 5 Nov 2025).
• Tensorial dissipation (thermoviscoelasticity): $$Q = \langle \Gamma(\theta) : \varepsilon(u_t),\,\varepsilon(u_t) \rangle$$ where $\Gamma$ depends on temperature (Kelvin-Voigt dissipation) (Claes et al., 22 Nov 2024).
• Strongly nonlinear/nuclear law: $Q(T) \propto T^{-3} \exp(-B/T)$ representing triple-$\alpha$ reaction rates (Garcia et al., 2018).
• Plastic work (geomechanics): $$Q = s_{ij} \dot\varepsilon_{ij}^{\rm inel} + P\,\dot\varepsilon_{kk}^{\rm vp}$$, involving both deviatoric and volumetric (dilatant) terms with temperature-dependent creep rates (Momoh et al., 6 Dec 2024).

Boundary and interface modeling requires the careful treatment of $Q(T)$ to maintain physical and numerical fidelity, with analytical or finite difference discretizations adapted according to the governing equations.

2. Thermomechanical Models: Coupled PDE Systems

In temperature-dependent Kelvin–Voigt viscoelasticity (Claes et al., 22 Nov 2024), heat generation arises from mechanical energy dissipation during acoustic wave propagation. The coupled PDEs are:

$$\begin{cases} u_{tt} = \operatorname{div}(\gamma(\theta)\nabla u_t) + a\,\operatorname{div}(\gamma(\theta)\nabla u) \ \theta_t = D\Delta \theta + \Gamma(\theta):\varepsilon(u_t):\varepsilon(u_t) \end{cases}$$

Here, the volumetric heat generation term

$$Q(\theta, \nabla u_t) = \Gamma(\theta):\varepsilon(u_t):\varepsilon(u_t)$$

inherits its temperature dependence from the (fourth-order) elastic modulus tensor $C(\theta)$. This coupling introduces non-monotone feedback between thermal and mechanical energy, breaking classical energy identities and requiring generalized solution concepts to guarantee global existence.

For plastic geomechanics (Momoh et al., 6 Dec 2024), internal heat generation incorporates both shear and volumetric (dilatant) mechanical work:

$$\Phi = s_{ij}(\dot\varepsilon_{ij}^{v} + \dot\varepsilon_{ij}^{vp}) + P\,\dot\varepsilon_{kk}^{vp}$$

with temperature-sensitive viscoplastic and creep rates modulated by Arrhenius-like dependencies.

3. Nonlinear, Exponential Source Terms in Astrophysical and High-Energy Contexts

Models of nuclear burning, such as triple-$\alpha$ reactions in accreting neutron star oceans, demand volumetric source terms with extreme temperature sensitivity (Garcia et al., 2018). The standard fitting law is

$$q(T) = 5.3 \times 10^{18}\,\rho_5^2\,Y^3\,T_9^{-3}\,\exp\left(-\frac{4.4}{T_9}\right)$$

with $\rho_5 = \rho / (10^5\;\mathrm{g\,cm}^{-3})$, $T_9 = T / (10^9\;\mathrm K)$, and $Y$ the He mass fraction. These source terms induce regime transitions in thermal convection modeled using the Boussinesq approximation, leading to supercritical behavior, restabilization of conductive states, and the eventual onset of chaotic and turbulent flows as the nuclear heating control parameter ($\mathrm{Ra}_n$) is varied.

The dimensionless energy equation takes the form:

$$\Pr\,\partial_t\Theta + \mathbf u\cdot\nabla\Theta = \nabla^2\Theta + \mathrm{Ra}\,\frac{\eta}{(1-\eta)^2}\,r^{-3}u_r + \mathrm{Ra}_n f(\Theta, r)$$

with $$f(\Theta, r) = (\Theta + T_c(r))^{-3} \exp[-B_n / (\Theta + T_c(r))]$$. This high nonlinearity captures ignition thresholds and explosive growth encountered in astrophysical processes.

4. Linear Temperature Dependence: Analytical and Numerical Solutions

Problems with $Q(T) = \nu T$ retain greater analytical tractability. In bilayer convection–diffusion–reaction–source (CDRS) models (Umbricht et al., 5 Nov 2025), the governing equation for each layer is

$$\frac{\partial T_m}{\partial t} = \alpha_m \Delta T_m - \mathbf\beta_m \cdot \nabla T_m + \nu_m T_m + s_m(x, y, t)$$

Here, the sign of $\nu_m$ determines whether the term models heating ($\nu_m > 0$) or internal heat loss ($\nu_m < 0$). Analytical solutions follow by eigenfunction expansion and Fourier series, with each mode's decay rate shifted by $\psi_m = \bar\nu_m - \|\chi_m\|^2$, thereby controlling the long-term stability and steady-state profiles. The tractable linearity facilitates closed-form analysis and eigenmode decomposition but limits the physical realism when strong nonlinearities dominate.

