Relativistic heat conduction

Fourier, hyperbolic and relativistic heat transfer equations: a comparative analytical study

“…To compare with the HHE, we chose C in such a way that C 2 = α/τ so that the coefficients of the governing equation would be the same in both models, even though this procedure did not ensure the exact replication of the physical reality. It is noteworthy that with the values of the parameters considered in this study, C takes a theoretical value as small as C = 9.07 × 10 −5 m s −1 , in good agreement with the qualitative indications given in Ali & Zhang [12]. To date, there are no experimental data about the value of C.…”

“…Our study did not try to elucidate which model is true, but to qualitatively describe how each approach works for the mathematical modelling of heating of biological tissue, and especially to identify the differences in the computed temperature profiles. In fact, Ali & Zhang [12] did not compare the thermal performance of the relativistic model with the others models. In this respect, our results showed that RHE temperature values were always lower than those of the FHE, where in general the computed temperature in the HHE is higher than the FHE, except at the early stages of heating and at points away from the electrode.…”

“…This RHE in a four-dimensional space-time, turns out to be identical to the Maxwell equation, is derived without any microscopic or material-specific considerations, and overcomes [12] . Moreover, except for the terms involving the heat source S(x, t), RHE (1.10) is mathematically identical to HHE (1.7), but the HHE heat flux is based on (1.4) and the RHE heat flux on (1.7).…”

“…This is because in metals, both heat and sound are propagated by mechanisms, such as lattice vibrations, which are all decelerated by the structural imperfections of biological tissues and non-metal materials. In addition, some studies have shown that in many cases the values of C and α imply that the first term of equation (1.3) (secondorder derivative) does not have to be necessarily negligible with respect to the second term (first-order derivative) [12,13]. With the same purpose, Cattaneo [14] and Vernotte [15] proposed a modification of Fourier's linear flux as follows:…”

“…To overcome the problems involved in FHE and HHE, Ali & Zhang [12] proposed a theoretical model based on a weak interpretation of the theory of relativity. In this setting, relativistic does not mean 'concerned with objects moving at speeds comparable to the speed of light', but 'concerning objects that move or propagate at speed comparable with a limiting speed characteristic of the field or medium involved, denoted by C'.…”