Nonlinear waves and "negative heat capacity" in a medium with competitive sources

O. N. Shablovskii

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 66 (2020), pp. 64-76

Résumé

For a wave equation with sources, new running-wave type solutions are built. The results are expressed in terms of the heat transfer theory. We study two types of alternating volume energy sources $q_\upsilon$ with a nonlinear temperature dependence $T$. Let $q_\upsilon(T=T^1)=0$ where $T^1$ is the temperature of the source sign change. The source is positive at $T>T^1$ (heat input) and negative at $T$ (heat output) when is has technical origin. A source of biological origin differs from technical ones. It serves as a compensator: at $T>T^1$ it takes the heat in; at $T$, it gives the heat out. Three types of analytical solutions are obtained: the sole wave, the kink structure, and the wave chain. Subsonic and supersonic wave processes are studied with respect to the rate of heat perturbations. The examples for a non-classical phenomenon of "negative heat capacity" are given when heat input/output leads to a temperature decrease/increase. We have considered a nonlinear medium liable to an exact analytical description of a wave problem with a having a resonance type of the temperature dependence: its oscillations have a crescent amplitude. As an example of physical interpretation for one solution, the rate of crystal growth is calculated as a function of the melt undercooling.

Détail

Keywords: wave equation, nonlinear energy source, temperature response of the medium, undercooled melt.