Numerical validation employs explicit finite-difference schemes with CFL-type stability conditions, careful treatment of interface conditions, and the ability to identify the influence of $\nu_m$ on both dynamic and steady-state temperature fields.

5. Physical and Mathematical Consequences of Temperature Dependence

Allowing volumetric heating rates to depend on temperature leads to several key effects:

• Energy budget modification: In temperature-dependent Kelvin–Voigt models, the dissipation term $Q(\theta, \nabla u_t)$ increases or decreases with $C(\theta)$, affecting the risk of thermal runaway or stabilization depending on the sign and magnitude of the dependence (Claes et al., 22 Nov 2024).
• Loss of monotonicity/coercivity: For temperature-dependent $C(\theta)$, energy identities used in classical existence proofs fail or become non-monotone, necessitating new mathematical strategies such as composite energy densities $$\Psi = \frac12 |u_t|^2 + \kappa \frac12 |u|^2 + \theta$$.
• Feedback mechanisms: An increase in temperature elevates dissipation in creep or thermoviscoelastic settings (e.g., via Arrhenius dependence), leading to localized heating, possible thermal runaway, and complex pattern formation as in convection driven by nuclear processes (Garcia et al., 2018).
• Volumetric vs. shear heating: In geodynamic models, inclusion of dilatant plastic work adds a volumetric term with a negative sign, manifesting as cooling due to plastic dilation against confining pressure. Numerical experiments demonstrate that this effect can reduce peak temperatures by a few to tens of kelvin (e.g., reduction from $35.3\,\mathrm K$ to $32.7\,\mathrm K$ in constant-strain-rate tests) (Momoh et al., 6 Dec 2024).
• Layered systems behavior: In bilayer and composite systems, frameworks with varying $\nu_m$ between layers lead to pronounced interfacial jumps in temperature and distinct zones of heating or absorption, directly informing the engineering optimization of composite materials (Umbricht et al., 5 Nov 2025).

6. Applications and Regime-Specific Insights

Temperature-dependent volumetric heat generation is essential in the following scenarios:

Domain	Source Term Form	Notable Effects
Piezoelectric and viscoelastic solids	$$Q = \Gamma(\theta):\varepsilon(u_t):\varepsilon(u_t)$$	Feedback in acoustic attenuation; altered wave speed
Accreting neutron star oceans	$Q \propto T^{-3}\exp(-B/T)$	Onset of burning-driven convection, turbulent regimes
Layered composite materials	$Q = \nu_m T$	Layer-specific heating rates, sharp interface effects
Geodynamic plasticity (dilatant flow)	$Q = P\,\alpha_3\,\dot\gamma^{vp}$	Reduction in peak $\Delta T$; thickening of brittle lid

• In engineering design, these effects guide strategies for managing heat in composite structures and electronic devices.
• In planetary and astrophysical modeling, precise source modeling underpins the simulation of convection, ignition, and pattern formation on a global scale.
• In computational geodynamics, the inclusion of volumetric dissipation clarifies the interplay between fault localization and cooling under different boundary and loading conditions.

7. Modeling Challenges and Outlook

Accurate modeling of temperature-dependent volumetric heat generation introduces mathematical and computational complexity:

• Strong temperature dependence and nonlinearity can precipitate instabilities or require stabilized numerics.
• In regimes with exponential sensitivity (e.g., nuclear burning), fine spatial and temporal resolution is required to capture transitions between conductive, convective, and turbulent states.
• For coupled thermomechanical systems, ensuring physically meaningful and globally existent solutions necessitates uniform ellipticity and regularity assumptions; for instance, assumptions of $C(\theta)$ and $\Gamma(\theta)$ in $L^\infty \cap C^2$, and uniform positivity, are pivotal (Claes et al., 22 Nov 2024).
• In multi-material and layered contexts, interface thermal resistance and matching of advective/diffusive balances demand careful boundary condition implementation (Umbricht et al., 5 Nov 2025).

A plausible implication is that progress toward efficient simulation and analysis will rely on hybrid methods blending analytical tractability (for cases with linear or weakly nonlinear $Q(T)$) and robust, highly resolved numerical schemes for cases with strong $T$-dependence or tensorial structure. Future directions also include refinement of experimental parameterizations of $Q(T)$ in novel material systems and improved coupling to large-scale multiphysics models.