Mathematical Simulation of Thermal Shock Wave Dynamics in Nonlinear Local Non-Equilibrium Media

Abstract

We performed a mathematical simulation of heat transfer in a local non-equilibrium medium whose transfer characteristics are functions of the temperature distribution. A homogeneous polynomial of arbitrary degree represents nonlinearities in thermal conductivity and thermal diffusivity. The mathematical model consists of a hyperbolic nonlinear heat transfer wave equation, initial conditions and nonlinear boundary conditions of the second and first kind. To solve this problem, we used a conservative homogeneous finite-difference scheme along the upper time grid line (implicitly). We then used the tridiagonal matrix algorithm of the second order in the spatial variable and of the first order in time to solve the resulting system of linearised algebraic equations. A periodic series of rectangular temperature or heat flux pulses form the boundary conditions of the first and second kind. Computation results reveal ultimate propagation rates of temperature and heat fronts featuring pronounced first-kind discontinuities with attenuating magnitudes. As the process unfolds, the initial pulses heat the region between the boundary and the heat wave front, while the subsequent pulses traverse this region at a higher velocity due to thermal diffusivity being a function of temperature, their fronts "catching up" with the previous fronts, increasing the discontinuity magnitude at the initial pulse front, that is, forming a thermal shock wave front similar to that of a shock wave in gas dynamics. We obtained such thermal shock waves for boundary conditions of both the first and the second kind. We also analysed kinematic and dynamic characteristics of thermal waves