Method for solving linearized thermal problems taking into account the phenomenon of heat flow relaxation

Abstract

A method is proposed for computational modeling of the features of motion in a material of a thermal wave within the framework of a linearized problem. Such problems arise when applying heat transfer models with allowance for heat flux relaxation. It is assumed that the proportional relationship between the temperature gradient and the heat flux (Fourier's law) cannot appear instantly when the temperature field changes in the medium. It takes time to change the heat flow. This approach refers to the description of the time variations of the heat flux in the Cattaneo and Vernotte models. The method for solving thermal problems within this scope is reflected in a large number of publications, but they deal mainly with the redistribution of heat in the material. However, there are tasks the solution of which requires understanding of how exactly the change in heat fluxes in the medium occurs. This is especially important when the data are used to find solutions to the problems of determining stress (depending not only on temperature and strain, but also on the value of the heat flux) fields in media. The paper shows how to construct a system of equations convenient for calculations the temperature changes and heat flux fields. It has been established that in a linear problem the initial equations can be transformed into a system of two hyperbolic equations with developed solution algorithms. As an example, computational modeling of the emergence and movement of a thermal wave during the ion-plasma treatment of a material surface is considered. The conditions required for solving the boundary value problem are formulated. The important results of the calculations were the pattern of formation, detachment from the surface of the sample after the completion of the impulse, and the beginning of the movement of a thermal wave into the depth of the material during ion-plasma treatment. The dependence of solutions on the characteristic relaxation time of the heat flux is studied.