Document Type


Date of Award

Summer 8-31-1998

Degree Name

Doctor of Philosophy in Applied Mathematics - (Ph.D.)



First Advisor

Gregory A. Kriegsmann

Second Advisor

Demetrius T. Papageorgiou

Third Advisor

Jonathan H.C. Luke

Fourth Advisor

Yuriko Y. Renardy

Fifth Advisor

Burt S. Tilley


In this work we study the effects of externally induced heating on the dynamics of fluid layers, and materials composed of two phases separated by a thermally driven moving front. One novel aspect of our study is in the nature of the external source, which is provided by the action of microwaves acting on dielectric materials. The main challenge is to model and solve systems of differential equations, which couple fluid dynamical motions (the Navie- Stokes equations for nonisothermal flows) and electromagnetic wave propagation (governed by Maxwell's equations).

When an electromagnetic wave impinges on a material, energy is generated within the material due to dipolar and ohmic heating. The electrical and thermal properties of the material dictate the dynamics of the heating process, as well as steady state temperature profiles. Such forms of heating have received little attention in studies of hydrodynamic instabilities of non-isothermal flows, such as the classical Benard problem, for instance. The novel feature, which allows possibilities for fluid management and control, is the non-local coupling between the electromagnetic field and the temperature distribution within the fluid. In the first part of the thesis, we consider hydrodynamic instabilities of such systems with particular emphasis on conditions for onset of convection. This is achieved by solving the linear stability equations in order to identify parameter values, which produce instability. The analysis and subsequent numerical solutions are carried out both for materials with constant dielectric attributes (in such cases the electric field equations decouple and they can be solved in closed form), and materials with temperature dependent Conductivities, dielectric permittivities and dielectric loss factors. In the latter case we incorporate known data for water or ethanol into our numerical solutions. Our solutions provide a complete picture of onset conditions as a function of input power levels and microwave frequency (or equivalently fluid layer thickness). In addition, in the case of water, the flow is found to be more stable for constant attributes as compared with temperature dependent attributes; that is, a higher power is required to set the fluid layer into convective motions in the latter case. We have also established that onset is obtained at power levels well below those needed to cause thermal runaway and consequently boiling of the water layer, for instance. Our results also identify different parameter ranges, which can produce convection cells of different sizes with the same power input. Such results are directly related to the micro-wave radiation, which provides the heating, and in particular the distribution of the electric field within the fluid layer. Several interesting experiments are suggested by the theoretical predictions.

The second problem studied is concerned with the use of microwave radiation in the processing of materials, which contain two phases separated by a moving front, which forms and propagates due to a jump in temperature flux across the interface separating the phases. The problem is an extension of the classical Stefan problem with the propagation caused by temperature gradients induced by the electromagnetic radiation. 'We have modeled and solved the problem of two phases separated by a planar interface and in the absence of fluid motion if melting is involved. The boundary conditions are those of convective cooling at the top surface and either a heat sink (to maintain a frozen state for ice, for example) or a perfectly electro magnetically reflective bounding surface at the bottom. Known data modeling a water-ice system have been used, but the methods are the same for other materials. We have addressed the cases of constant and temperature obtained by solving a coupled system of nonlinear differential equations leading to an eigenvalue problem for the interfacial position. In addition, a time-dependent code was developed in order to study transient motions towards steady-state, starting from initial configurations of a thin water layer on the ice, for example. Our results indicate that for a given power level there can be two stable steady-state positions for the melting front as well as an unstable one. Existence of multiple states is a consequence of electromagnetic wave resonances within the material and their global effects on the thermal distribution. Such behavior leads to a theoretical framework in efforts to control the position of phase separation interfaces in the processing of materials.



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