Nonlinear heat wave propagation in a rigid thermal conductor

Abstract

We present new nonlinear, one-dimensional equations of extended thermodynamics for temperature and heat flux that describe damped heat wave propagation and predict dependence of the second sound velocity on temperature and heat flux in a rigid thermal conductor. The aim of the present work is to investigate the implications of the considered nonlinearities on the process of heat propagation, especially that such nonlinear effects cannot be disregarded for nanosystems and in small-sized heat conductors, when temperature or heat flux variations are not always proportional to gradients. The characteristics of the nonlinear system of equations are investigated, and a formula for the breaking distance is obtained. For illustration, these equations are solved in the linear approximation by separation of variables, then in the nonlinear case by the homotopy perturbation method in the half-space, and numerically in a bounded interval under concrete initial and boundary conditions. The solutions are discussed in detail and presented graphically. They illustrate the effect of the different material parameters on the propagation of thermal waves and may be used to evaluate some material parameters. The proposed equations provide a closer insight into the thermal interactions in rigid thermal conductors.