The investigator models and analyzes two classes of physical problems arising in ceramic processing by microwave and ohmic heating. These problems give rise to challenging nonlinear initial boundary value problems, the mathematical statements of which involve one or more nonlinear parabolic equations, governing the diffusion of heat and the diffusion of a chemical concentration, the time harmonic version of Maxwell's equations, and in the case of ohmic heating, a nonlinear potential problem. These equations are coupled together in a nonlinear fashion and, in addition, nonlinear heat balances are required in certain applications where surface temperatures produce significant thermal radiation. The investigator develops and uses asymptotic methods, dictated by certain physical limits, to mathematically model these physical problems. These models are analyzed, using both analytical and numerical methods, to obtain accurate descriptions of the processes.
The rapid heating of ceramics by microwave and other forms of electrical energy is the basis of several emerging industrial processes such as sintering, fiber coating, joining, and chemical vapor infiltration. These processes may be capable of efficiently producing high quality materials for use in high temperature applications, such as heat exchangers, combustion liners, pump seals, and rocket nozzles. The development and exploitation of these promising technologies remain hindered by the basic physics of the process. The first problem is the occurrence of nonuniform heating, which causes the deleterious effect of nonuniform material properties. The second, in the case of microwave heating, is the catastrophic phenomenon of thermal runaway, in which a slight change of power causes the temperature to increase rapidly to the melting point of the ceramic. It is the understanding, control, and use of these phenomena and the development of accurate and efficient mathematical tools to quantify them that is addressed by the investigator and his students.
