Pattern formation in microwave heated ceramics: cylinders and slabs

Analyses of microwave heating of a thin ceramic cylinder and a thin ceramic slab in a single mode, highly resonant cavity are presented. Realistic ass umptions regarding the effective electrical conductivity, thermal parameter s, and physical dimensions are adhered to throughout. Consequently, the mod els developed herein incorporate most of the features of actual experiments . They incorporate both the effects of cavity detuning and a local electric field perturbation on the heating process. The models presented take the form of one- and two-dimensional reaction-dif fusion equations which contain a functional and an inhomogeneous source ter m for the cylinder and slab, respectively. The development of these equatio ns is the product of a systematic modelling process that involves S-matrix theory, a small Blot number asymptotic analysis, and a matched asymptotic a nalysis of a non-standard electromagnetic scattering problem. The one-dimen sional equation for the cylinder reveals both the mathematical structure an d physical mechanism far the formation of hot-spots. The two-dimensional eq uation supports a hot stripe pattern, due to preferential electromagnetic h eating, which becomes unstable and evolves into an oval-like spot. Accurate numerical methods which approximate the solutions of these equations and t heir stability are presented and these agree qualitatively with experiments and predict observed trends.