Relativistic heat conduction

Relativistic heat conduction refers to the modelling of heat conduction (and similar diffusion processes) in a way compatible with special relativity. This article discusses models using a wave equation with a dissipative term.

Heat conduction in a Newtonian context is modelled by the Fourier equation:[1]

${\displaystyle {\frac {\partial \theta }{\partial t}}~=~\alpha ~\nabla ^{2}\theta ,}$

where θ is temperature,[2] t is time, α = k/(ρ c) is thermal diffusivity, k is thermal conductivity, ρ is density, and c is specific heat capacity. The Laplace operator,${\displaystyle \scriptstyle \nabla ^{2}}$, is defined in Cartesian coordinates as

${\displaystyle \nabla ^{2}~=~{\frac {\partial ^{2}}{\partial x^{2}}}~+~{\frac {\partial ^{2}}{\partial y^{2}}}~+~{\frac {\partial ^{2}}{\partial z^{2}}}.}$

This Fourier equation can be derived by substituting Fourier’s linear approximation of the heat flux vector, q, as a function of temperature gradient,

${\displaystyle \mathbf {q} ~=~-k~\nabla \theta ,}$

into the first law of thermodynamics

${\displaystyle \rho ~c~{\frac {\partial \theta }{\partial t}}~+~\nabla \cdot \mathbf {q} ~=~0,}$

where the del operator, ∇, is defined in 3D as

${\displaystyle \nabla ~=~{\frac {\partial }{\partial x}}~\mathbf {i} ~+~{\frac {\partial }{\partial y}}~\mathbf {j} ~+~{\frac {\partial }{\partial z}}~\mathbf {k} .}$

It can be shown that this definition of the heat flux vector also satisfies the second law of thermodynamics,[3]

${\displaystyle \nabla \cdot \left({\frac {\mathbf {q} }{\theta }}\right)~+~\rho ~{\frac {\partial s}{\partial t}}~=~\sigma ,}$

where s is specific entropy and σ is entropy production.

Notes

1. Carslaw, H. S.; Jaeger, J. C. (1959). Conduction of Heat in Solids (Second ed.). Oxford: University Press.
2. Some authors also use T, φ,...
3. Barletta, A.; Zanchini, E. (1997). "Hyperbolic heat conduction and local equilibrium: a second law analysis". International Journal of Heat and Mass Transfer. 40 (5): 1007–1016. doi:10.1016/0017-9310(96)00211-6.
4. Eckert, E. R. G.; Drake, R. M. (1972). Analysis of Heat and Mass Transfer. Tokyo: McGraw-Hill, Kogakusha.
5. Cattaneo, C. R. (1958). "Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée". Comptes Rendus. 247 (4): 431.
6. Vernotte, P. (1958). "Les paradoxes de la theorie continue de l'équation de la chaleur". Comptes Rendus. 246 (22): 3154.
7. Chester, M. (1963). "Second sound in solids". Physical Review. 131 (15): 2013–2015. Bibcode:1963PhRv..131.2013C. doi:10.1103/PhysRev.131.2013.
8. Morse, P. M.; Feshbach, H. (1953). Methods of Theoretical Physics. New York: McGraw-Hill.
9. Mandrusiak, G. D. (1997). "Analysis of non-Fourier conduction waves from a reciprocating heat source". Journal of Thermophysics and Heat Transfer. 11 (1): 82–89. doi:10.2514/2.6204.
10. Xu, M.; Wang, L. (2002). "Thermal oscillation and resonance in dual-phase-lagging heat conduction". International Journal of Heat and Mass Transfer. 45 (5): 1055–1061. doi:10.1016/S0017-9310(01)00199-5.
11. Barletta, A.; Zanchini, E. (1996). "Hyperbolic heat conduction and thermal resonances in a cylindrical solid carrying a steady periodic electric field". International Journal of Heat and Mass Transfer. 39 (6): 1307–1315. doi:10.1016/0017-9310(95)00202-2.
12. Tzou, D. Y. (1989). "Shock wave formation around a moving heat source in a solid with finite speed of heat propagation". International Journal of Heat and Mass Transfer. 32 (10): 1979–1987. doi:10.1016/0017-9310(89)90166-X.

References