Modelling thermal front dynamics in microwave heating

The formation and propagation of thermal fronts in a cylindrical medium that is undergoing microwave heating is studied in detail. The model consists of Maxwell's wave equation coupled to a temperature diffusion equation containing a bistable nonlinear term. When the thermal diffusivity is sufficiently small the leading‐order temperature solution of a singular perturbation analysis is used to reduce the system to a free boundary problem. This approximation is then used to derive predictions for the steady‐state penetration and profiles of the temperature and electric fields. These solutions are valid for arbitrary values of the electric conductivity, and thus extend the previous (small conductivity) results found in the literature. A quasi‐static approximation for the electric field is then used to obtain an ordinary differential equation for the relaxation dynamics to the steady state. This equation appears to accurately describe the time scale of the electric field's evolution both with and without the presence of a strongly coupled temperature front, and may be of wider interest than the model for microwave heating studied here.