Document Type


Date of Award

Fall 1-31-2001

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)


Mathematical Sciences

First Advisor

Gregory A. Kriegsmann

Second Advisor

David C. Stickler

Third Advisor

Jonathan H.C. Luke

Fourth Advisor

Peter G. Petropoulos

Fifth Advisor

Edip Niver


In order to gain insight into hot spot development in microwave heated ceramics, a partially insulated, two dimensional ceramic slab situated in a TEM01 cavity is modeled in the small Biot number limit. If the electrical conductivity is an exponential function of temperture and E0 is the strength of the incident mode, then the relationship between the spatially uniform, steady state leading order temperature, v8, and E01 is characterized by the well known bi-stable, or S shaped, response curve. The steady state second order temperature, v1, is described by a boundary value problem consisting of a Poisson equation with a non linear source term and non linear boundary flux conditions at two of the slab faces; the remaining two slab faces are assumed to be insulated. The source term depends on the electrical conductivity function and the spatially dependent leading order electric field, both of which are parametrized by v8. The boundary conditions demonstrate fourth power dependence on v8 as a result of radiated heat loss. It is the spatial dependence of v1, that is the source of temperature non-uniformities that can cause deleterious density fluctuations in a sintered ceramic.

The main results are as follows: First, for cavities of increasing width (i.e., increasing M) with v8 chosen on the lower branches of their corresponding S curves where the TEM01 mode remains dominant, it is shown that detuning of the cavity is the primary reason for the decrease of electric field strengths from their initial resonant values. For v8, values chosen on the upper branches of the S curves where the TEM01 mode ceases to be dominant, it is demonstrated that skin effect shielding is the primary reason for the decrease of electric field strengths, while cavity detuning plays a secondary role as evidenced by the distortion of the electric field pattern. This distortion results from the prominent influence of other field modes.

Secondly, it is shown that, for lower branch values of v8, the dominant TEM01 mode determines a similar sinusoidal mode structure for v1. As the value of M increases for M ≥ 3, the steady state second order temperature transitions from a state in which its double maxima are located near the insulated edges of the slab for M = 3 to a state in which a single maximum is located in the center of the slab for M ≥ 7. Preliminary data indicate that this trend is not observed for upper branch v8 values due to the unpredictable behavior of the electric field. Finally, data are presented which suggest a trend toward uniformity of temperature for lower branch v8 values. This supports the experimental observation that the temperature of a microwave heated object becomes more uniform as the number of electric field modes propagating within the object increases.

Included in

Mathematics Commons



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